# Properties

 Label 4840.2.a.bh.1.2 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.a (trivial)

## 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: yes Analytic conductor: $$38.6475945783$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100$$ x^8 - x^7 - 21*x^6 + 15*x^5 + 140*x^4 - 60*x^3 - 295*x^2 + 50*x + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.77190$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.77190 q^{3} +1.00000 q^{5} +4.55506 q^{7} +4.68342 q^{9} +O(q^{10})$$ $$q-2.77190 q^{3} +1.00000 q^{5} +4.55506 q^{7} +4.68342 q^{9} -2.90026 q^{13} -2.77190 q^{15} +7.08802 q^{17} +7.14347 q^{19} -12.6262 q^{21} -1.86699 q^{23} +1.00000 q^{25} -4.66626 q^{27} +1.01070 q^{29} +4.41034 q^{31} +4.55506 q^{35} -5.39094 q^{37} +8.03922 q^{39} +4.48790 q^{41} +6.73002 q^{43} +4.68342 q^{45} +7.85017 q^{47} +13.7486 q^{49} -19.6473 q^{51} +0.714458 q^{53} -19.8010 q^{57} -3.80330 q^{59} -2.11595 q^{61} +21.3332 q^{63} -2.90026 q^{65} -12.8384 q^{67} +5.17511 q^{69} +4.21095 q^{71} -1.36576 q^{73} -2.77190 q^{75} +12.2375 q^{79} -1.11585 q^{81} +16.3351 q^{83} +7.08802 q^{85} -2.80156 q^{87} -11.8165 q^{89} -13.2108 q^{91} -12.2250 q^{93} +7.14347 q^{95} -10.8544 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9}+O(q^{10})$$ 8 * q + q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 $$8 q + q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} - 12 q^{13} + q^{15} - 2 q^{17} + 6 q^{19} - 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} - 8 q^{29} + 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} + 3 q^{41} + 8 q^{43} + 19 q^{45} + 10 q^{47} + 22 q^{49} + 7 q^{51} + 28 q^{53} - 25 q^{57} + 25 q^{59} - 10 q^{61} + 64 q^{63} - 12 q^{65} - 2 q^{67} + 18 q^{69} + 25 q^{71} - 38 q^{73} + q^{75} + 38 q^{79} + 32 q^{81} + 28 q^{83} - 2 q^{85} - 15 q^{87} + 12 q^{89} + 8 q^{91} - 15 q^{93} + 6 q^{95} + 21 q^{97}+O(q^{100})$$ 8 * q + q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 - 12 * q^13 + q^15 - 2 * q^17 + 6 * q^19 - 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 - 8 * q^29 + 19 * q^31 + 6 * q^35 + 12 * q^37 + 21 * q^39 + 3 * q^41 + 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 + 7 * q^51 + 28 * q^53 - 25 * q^57 + 25 * q^59 - 10 * q^61 + 64 * q^63 - 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 - 38 * q^73 + q^75 + 38 * q^79 + 32 * q^81 + 28 * q^83 - 2 * q^85 - 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 + 6 * q^95 + 21 * q^97

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.77190 −1.60036 −0.800178 0.599763i $$-0.795262\pi$$
−0.800178 + 0.599763i $$0.795262\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.55506 1.72165 0.860825 0.508901i $$-0.169948\pi$$
0.860825 + 0.508901i $$0.169948\pi$$
$$8$$ 0 0
$$9$$ 4.68342 1.56114
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −2.90026 −0.804387 −0.402193 0.915555i $$-0.631752\pi$$
−0.402193 + 0.915555i $$0.631752\pi$$
$$14$$ 0 0
$$15$$ −2.77190 −0.715701
$$16$$ 0 0
$$17$$ 7.08802 1.71910 0.859548 0.511054i $$-0.170745\pi$$
0.859548 + 0.511054i $$0.170745\pi$$
$$18$$ 0 0
$$19$$ 7.14347 1.63882 0.819412 0.573205i $$-0.194300\pi$$
0.819412 + 0.573205i $$0.194300\pi$$
$$20$$ 0 0
$$21$$ −12.6262 −2.75525
$$22$$ 0 0
$$23$$ −1.86699 −0.389295 −0.194647 0.980873i $$-0.562356\pi$$
−0.194647 + 0.980873i $$0.562356\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −4.66626 −0.898023
$$28$$ 0 0
$$29$$ 1.01070 0.187682 0.0938412 0.995587i $$-0.470085\pi$$
0.0938412 + 0.995587i $$0.470085\pi$$
$$30$$ 0 0
$$31$$ 4.41034 0.792121 0.396061 0.918224i $$-0.370377\pi$$
0.396061 + 0.918224i $$0.370377\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.55506 0.769945
$$36$$ 0 0
$$37$$ −5.39094 −0.886265 −0.443132 0.896456i $$-0.646133\pi$$
−0.443132 + 0.896456i $$0.646133\pi$$
$$38$$ 0 0
$$39$$ 8.03922 1.28730
$$40$$ 0 0
$$41$$ 4.48790 0.700892 0.350446 0.936583i $$-0.386030\pi$$
0.350446 + 0.936583i $$0.386030\pi$$
$$42$$ 0 0
$$43$$ 6.73002 1.02632 0.513159 0.858293i $$-0.328475\pi$$
0.513159 + 0.858293i $$0.328475\pi$$
$$44$$ 0 0
$$45$$ 4.68342 0.698163
$$46$$ 0 0
$$47$$ 7.85017 1.14507 0.572533 0.819882i $$-0.305961\pi$$
0.572533 + 0.819882i $$0.305961\pi$$
$$48$$ 0 0
$$49$$ 13.7486 1.96408
$$50$$ 0 0
$$51$$ −19.6473 −2.75117
$$52$$ 0 0
$$53$$ 0.714458 0.0981383 0.0490692 0.998795i $$-0.484375\pi$$
0.0490692 + 0.998795i $$0.484375\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −19.8010 −2.62270
$$58$$ 0 0
$$59$$ −3.80330 −0.495147 −0.247574 0.968869i $$-0.579633\pi$$
−0.247574 + 0.968869i $$0.579633\pi$$
$$60$$ 0 0
$$61$$ −2.11595 −0.270920 −0.135460 0.990783i $$-0.543251\pi$$
−0.135460 + 0.990783i $$0.543251\pi$$
$$62$$ 0 0
$$63$$ 21.3332 2.68774
$$64$$ 0 0
$$65$$ −2.90026 −0.359733
$$66$$ 0 0
$$67$$ −12.8384 −1.56847 −0.784233 0.620467i $$-0.786943\pi$$
−0.784233 + 0.620467i $$0.786943\pi$$
$$68$$ 0 0
$$69$$ 5.17511 0.623010
$$70$$ 0 0
$$71$$ 4.21095 0.499747 0.249874 0.968278i $$-0.419611\pi$$
0.249874 + 0.968278i $$0.419611\pi$$
$$72$$ 0 0
$$73$$ −1.36576 −0.159850 −0.0799249 0.996801i $$-0.525468\pi$$
−0.0799249 + 0.996801i $$0.525468\pi$$
$$74$$ 0 0
$$75$$ −2.77190 −0.320071
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.2375 1.37682 0.688411 0.725320i $$-0.258308\pi$$
0.688411 + 0.725320i $$0.258308\pi$$
$$80$$ 0 0
$$81$$ −1.11585 −0.123983
$$82$$ 0 0
$$83$$ 16.3351 1.79301 0.896503 0.443037i $$-0.146099\pi$$
0.896503 + 0.443037i $$0.146099\pi$$
$$84$$ 0 0
$$85$$ 7.08802 0.768803
$$86$$ 0 0
$$87$$ −2.80156 −0.300359
$$88$$ 0 0
$$89$$ −11.8165 −1.25255 −0.626273 0.779604i $$-0.715420\pi$$
−0.626273 + 0.779604i $$0.715420\pi$$
$$90$$ 0 0
$$91$$ −13.2108 −1.38487
$$92$$ 0 0
$$93$$ −12.2250 −1.26768
$$94$$ 0 0
$$95$$ 7.14347 0.732904
$$96$$ 0 0
$$97$$ −10.8544 −1.10210 −0.551048 0.834473i $$-0.685772\pi$$
−0.551048 + 0.834473i $$0.685772\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −12.6665 −1.26037 −0.630184 0.776446i $$-0.717021\pi$$
−0.630184 + 0.776446i $$0.717021\pi$$
$$102$$ 0 0
$$103$$ −6.18162 −0.609093 −0.304547 0.952497i $$-0.598505\pi$$
−0.304547 + 0.952497i $$0.598505\pi$$
$$104$$ 0 0
$$105$$ −12.6262 −1.23219
$$106$$ 0 0
$$107$$ −1.00774 −0.0974223 −0.0487111 0.998813i $$-0.515511\pi$$
−0.0487111 + 0.998813i $$0.515511\pi$$
$$108$$ 0 0
$$109$$ −6.60399 −0.632548 −0.316274 0.948668i $$-0.602432\pi$$
−0.316274 + 0.948668i $$0.602432\pi$$
$$110$$ 0 0
$$111$$ 14.9431 1.41834
$$112$$ 0 0
$$113$$ −10.2611 −0.965279 −0.482640 0.875819i $$-0.660322\pi$$
−0.482640 + 0.875819i $$0.660322\pi$$
$$114$$ 0 0
$$115$$ −1.86699 −0.174098
$$116$$ 0 0
$$117$$ −13.5831 −1.25576
$$118$$ 0 0
$$119$$ 32.2863 2.95968
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −12.4400 −1.12168
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −17.7211 −1.57249 −0.786245 0.617914i $$-0.787978\pi$$
−0.786245 + 0.617914i $$0.787978\pi$$
$$128$$ 0 0
$$129$$ −18.6549 −1.64247
$$130$$ 0 0
$$131$$ −14.3999 −1.25812 −0.629061 0.777356i $$-0.716560\pi$$
−0.629061 + 0.777356i $$0.716560\pi$$
$$132$$ 0 0
$$133$$ 32.5389 2.82148
$$134$$ 0 0
$$135$$ −4.66626 −0.401608
$$136$$ 0 0
$$137$$ 4.46970 0.381872 0.190936 0.981603i $$-0.438848\pi$$
0.190936 + 0.981603i $$0.438848\pi$$
$$138$$ 0 0
$$139$$ −4.84523 −0.410967 −0.205483 0.978661i $$-0.565877\pi$$
−0.205483 + 0.978661i $$0.565877\pi$$
$$140$$ 0 0
$$141$$ −21.7599 −1.83251
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1.01070 0.0839341
$$146$$ 0 0
$$147$$ −38.1096 −3.14323
$$148$$ 0 0
$$149$$ 10.7263 0.878734 0.439367 0.898308i $$-0.355203\pi$$
0.439367 + 0.898308i $$0.355203\pi$$
$$150$$ 0 0
$$151$$ 0.201814 0.0164234 0.00821172 0.999966i $$-0.497386\pi$$
0.00821172 + 0.999966i $$0.497386\pi$$
$$152$$ 0 0
$$153$$ 33.1961 2.68375
$$154$$ 0 0
$$155$$ 4.41034 0.354247
$$156$$ 0 0
$$157$$ −2.92198 −0.233199 −0.116600 0.993179i $$-0.537199\pi$$
−0.116600 + 0.993179i $$0.537199\pi$$
$$158$$ 0 0
$$159$$ −1.98040 −0.157056
$$160$$ 0 0
$$161$$ −8.50425 −0.670229
$$162$$ 0 0
$$163$$ 8.48158 0.664329 0.332164 0.943222i $$-0.392221\pi$$
0.332164 + 0.943222i $$0.392221\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.0642 0.778790 0.389395 0.921071i $$-0.372684\pi$$
0.389395 + 0.921071i $$0.372684\pi$$
$$168$$ 0 0
$$169$$ −4.58851 −0.352962
$$170$$ 0 0
$$171$$ 33.4558 2.55843
$$172$$ 0 0
$$173$$ −5.77262 −0.438884 −0.219442 0.975626i $$-0.570424\pi$$
−0.219442 + 0.975626i $$0.570424\pi$$
$$174$$ 0 0
$$175$$ 4.55506 0.344330
$$176$$ 0 0
$$177$$ 10.5424 0.792412
$$178$$ 0 0
$$179$$ 16.8671 1.26071 0.630353 0.776309i $$-0.282910\pi$$
0.630353 + 0.776309i $$0.282910\pi$$
$$180$$ 0 0
$$181$$ 15.1054 1.12277 0.561387 0.827553i $$-0.310268\pi$$
0.561387 + 0.827553i $$0.310268\pi$$
$$182$$ 0 0
$$183$$ 5.86520 0.433568
$$184$$ 0 0
$$185$$ −5.39094 −0.396350
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −21.2551 −1.54608
$$190$$ 0 0
$$191$$ 10.4338 0.754966 0.377483 0.926016i $$-0.376790\pi$$
0.377483 + 0.926016i $$0.376790\pi$$
$$192$$ 0 0
$$193$$ −13.9250 −1.00234 −0.501172 0.865348i $$-0.667098\pi$$
−0.501172 + 0.865348i $$0.667098\pi$$
$$194$$ 0 0
$$195$$ 8.03922 0.575700
$$196$$ 0 0
$$197$$ 2.25234 0.160473 0.0802363 0.996776i $$-0.474433\pi$$
0.0802363 + 0.996776i $$0.474433\pi$$
$$198$$ 0 0
$$199$$ 22.1246 1.56837 0.784184 0.620528i $$-0.213082\pi$$
0.784184 + 0.620528i $$0.213082\pi$$
$$200$$ 0 0
$$201$$ 35.5869 2.51010
$$202$$ 0 0
$$203$$ 4.60380 0.323123
$$204$$ 0 0
$$205$$ 4.48790 0.313448
$$206$$ 0 0
$$207$$ −8.74390 −0.607743
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −7.77119 −0.534991 −0.267495 0.963559i $$-0.586196\pi$$
−0.267495 + 0.963559i $$0.586196\pi$$
$$212$$ 0 0
$$213$$ −11.6723 −0.799774
$$214$$ 0 0
$$215$$ 6.73002 0.458984
$$216$$ 0 0
$$217$$ 20.0894 1.36376
$$218$$ 0 0
$$219$$ 3.78574 0.255817
$$220$$ 0 0
$$221$$ −20.5571 −1.38282
$$222$$ 0 0
$$223$$ −7.92084 −0.530419 −0.265209 0.964191i $$-0.585441\pi$$
−0.265209 + 0.964191i $$0.585441\pi$$
$$224$$ 0 0
$$225$$ 4.68342 0.312228
$$226$$ 0 0
$$227$$ 20.2672 1.34518 0.672590 0.740015i $$-0.265182\pi$$
0.672590 + 0.740015i $$0.265182\pi$$
$$228$$ 0 0
$$229$$ 9.88050 0.652922 0.326461 0.945211i $$-0.394144\pi$$
0.326461 + 0.945211i $$0.394144\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −9.79865 −0.641931 −0.320966 0.947091i $$-0.604007\pi$$
−0.320966 + 0.947091i $$0.604007\pi$$
$$234$$ 0 0
$$235$$ 7.85017 0.512089
$$236$$ 0 0
$$237$$ −33.9210 −2.20341
$$238$$ 0 0
$$239$$ 11.0149 0.712493 0.356247 0.934392i $$-0.384056\pi$$
0.356247 + 0.934392i $$0.384056\pi$$
$$240$$ 0 0
$$241$$ 13.3108 0.857421 0.428711 0.903442i $$-0.358968\pi$$
0.428711 + 0.903442i $$0.358968\pi$$
$$242$$ 0 0
$$243$$ 17.0918 1.09644
$$244$$ 0 0
$$245$$ 13.7486 0.878363
$$246$$ 0 0
$$247$$ −20.7179 −1.31825
$$248$$ 0 0
$$249$$ −45.2791 −2.86945
$$250$$ 0 0
$$251$$ −14.5916 −0.921011 −0.460506 0.887657i $$-0.652332\pi$$
−0.460506 + 0.887657i $$0.652332\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −19.6473 −1.23036
$$256$$ 0 0
$$257$$ 0.965528 0.0602280 0.0301140 0.999546i $$-0.490413\pi$$
0.0301140 + 0.999546i $$0.490413\pi$$
$$258$$ 0 0
$$259$$ −24.5560 −1.52584
$$260$$ 0 0
$$261$$ 4.73353 0.292998
$$262$$ 0 0
$$263$$ −20.7753 −1.28106 −0.640529 0.767934i $$-0.721285\pi$$
−0.640529 + 0.767934i $$0.721285\pi$$
$$264$$ 0 0
$$265$$ 0.714458 0.0438888
$$266$$ 0 0
$$267$$ 32.7541 2.00452
$$268$$ 0 0
$$269$$ −3.91387 −0.238633 −0.119317 0.992856i $$-0.538070\pi$$
−0.119317 + 0.992856i $$0.538070\pi$$
$$270$$ 0 0
$$271$$ 14.2782 0.867338 0.433669 0.901072i $$-0.357219\pi$$
0.433669 + 0.901072i $$0.357219\pi$$
$$272$$ 0 0
$$273$$ 36.6191 2.21629
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3.76298 −0.226095 −0.113048 0.993590i $$-0.536061\pi$$
−0.113048 + 0.993590i $$0.536061\pi$$
$$278$$ 0 0
$$279$$ 20.6555 1.23661
$$280$$ 0 0
$$281$$ 10.3189 0.615576 0.307788 0.951455i $$-0.400411\pi$$
0.307788 + 0.951455i $$0.400411\pi$$
$$282$$ 0 0
$$283$$ −30.4565 −1.81045 −0.905224 0.424934i $$-0.860297\pi$$
−0.905224 + 0.424934i $$0.860297\pi$$
$$284$$ 0 0
$$285$$ −19.8010 −1.17291
$$286$$ 0 0
$$287$$ 20.4426 1.20669
$$288$$ 0 0
$$289$$ 33.2400 1.95529
$$290$$ 0 0
$$291$$ 30.0873 1.76375
$$292$$ 0 0
$$293$$ 7.90548 0.461843 0.230922 0.972972i $$-0.425826\pi$$
0.230922 + 0.972972i $$0.425826\pi$$
$$294$$ 0 0
$$295$$ −3.80330 −0.221437
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 5.41475 0.313143
$$300$$ 0 0
$$301$$ 30.6556 1.76696
$$302$$ 0 0
$$303$$ 35.1104 2.01704
$$304$$ 0 0
$$305$$ −2.11595 −0.121159
$$306$$ 0 0
$$307$$ −14.2585 −0.813777 −0.406889 0.913478i $$-0.633386\pi$$
−0.406889 + 0.913478i $$0.633386\pi$$
$$308$$ 0 0
$$309$$ 17.1348 0.974766
$$310$$ 0 0
$$311$$ 17.2118 0.975991 0.487995 0.872846i $$-0.337728\pi$$
0.487995 + 0.872846i $$0.337728\pi$$
$$312$$ 0 0
$$313$$ −7.96400 −0.450152 −0.225076 0.974341i $$-0.572263\pi$$
−0.225076 + 0.974341i $$0.572263\pi$$
$$314$$ 0 0
$$315$$ 21.3332 1.20199
$$316$$ 0 0
$$317$$ −13.7307 −0.771191 −0.385595 0.922668i $$-0.626004\pi$$
−0.385595 + 0.922668i $$0.626004\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 2.79336 0.155910
$$322$$ 0 0
$$323$$ 50.6330 2.81730
$$324$$ 0 0
$$325$$ −2.90026 −0.160877
$$326$$ 0 0
$$327$$ 18.3056 1.01230
$$328$$ 0 0
$$329$$ 35.7580 1.97140
$$330$$ 0 0
$$331$$ 10.6971 0.587967 0.293983 0.955811i $$-0.405019\pi$$
0.293983 + 0.955811i $$0.405019\pi$$
$$332$$ 0 0
$$333$$ −25.2480 −1.38358
$$334$$ 0 0
$$335$$ −12.8384 −0.701439
$$336$$ 0 0
$$337$$ −11.4797 −0.625338 −0.312669 0.949862i $$-0.601223\pi$$
−0.312669 + 0.949862i $$0.601223\pi$$
$$338$$ 0 0
$$339$$ 28.4426 1.54479
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 30.7401 1.65981
$$344$$ 0 0
$$345$$ 5.17511 0.278618
$$346$$ 0 0
$$347$$ 0.423181 0.0227176 0.0113588 0.999935i $$-0.496384\pi$$
0.0113588 + 0.999935i $$0.496384\pi$$
$$348$$ 0 0
$$349$$ −0.847568 −0.0453693 −0.0226846 0.999743i $$-0.507221\pi$$
−0.0226846 + 0.999743i $$0.507221\pi$$
$$350$$ 0 0
$$351$$ 13.5334 0.722357
$$352$$ 0 0
$$353$$ 9.79855 0.521524 0.260762 0.965403i $$-0.416026\pi$$
0.260762 + 0.965403i $$0.416026\pi$$
$$354$$ 0 0
$$355$$ 4.21095 0.223494
$$356$$ 0 0
$$357$$ −89.4944 −4.73655
$$358$$ 0 0
$$359$$ −7.60668 −0.401465 −0.200733 0.979646i $$-0.564332\pi$$
−0.200733 + 0.979646i $$0.564332\pi$$
$$360$$ 0 0
$$361$$ 32.0291 1.68574
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1.36576 −0.0714870
$$366$$ 0 0
$$367$$ −1.86780 −0.0974981 −0.0487491 0.998811i $$-0.515523\pi$$
−0.0487491 + 0.998811i $$0.515523\pi$$
$$368$$ 0 0
$$369$$ 21.0187 1.09419
$$370$$ 0 0
$$371$$ 3.25440 0.168960
$$372$$ 0 0
$$373$$ −20.0214 −1.03667 −0.518335 0.855178i $$-0.673448\pi$$
−0.518335 + 0.855178i $$0.673448\pi$$
$$374$$ 0 0
$$375$$ −2.77190 −0.143140
$$376$$ 0 0
$$377$$ −2.93129 −0.150969
$$378$$ 0 0
$$379$$ 24.0052 1.23307 0.616533 0.787329i $$-0.288537\pi$$
0.616533 + 0.787329i $$0.288537\pi$$
$$380$$ 0 0
$$381$$ 49.1210 2.51655
$$382$$ 0 0
$$383$$ 19.6767 1.00543 0.502716 0.864452i $$-0.332334\pi$$
0.502716 + 0.864452i $$0.332334\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 31.5195 1.60223
$$388$$ 0 0
$$389$$ 25.0051 1.26781 0.633904 0.773412i $$-0.281451\pi$$
0.633904 + 0.773412i $$0.281451\pi$$
$$390$$ 0 0
$$391$$ −13.2333 −0.669235
$$392$$ 0 0
$$393$$ 39.9149 2.01344
$$394$$ 0 0
$$395$$ 12.2375 0.615734
$$396$$ 0 0
$$397$$ −19.3120 −0.969240 −0.484620 0.874725i $$-0.661042\pi$$
−0.484620 + 0.874725i $$0.661042\pi$$
$$398$$ 0 0
$$399$$ −90.1946 −4.51538
$$400$$ 0 0
$$401$$ 23.2469 1.16090 0.580448 0.814297i $$-0.302877\pi$$
0.580448 + 0.814297i $$0.302877\pi$$
$$402$$ 0 0
$$403$$ −12.7911 −0.637172
$$404$$ 0 0
$$405$$ −1.11585 −0.0554470
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 5.92254 0.292851 0.146425 0.989222i $$-0.453223\pi$$
0.146425 + 0.989222i $$0.453223\pi$$
$$410$$ 0 0
$$411$$ −12.3895 −0.611131
$$412$$ 0 0
$$413$$ −17.3242 −0.852470
$$414$$ 0 0
$$415$$ 16.3351 0.801857
$$416$$ 0 0
$$417$$ 13.4305 0.657693
$$418$$ 0 0
$$419$$ −1.11739 −0.0545881 −0.0272940 0.999627i $$-0.508689\pi$$
−0.0272940 + 0.999627i $$0.508689\pi$$
$$420$$ 0 0
$$421$$ 17.2344 0.839956 0.419978 0.907534i $$-0.362038\pi$$
0.419978 + 0.907534i $$0.362038\pi$$
$$422$$ 0 0
$$423$$ 36.7656 1.78761
$$424$$ 0 0
$$425$$ 7.08802 0.343819
$$426$$ 0 0
$$427$$ −9.63828 −0.466429
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −7.74895 −0.373254 −0.186627 0.982431i $$-0.559756\pi$$
−0.186627 + 0.982431i $$0.559756\pi$$
$$432$$ 0 0
$$433$$ 3.93864 0.189279 0.0946395 0.995512i $$-0.469830\pi$$
0.0946395 + 0.995512i $$0.469830\pi$$
$$434$$ 0 0
$$435$$ −2.80156 −0.134324
$$436$$ 0 0
$$437$$ −13.3368 −0.637985
$$438$$ 0 0
$$439$$ 34.7505 1.65855 0.829275 0.558841i $$-0.188754\pi$$
0.829275 + 0.558841i $$0.188754\pi$$
$$440$$ 0 0
$$441$$ 64.3903 3.06620
$$442$$ 0 0
$$443$$ 24.3227 1.15560 0.577802 0.816177i $$-0.303910\pi$$
0.577802 + 0.816177i $$0.303910\pi$$
$$444$$ 0 0
$$445$$ −11.8165 −0.560155
$$446$$ 0 0
$$447$$ −29.7322 −1.40629
$$448$$ 0 0
$$449$$ −18.9235 −0.893057 −0.446528 0.894770i $$-0.647340\pi$$
−0.446528 + 0.894770i $$0.647340\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −0.559409 −0.0262833
$$454$$ 0 0
$$455$$ −13.2108 −0.619334
$$456$$ 0 0
$$457$$ 16.2210 0.758787 0.379393 0.925235i $$-0.376133\pi$$
0.379393 + 0.925235i $$0.376133\pi$$
$$458$$ 0 0
$$459$$ −33.0745 −1.54379
$$460$$ 0 0
$$461$$ 11.3851 0.530258 0.265129 0.964213i $$-0.414585\pi$$
0.265129 + 0.964213i $$0.414585\pi$$
$$462$$ 0 0
$$463$$ −36.4523 −1.69408 −0.847040 0.531529i $$-0.821618\pi$$
−0.847040 + 0.531529i $$0.821618\pi$$
$$464$$ 0 0
$$465$$ −12.2250 −0.566922
$$466$$ 0 0
$$467$$ 28.0455 1.29779 0.648895 0.760878i $$-0.275232\pi$$
0.648895 + 0.760878i $$0.275232\pi$$
$$468$$ 0 0
$$469$$ −58.4799 −2.70035
$$470$$ 0 0
$$471$$ 8.09943 0.373202
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 7.14347 0.327765
$$476$$ 0 0
$$477$$ 3.34610 0.153208
$$478$$ 0 0
$$479$$ 36.3433 1.66057 0.830284 0.557340i $$-0.188178\pi$$
0.830284 + 0.557340i $$0.188178\pi$$
$$480$$ 0 0
$$481$$ 15.6351 0.712899
$$482$$ 0 0
$$483$$ 23.5729 1.07261
$$484$$ 0 0
$$485$$ −10.8544 −0.492873
$$486$$ 0 0
$$487$$ 41.8132 1.89474 0.947368 0.320146i $$-0.103732\pi$$
0.947368 + 0.320146i $$0.103732\pi$$
$$488$$ 0 0
$$489$$ −23.5101 −1.06316
$$490$$ 0 0
$$491$$ −19.3555 −0.873501 −0.436751 0.899583i $$-0.643871\pi$$
−0.436751 + 0.899583i $$0.643871\pi$$
$$492$$ 0 0
$$493$$ 7.16386 0.322644
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 19.1811 0.860390
$$498$$ 0 0
$$499$$ 13.3514 0.597691 0.298846 0.954301i $$-0.403398\pi$$
0.298846 + 0.954301i $$0.403398\pi$$
$$500$$ 0 0
$$501$$ −27.8969 −1.24634
$$502$$ 0 0
$$503$$ −35.8888 −1.60020 −0.800100 0.599866i $$-0.795220\pi$$
−0.800100 + 0.599866i $$0.795220\pi$$
$$504$$ 0 0
$$505$$ −12.6665 −0.563654
$$506$$ 0 0
$$507$$ 12.7189 0.564865
$$508$$ 0 0
$$509$$ 17.9153 0.794080 0.397040 0.917801i $$-0.370037\pi$$
0.397040 + 0.917801i $$0.370037\pi$$
$$510$$ 0 0
$$511$$ −6.22111 −0.275206
$$512$$ 0 0
$$513$$ −33.3333 −1.47170
$$514$$ 0 0
$$515$$ −6.18162 −0.272395
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 16.0011 0.702370
$$520$$ 0 0
$$521$$ 7.25941 0.318040 0.159020 0.987275i $$-0.449167\pi$$
0.159020 + 0.987275i $$0.449167\pi$$
$$522$$ 0 0
$$523$$ −34.0176 −1.48748 −0.743742 0.668467i $$-0.766951\pi$$
−0.743742 + 0.668467i $$0.766951\pi$$
$$524$$ 0 0
$$525$$ −12.6262 −0.551051
$$526$$ 0 0
$$527$$ 31.2606 1.36173
$$528$$ 0 0
$$529$$ −19.5143 −0.848450
$$530$$ 0 0
$$531$$ −17.8124 −0.772994
$$532$$ 0 0
$$533$$ −13.0161 −0.563788
$$534$$ 0 0
$$535$$ −1.00774 −0.0435686
$$536$$ 0 0
$$537$$ −46.7539 −2.01758
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −19.0875 −0.820637 −0.410319 0.911942i $$-0.634583\pi$$
−0.410319 + 0.911942i $$0.634583\pi$$
$$542$$ 0 0
$$543$$ −41.8706 −1.79684
$$544$$ 0 0
$$545$$ −6.60399 −0.282884
$$546$$ 0 0
$$547$$ 23.3090 0.996621 0.498311 0.866999i $$-0.333954\pi$$
0.498311 + 0.866999i $$0.333954\pi$$
$$548$$ 0 0
$$549$$ −9.90988 −0.422944
$$550$$ 0 0
$$551$$ 7.21991 0.307578
$$552$$ 0 0
$$553$$ 55.7424 2.37041
$$554$$ 0 0
$$555$$ 14.9431 0.634300
$$556$$ 0 0
$$557$$ 28.3838 1.20266 0.601329 0.799001i $$-0.294638\pi$$
0.601329 + 0.799001i $$0.294638\pi$$
$$558$$ 0 0
$$559$$ −19.5188 −0.825557
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 8.13215 0.342729 0.171365 0.985208i $$-0.445182\pi$$
0.171365 + 0.985208i $$0.445182\pi$$
$$564$$ 0 0
$$565$$ −10.2611 −0.431686
$$566$$ 0 0
$$567$$ −5.08276 −0.213456
$$568$$ 0 0
$$569$$ −16.3414 −0.685067 −0.342534 0.939506i $$-0.611285\pi$$
−0.342534 + 0.939506i $$0.611285\pi$$
$$570$$ 0 0
$$571$$ −33.1999 −1.38937 −0.694687 0.719312i $$-0.744457\pi$$
−0.694687 + 0.719312i $$0.744457\pi$$
$$572$$ 0 0
$$573$$ −28.9215 −1.20822
$$574$$ 0 0
$$575$$ −1.86699 −0.0778589
$$576$$ 0 0
$$577$$ −45.0285 −1.87456 −0.937281 0.348576i $$-0.886665\pi$$
−0.937281 + 0.348576i $$0.886665\pi$$
$$578$$ 0 0
$$579$$ 38.5987 1.60411
$$580$$ 0 0
$$581$$ 74.4072 3.08693
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −13.5831 −0.561593
$$586$$ 0 0
$$587$$ −44.6981 −1.84489 −0.922444 0.386131i $$-0.873811\pi$$
−0.922444 + 0.386131i $$0.873811\pi$$
$$588$$ 0 0
$$589$$ 31.5052 1.29815
$$590$$ 0 0
$$591$$ −6.24326 −0.256813
$$592$$ 0 0
$$593$$ 18.7395 0.769538 0.384769 0.923013i $$-0.374281\pi$$
0.384769 + 0.923013i $$0.374281\pi$$
$$594$$ 0 0
$$595$$ 32.2863 1.32361
$$596$$ 0 0
$$597$$ −61.3270 −2.50995
$$598$$ 0 0
$$599$$ −19.5241 −0.797735 −0.398867 0.917009i $$-0.630597\pi$$
−0.398867 + 0.917009i $$0.630597\pi$$
$$600$$ 0 0
$$601$$ 31.3146 1.27735 0.638674 0.769477i $$-0.279483\pi$$
0.638674 + 0.769477i $$0.279483\pi$$
$$602$$ 0 0
$$603$$ −60.1278 −2.44859
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 22.0334 0.894309 0.447154 0.894457i $$-0.352437\pi$$
0.447154 + 0.894457i $$0.352437\pi$$
$$608$$ 0 0
$$609$$ −12.7613 −0.517112
$$610$$ 0 0
$$611$$ −22.7675 −0.921075
$$612$$ 0 0
$$613$$ −37.8937 −1.53051 −0.765256 0.643726i $$-0.777388\pi$$
−0.765256 + 0.643726i $$0.777388\pi$$
$$614$$ 0 0
$$615$$ −12.4400 −0.501629
$$616$$ 0 0
$$617$$ −34.0287 −1.36995 −0.684973 0.728569i $$-0.740186\pi$$
−0.684973 + 0.728569i $$0.740186\pi$$
$$618$$ 0 0
$$619$$ −26.4162 −1.06176 −0.530878 0.847448i $$-0.678138\pi$$
−0.530878 + 0.847448i $$0.678138\pi$$
$$620$$ 0 0
$$621$$ 8.71187 0.349595
$$622$$ 0 0
$$623$$ −53.8248 −2.15645
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −38.2111 −1.52357
$$630$$ 0 0
$$631$$ −38.5953 −1.53645 −0.768226 0.640178i $$-0.778861\pi$$
−0.768226 + 0.640178i $$0.778861\pi$$
$$632$$ 0 0
$$633$$ 21.5410 0.856176
$$634$$ 0 0
$$635$$ −17.7211 −0.703239
$$636$$ 0 0
$$637$$ −39.8744 −1.57988
$$638$$ 0 0
$$639$$ 19.7216 0.780175
$$640$$ 0 0
$$641$$ −18.2250 −0.719843 −0.359921 0.932983i $$-0.617196\pi$$
−0.359921 + 0.932983i $$0.617196\pi$$
$$642$$ 0 0
$$643$$ 10.5456 0.415879 0.207940 0.978142i $$-0.433324\pi$$
0.207940 + 0.978142i $$0.433324\pi$$
$$644$$ 0 0
$$645$$ −18.6549 −0.734537
$$646$$ 0 0
$$647$$ 16.9165 0.665057 0.332529 0.943093i $$-0.392098\pi$$
0.332529 + 0.943093i $$0.392098\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −55.6857 −2.18249
$$652$$ 0 0
$$653$$ 47.9383 1.87597 0.937986 0.346673i $$-0.112689\pi$$
0.937986 + 0.346673i $$0.112689\pi$$
$$654$$ 0 0
$$655$$ −14.3999 −0.562649
$$656$$ 0 0
$$657$$ −6.39641 −0.249548
$$658$$ 0 0
$$659$$ 5.96793 0.232477 0.116239 0.993221i $$-0.462916\pi$$
0.116239 + 0.993221i $$0.462916\pi$$
$$660$$ 0 0
$$661$$ −22.5701 −0.877875 −0.438937 0.898518i $$-0.644645\pi$$
−0.438937 + 0.898518i $$0.644645\pi$$
$$662$$ 0 0
$$663$$ 56.9821 2.21300
$$664$$ 0 0
$$665$$ 32.5389 1.26181
$$666$$ 0 0
$$667$$ −1.88697 −0.0730637
$$668$$ 0 0
$$669$$ 21.9558 0.848858
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −6.69924 −0.258237 −0.129118 0.991629i $$-0.541215\pi$$
−0.129118 + 0.991629i $$0.541215\pi$$
$$674$$ 0 0
$$675$$ −4.66626 −0.179605
$$676$$ 0 0
$$677$$ −41.5199 −1.59574 −0.797870 0.602830i $$-0.794040\pi$$
−0.797870 + 0.602830i $$0.794040\pi$$
$$678$$ 0 0
$$679$$ −49.4424 −1.89743
$$680$$ 0 0
$$681$$ −56.1786 −2.15277
$$682$$ 0 0
$$683$$ −13.4576 −0.514940 −0.257470 0.966286i $$-0.582889\pi$$
−0.257470 + 0.966286i $$0.582889\pi$$
$$684$$ 0 0
$$685$$ 4.46970 0.170778
$$686$$ 0 0
$$687$$ −27.3877 −1.04491
$$688$$ 0 0
$$689$$ −2.07211 −0.0789411
$$690$$ 0 0
$$691$$ −29.4815 −1.12153 −0.560765 0.827975i $$-0.689493\pi$$
−0.560765 + 0.827975i $$0.689493\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4.84523 −0.183790
$$696$$ 0 0
$$697$$ 31.8103 1.20490
$$698$$ 0 0
$$699$$ 27.1609 1.02732
$$700$$ 0 0
$$701$$ 9.03034 0.341071 0.170536 0.985352i $$-0.445450\pi$$
0.170536 + 0.985352i $$0.445450\pi$$
$$702$$ 0 0
$$703$$ −38.5100 −1.45243
$$704$$ 0 0
$$705$$ −21.7599 −0.819524
$$706$$ 0 0
$$707$$ −57.6969 −2.16991
$$708$$ 0 0
$$709$$ 49.0834 1.84336 0.921682 0.387945i $$-0.126815\pi$$
0.921682 + 0.387945i $$0.126815\pi$$
$$710$$ 0 0
$$711$$ 57.3132 2.14941
$$712$$ 0 0
$$713$$ −8.23407 −0.308368
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −30.5321 −1.14024
$$718$$ 0 0
$$719$$ −48.7090 −1.81654 −0.908269 0.418387i $$-0.862596\pi$$
−0.908269 + 0.418387i $$0.862596\pi$$
$$720$$ 0 0
$$721$$ −28.1576 −1.04865
$$722$$ 0 0
$$723$$ −36.8961 −1.37218
$$724$$ 0 0
$$725$$ 1.01070 0.0375365
$$726$$ 0 0
$$727$$ −50.6964 −1.88023 −0.940113 0.340864i $$-0.889281\pi$$
−0.940113 + 0.340864i $$0.889281\pi$$
$$728$$ 0 0
$$729$$ −44.0292 −1.63071
$$730$$ 0 0
$$731$$ 47.7025 1.76434
$$732$$ 0 0
$$733$$ 47.1585 1.74184 0.870920 0.491425i $$-0.163524\pi$$
0.870920 + 0.491425i $$0.163524\pi$$
$$734$$ 0 0
$$735$$ −38.1096 −1.40569
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −18.1569 −0.667912 −0.333956 0.942589i $$-0.608384\pi$$
−0.333956 + 0.942589i $$0.608384\pi$$
$$740$$ 0 0
$$741$$ 57.4279 2.10967
$$742$$ 0 0
$$743$$ 3.73625 0.137070 0.0685349 0.997649i $$-0.478168\pi$$
0.0685349 + 0.997649i $$0.478168\pi$$
$$744$$ 0 0
$$745$$ 10.7263 0.392982
$$746$$ 0 0
$$747$$ 76.5040 2.79913
$$748$$ 0 0
$$749$$ −4.59033 −0.167727
$$750$$ 0 0
$$751$$ −27.8430 −1.01600 −0.508002 0.861356i $$-0.669616\pi$$
−0.508002 + 0.861356i $$0.669616\pi$$
$$752$$ 0 0
$$753$$ 40.4463 1.47395
$$754$$ 0 0
$$755$$ 0.201814 0.00734478
$$756$$ 0 0
$$757$$ 45.0313 1.63669 0.818346 0.574726i $$-0.194891\pi$$
0.818346 + 0.574726i $$0.194891\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.7068 0.678120 0.339060 0.940765i $$-0.389891\pi$$
0.339060 + 0.940765i $$0.389891\pi$$
$$762$$ 0 0
$$763$$ −30.0816 −1.08903
$$764$$ 0 0
$$765$$ 33.1961 1.20021
$$766$$ 0 0
$$767$$ 11.0305 0.398290
$$768$$ 0 0
$$769$$ 13.5701 0.489351 0.244675 0.969605i $$-0.421319\pi$$
0.244675 + 0.969605i $$0.421319\pi$$
$$770$$ 0 0
$$771$$ −2.67635 −0.0963862
$$772$$ 0 0
$$773$$ −3.27790 −0.117898 −0.0589489 0.998261i $$-0.518775\pi$$
−0.0589489 + 0.998261i $$0.518775\pi$$
$$774$$ 0 0
$$775$$ 4.41034 0.158424
$$776$$ 0 0
$$777$$ 68.0668 2.44188
$$778$$ 0 0
$$779$$ 32.0592 1.14864
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −4.71619 −0.168543
$$784$$ 0 0
$$785$$ −2.92198 −0.104290
$$786$$ 0 0
$$787$$ −2.54160 −0.0905981 −0.0452991 0.998973i $$-0.514424\pi$$
−0.0452991 + 0.998973i $$0.514424\pi$$
$$788$$ 0 0
$$789$$ 57.5870 2.05015
$$790$$ 0 0
$$791$$ −46.7397 −1.66187
$$792$$ 0 0
$$793$$ 6.13680 0.217924
$$794$$ 0 0
$$795$$ −1.98040 −0.0702377
$$796$$ 0 0
$$797$$ 26.2914 0.931290 0.465645 0.884972i $$-0.345822\pi$$
0.465645 + 0.884972i $$0.345822\pi$$
$$798$$ 0 0
$$799$$ 55.6422 1.96848
$$800$$ 0 0
$$801$$ −55.3416 −1.95540
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −8.50425 −0.299736
$$806$$ 0 0
$$807$$ 10.8489 0.381898
$$808$$ 0 0
$$809$$ 7.94284 0.279255 0.139628 0.990204i $$-0.455409\pi$$
0.139628 + 0.990204i $$0.455409\pi$$
$$810$$ 0 0
$$811$$ −19.4167 −0.681811 −0.340906 0.940098i $$-0.610734\pi$$
−0.340906 + 0.940098i $$0.610734\pi$$
$$812$$ 0 0
$$813$$ −39.5777 −1.38805
$$814$$ 0 0
$$815$$ 8.48158 0.297097
$$816$$ 0 0
$$817$$ 48.0757 1.68196
$$818$$ 0 0
$$819$$ −61.8719 −2.16198
$$820$$ 0 0
$$821$$ 13.0819 0.456561 0.228281 0.973595i $$-0.426690\pi$$
0.228281 + 0.973595i $$0.426690\pi$$
$$822$$ 0 0
$$823$$ 31.4670 1.09687 0.548435 0.836193i $$-0.315224\pi$$
0.548435 + 0.836193i $$0.315224\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 6.47448 0.225140 0.112570 0.993644i $$-0.464092\pi$$
0.112570 + 0.993644i $$0.464092\pi$$
$$828$$ 0 0
$$829$$ −50.0421 −1.73804 −0.869018 0.494781i $$-0.835248\pi$$
−0.869018 + 0.494781i $$0.835248\pi$$
$$830$$ 0 0
$$831$$ 10.4306 0.361833
$$832$$ 0 0
$$833$$ 97.4500 3.37644
$$834$$ 0 0
$$835$$ 10.0642 0.348286
$$836$$ 0 0
$$837$$ −20.5798 −0.711343
$$838$$ 0 0
$$839$$ 10.0716 0.347709 0.173855 0.984771i $$-0.444378\pi$$
0.173855 + 0.984771i $$0.444378\pi$$
$$840$$ 0 0
$$841$$ −27.9785 −0.964775
$$842$$ 0 0
$$843$$ −28.6030 −0.985141
$$844$$ 0 0
$$845$$ −4.58851 −0.157850
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 84.4222 2.89736
$$850$$ 0 0
$$851$$ 10.0648 0.345018
$$852$$ 0 0
$$853$$ 24.2805 0.831348 0.415674 0.909514i $$-0.363546\pi$$
0.415674 + 0.909514i $$0.363546\pi$$
$$854$$ 0 0
$$855$$ 33.4558 1.14417
$$856$$ 0 0
$$857$$ −23.8353 −0.814200 −0.407100 0.913384i $$-0.633460\pi$$
−0.407100 + 0.913384i $$0.633460\pi$$
$$858$$ 0 0
$$859$$ 27.1693 0.927004 0.463502 0.886096i $$-0.346593\pi$$
0.463502 + 0.886096i $$0.346593\pi$$
$$860$$ 0 0
$$861$$ −56.6649 −1.93114
$$862$$ 0 0
$$863$$ 33.5003 1.14036 0.570182 0.821518i $$-0.306873\pi$$
0.570182 + 0.821518i $$0.306873\pi$$
$$864$$ 0 0
$$865$$ −5.77262 −0.196275
$$866$$ 0 0
$$867$$ −92.1378 −3.12917
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 37.2348 1.26165
$$872$$ 0 0
$$873$$ −50.8357 −1.72053
$$874$$ 0 0
$$875$$ 4.55506 0.153989
$$876$$ 0 0
$$877$$ 4.10027 0.138456 0.0692281 0.997601i $$-0.477946\pi$$
0.0692281 + 0.997601i $$0.477946\pi$$
$$878$$ 0 0
$$879$$ −21.9132 −0.739113
$$880$$ 0 0
$$881$$ 34.8285 1.17340 0.586701 0.809804i $$-0.300426\pi$$
0.586701 + 0.809804i $$0.300426\pi$$
$$882$$ 0 0
$$883$$ −26.6934 −0.898304 −0.449152 0.893455i $$-0.648274\pi$$
−0.449152 + 0.893455i $$0.648274\pi$$
$$884$$ 0 0
$$885$$ 10.5424 0.354377
$$886$$ 0 0
$$887$$ −51.4083 −1.72612 −0.863060 0.505101i $$-0.831455\pi$$
−0.863060 + 0.505101i $$0.831455\pi$$
$$888$$ 0 0
$$889$$ −80.7205 −2.70728
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 56.0775 1.87656
$$894$$ 0 0
$$895$$ 16.8671 0.563805
$$896$$ 0 0
$$897$$ −15.0091 −0.501141
$$898$$ 0 0
$$899$$ 4.45754 0.148667
$$900$$ 0 0
$$901$$ 5.06409 0.168709
$$902$$ 0 0
$$903$$ −84.9743 −2.82777
$$904$$ 0 0
$$905$$ 15.1054 0.502120
$$906$$ 0 0
$$907$$ −6.75322 −0.224237 −0.112118 0.993695i $$-0.535764\pi$$
−0.112118 + 0.993695i $$0.535764\pi$$
$$908$$ 0 0
$$909$$ −59.3227 −1.96761
$$910$$ 0 0
$$911$$ −13.0900 −0.433692 −0.216846 0.976206i $$-0.569577\pi$$
−0.216846 + 0.976206i $$0.569577\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 5.86520 0.193898
$$916$$ 0 0
$$917$$ −65.5922 −2.16605
$$918$$ 0 0
$$919$$ −0.533564 −0.0176007 −0.00880033 0.999961i $$-0.502801\pi$$
−0.00880033 + 0.999961i $$0.502801\pi$$
$$920$$ 0 0
$$921$$ 39.5232 1.30233
$$922$$ 0 0
$$923$$ −12.2128 −0.401990
$$924$$ 0 0
$$925$$ −5.39094 −0.177253
$$926$$ 0 0
$$927$$ −28.9511 −0.950879
$$928$$ 0 0
$$929$$ −9.92150 −0.325514 −0.162757 0.986666i $$-0.552039\pi$$
−0.162757 + 0.986666i $$0.552039\pi$$
$$930$$ 0 0
$$931$$ 98.2124 3.21878
$$932$$ 0 0
$$933$$ −47.7093 −1.56193
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3.57002 0.116627 0.0583137 0.998298i $$-0.481428\pi$$
0.0583137 + 0.998298i $$0.481428\pi$$
$$938$$ 0 0
$$939$$ 22.0754 0.720403
$$940$$ 0 0
$$941$$ 48.8240 1.59162 0.795809 0.605548i $$-0.207046\pi$$
0.795809 + 0.605548i $$0.207046\pi$$
$$942$$ 0 0
$$943$$ −8.37887 −0.272853
$$944$$ 0 0
$$945$$ −21.2551 −0.691428
$$946$$ 0 0
$$947$$ 15.1463 0.492190 0.246095 0.969246i $$-0.420852\pi$$
0.246095 + 0.969246i $$0.420852\pi$$
$$948$$ 0 0
$$949$$ 3.96105 0.128581
$$950$$ 0 0
$$951$$ 38.0600 1.23418
$$952$$ 0 0
$$953$$ −25.6087 −0.829547 −0.414773 0.909925i $$-0.636139\pi$$
−0.414773 + 0.909925i $$0.636139\pi$$
$$954$$ 0 0
$$955$$ 10.4338 0.337631
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 20.3597 0.657450
$$960$$ 0 0
$$961$$ −11.5489 −0.372544
$$962$$ 0 0
$$963$$ −4.71969 −0.152090
$$964$$ 0 0
$$965$$ −13.9250 −0.448262
$$966$$ 0 0
$$967$$ −19.9187 −0.640544 −0.320272 0.947326i $$-0.603774\pi$$
−0.320272 + 0.947326i $$0.603774\pi$$
$$968$$ 0 0
$$969$$ −140.350 −4.50868
$$970$$ 0 0
$$971$$ −14.3646 −0.460980 −0.230490 0.973075i $$-0.574033\pi$$
−0.230490 + 0.973075i $$0.574033\pi$$
$$972$$ 0 0
$$973$$ −22.0703 −0.707541
$$974$$ 0 0
$$975$$ 8.03922 0.257461
$$976$$ 0 0
$$977$$ 48.4441 1.54987 0.774933 0.632044i $$-0.217784\pi$$
0.774933 + 0.632044i $$0.217784\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −30.9293 −0.987496
$$982$$ 0 0
$$983$$ 10.0332 0.320010 0.160005 0.987116i $$-0.448849\pi$$
0.160005 + 0.987116i $$0.448849\pi$$
$$984$$ 0 0
$$985$$ 2.25234 0.0717655
$$986$$ 0 0
$$987$$ −99.1175 −3.15495
$$988$$ 0 0
$$989$$ −12.5649 −0.399540
$$990$$ 0 0
$$991$$ −4.44920 −0.141333 −0.0706667 0.997500i $$-0.522513\pi$$
−0.0706667 + 0.997500i $$0.522513\pi$$
$$992$$ 0 0
$$993$$ −29.6513 −0.940956
$$994$$ 0 0
$$995$$ 22.1246 0.701396
$$996$$ 0 0
$$997$$ −42.4383 −1.34404 −0.672018 0.740535i $$-0.734572\pi$$
−0.672018 + 0.740535i $$0.734572\pi$$
$$998$$ 0 0
$$999$$ 25.1555 0.795886
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4840.2.a.bh.1.2 8
4.3 odd 2 9680.2.a.de.1.7 8
11.2 odd 10 440.2.y.d.81.1 16
11.6 odd 10 440.2.y.d.201.1 yes 16
11.10 odd 2 4840.2.a.bg.1.2 8
44.35 even 10 880.2.bo.k.81.4 16
44.39 even 10 880.2.bo.k.641.4 16
44.43 even 2 9680.2.a.df.1.7 8

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.1 16 11.2 odd 10
440.2.y.d.201.1 yes 16 11.6 odd 10
880.2.bo.k.81.4 16 44.35 even 10
880.2.bo.k.641.4 16 44.39 even 10
4840.2.a.bg.1.2 8 11.10 odd 2
4840.2.a.bh.1.2 8 1.1 even 1 trivial
9680.2.a.de.1.7 8 4.3 odd 2
9680.2.a.df.1.7 8 44.43 even 2