# Properties

 Label 2200.2.a.m.1.2 Level $2200$ Weight $2$ Character 2200.1 Self dual yes Analytic conductor $17.567$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$17.5670884447$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.56155 q^{3} -3.56155 q^{7} -0.561553 q^{9} +O(q^{10})$$ $$q+1.56155 q^{3} -3.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} +3.12311 q^{13} -5.56155 q^{17} +2.43845 q^{19} -5.56155 q^{21} -7.12311 q^{23} -5.56155 q^{27} -0.438447 q^{29} +8.68466 q^{31} +1.56155 q^{33} -9.80776 q^{37} +4.87689 q^{39} -10.0000 q^{41} -5.12311 q^{43} +7.12311 q^{47} +5.68466 q^{49} -8.68466 q^{51} +4.43845 q^{53} +3.80776 q^{57} -13.3693 q^{59} -3.56155 q^{61} +2.00000 q^{63} -11.1231 q^{69} -2.43845 q^{71} -4.87689 q^{73} -3.56155 q^{77} +0.876894 q^{79} -7.00000 q^{81} -10.0000 q^{83} -0.684658 q^{87} +9.80776 q^{89} -11.1231 q^{91} +13.5616 q^{93} -17.1231 q^{97} -0.561553 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 3 * q^7 + 3 * q^9 $$2 q - q^{3} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} - 7 q^{17} + 9 q^{19} - 7 q^{21} - 6 q^{23} - 7 q^{27} - 5 q^{29} + 5 q^{31} - q^{33} + q^{37} + 18 q^{39} - 20 q^{41} - 2 q^{43} + 6 q^{47} - q^{49} - 5 q^{51} + 13 q^{53} - 13 q^{57} - 2 q^{59} - 3 q^{61} + 4 q^{63} - 14 q^{69} - 9 q^{71} - 18 q^{73} - 3 q^{77} + 10 q^{79} - 14 q^{81} - 20 q^{83} + 11 q^{87} - q^{89} - 14 q^{91} + 23 q^{93} - 26 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q - q^3 - 3 * q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 - 7 * q^17 + 9 * q^19 - 7 * q^21 - 6 * q^23 - 7 * q^27 - 5 * q^29 + 5 * q^31 - q^33 + q^37 + 18 * q^39 - 20 * q^41 - 2 * q^43 + 6 * q^47 - q^49 - 5 * q^51 + 13 * q^53 - 13 * q^57 - 2 * q^59 - 3 * q^61 + 4 * q^63 - 14 * q^69 - 9 * q^71 - 18 * q^73 - 3 * q^77 + 10 * q^79 - 14 * q^81 - 20 * q^83 + 11 * q^87 - q^89 - 14 * q^91 + 23 * q^93 - 26 * q^97 + 3 * q^99

## 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
$$3$$ 1.56155 0.901563 0.450781 0.892634i $$-0.351145\pi$$
0.450781 + 0.892634i $$0.351145\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.56155 −1.34614 −0.673070 0.739579i $$-0.735025\pi$$
−0.673070 + 0.739579i $$0.735025\pi$$
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 3.12311 0.866194 0.433097 0.901347i $$-0.357421\pi$$
0.433097 + 0.901347i $$0.357421\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −5.56155 −1.34887 −0.674437 0.738332i $$-0.735614\pi$$
−0.674437 + 0.738332i $$0.735614\pi$$
$$18$$ 0 0
$$19$$ 2.43845 0.559418 0.279709 0.960085i $$-0.409762\pi$$
0.279709 + 0.960085i $$0.409762\pi$$
$$20$$ 0 0
$$21$$ −5.56155 −1.21363
$$22$$ 0 0
$$23$$ −7.12311 −1.48527 −0.742635 0.669696i $$-0.766424\pi$$
−0.742635 + 0.669696i $$0.766424\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.56155 −1.07032
$$28$$ 0 0
$$29$$ −0.438447 −0.0814176 −0.0407088 0.999171i $$-0.512962\pi$$
−0.0407088 + 0.999171i $$0.512962\pi$$
$$30$$ 0 0
$$31$$ 8.68466 1.55981 0.779905 0.625897i $$-0.215267\pi$$
0.779905 + 0.625897i $$0.215267\pi$$
$$32$$ 0 0
$$33$$ 1.56155 0.271831
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.80776 −1.61239 −0.806193 0.591652i $$-0.798476\pi$$
−0.806193 + 0.591652i $$0.798476\pi$$
$$38$$ 0 0
$$39$$ 4.87689 0.780928
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ −5.12311 −0.781266 −0.390633 0.920546i $$-0.627744\pi$$
−0.390633 + 0.920546i $$0.627744\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 7.12311 1.03901 0.519506 0.854467i $$-0.326116\pi$$
0.519506 + 0.854467i $$0.326116\pi$$
$$48$$ 0 0
$$49$$ 5.68466 0.812094
$$50$$ 0 0
$$51$$ −8.68466 −1.21610
$$52$$ 0 0
$$53$$ 4.43845 0.609668 0.304834 0.952406i $$-0.401399\pi$$
0.304834 + 0.952406i $$0.401399\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 3.80776 0.504351
$$58$$ 0 0
$$59$$ −13.3693 −1.74054 −0.870268 0.492578i $$-0.836055\pi$$
−0.870268 + 0.492578i $$0.836055\pi$$
$$60$$ 0 0
$$61$$ −3.56155 −0.456010 −0.228005 0.973660i $$-0.573220\pi$$
−0.228005 + 0.973660i $$0.573220\pi$$
$$62$$ 0 0
$$63$$ 2.00000 0.251976
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ −11.1231 −1.33906
$$70$$ 0 0
$$71$$ −2.43845 −0.289390 −0.144695 0.989476i $$-0.546220\pi$$
−0.144695 + 0.989476i $$0.546220\pi$$
$$72$$ 0 0
$$73$$ −4.87689 −0.570797 −0.285399 0.958409i $$-0.592126\pi$$
−0.285399 + 0.958409i $$0.592126\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.56155 −0.405877
$$78$$ 0 0
$$79$$ 0.876894 0.0986583 0.0493292 0.998783i $$-0.484292\pi$$
0.0493292 + 0.998783i $$0.484292\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ −10.0000 −1.09764 −0.548821 0.835940i $$-0.684923\pi$$
−0.548821 + 0.835940i $$0.684923\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −0.684658 −0.0734031
$$88$$ 0 0
$$89$$ 9.80776 1.03962 0.519810 0.854282i $$-0.326003\pi$$
0.519810 + 0.854282i $$0.326003\pi$$
$$90$$ 0 0
$$91$$ −11.1231 −1.16602
$$92$$ 0 0
$$93$$ 13.5616 1.40627
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −17.1231 −1.73859 −0.869294 0.494295i $$-0.835426\pi$$
−0.869294 + 0.494295i $$0.835426\pi$$
$$98$$ 0 0
$$99$$ −0.561553 −0.0564382
$$100$$ 0 0
$$101$$ −16.2462 −1.61656 −0.808279 0.588799i $$-0.799601\pi$$
−0.808279 + 0.588799i $$0.799601\pi$$
$$102$$ 0 0
$$103$$ 14.2462 1.40372 0.701860 0.712314i $$-0.252353\pi$$
0.701860 + 0.712314i $$0.252353\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −5.12311 −0.495269 −0.247635 0.968853i $$-0.579653\pi$$
−0.247635 + 0.968853i $$0.579653\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −15.3153 −1.45367
$$112$$ 0 0
$$113$$ −1.12311 −0.105653 −0.0528264 0.998604i $$-0.516823\pi$$
−0.0528264 + 0.998604i $$0.516823\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.75379 −0.162138
$$118$$ 0 0
$$119$$ 19.8078 1.81577
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −15.6155 −1.40800
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 7.36932 0.653921 0.326961 0.945038i $$-0.393976\pi$$
0.326961 + 0.945038i $$0.393976\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 0.684658 0.0598189 0.0299094 0.999553i $$-0.490478\pi$$
0.0299094 + 0.999553i $$0.490478\pi$$
$$132$$ 0 0
$$133$$ −8.68466 −0.753055
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 19.3693 1.65483 0.827416 0.561589i $$-0.189810\pi$$
0.827416 + 0.561589i $$0.189810\pi$$
$$138$$ 0 0
$$139$$ 16.4924 1.39887 0.699435 0.714697i $$-0.253435\pi$$
0.699435 + 0.714697i $$0.253435\pi$$
$$140$$ 0 0
$$141$$ 11.1231 0.936734
$$142$$ 0 0
$$143$$ 3.12311 0.261167
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 8.87689 0.732154
$$148$$ 0 0
$$149$$ −1.31534 −0.107757 −0.0538785 0.998547i $$-0.517158\pi$$
−0.0538785 + 0.998547i $$0.517158\pi$$
$$150$$ 0 0
$$151$$ −0.876894 −0.0713607 −0.0356803 0.999363i $$-0.511360\pi$$
−0.0356803 + 0.999363i $$0.511360\pi$$
$$152$$ 0 0
$$153$$ 3.12311 0.252488
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 12.9309 1.03200 0.515998 0.856590i $$-0.327421\pi$$
0.515998 + 0.856590i $$0.327421\pi$$
$$158$$ 0 0
$$159$$ 6.93087 0.549654
$$160$$ 0 0
$$161$$ 25.3693 1.99938
$$162$$ 0 0
$$163$$ −0.192236 −0.0150571 −0.00752854 0.999972i $$-0.502396\pi$$
−0.00752854 + 0.999972i $$0.502396\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1.31534 −0.101784 −0.0508921 0.998704i $$-0.516206\pi$$
−0.0508921 + 0.998704i $$0.516206\pi$$
$$168$$ 0 0
$$169$$ −3.24621 −0.249709
$$170$$ 0 0
$$171$$ −1.36932 −0.104714
$$172$$ 0 0
$$173$$ 21.3693 1.62468 0.812340 0.583185i $$-0.198194\pi$$
0.812340 + 0.583185i $$0.198194\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −20.8769 −1.56920
$$178$$ 0 0
$$179$$ 8.49242 0.634753 0.317377 0.948300i $$-0.397198\pi$$
0.317377 + 0.948300i $$0.397198\pi$$
$$180$$ 0 0
$$181$$ 14.4924 1.07721 0.538607 0.842557i $$-0.318951\pi$$
0.538607 + 0.842557i $$0.318951\pi$$
$$182$$ 0 0
$$183$$ −5.56155 −0.411122
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −5.56155 −0.406701
$$188$$ 0 0
$$189$$ 19.8078 1.44080
$$190$$ 0 0
$$191$$ 1.75379 0.126900 0.0634499 0.997985i $$-0.479790\pi$$
0.0634499 + 0.997985i $$0.479790\pi$$
$$192$$ 0 0
$$193$$ −19.8078 −1.42579 −0.712897 0.701269i $$-0.752617\pi$$
−0.712897 + 0.701269i $$0.752617\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −18.2462 −1.29999 −0.649994 0.759939i $$-0.725229\pi$$
−0.649994 + 0.759939i $$0.725229\pi$$
$$198$$ 0 0
$$199$$ −6.93087 −0.491316 −0.245658 0.969357i $$-0.579004\pi$$
−0.245658 + 0.969357i $$0.579004\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.56155 0.109600
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 2.43845 0.168671
$$210$$ 0 0
$$211$$ 0.684658 0.0471338 0.0235669 0.999722i $$-0.492498\pi$$
0.0235669 + 0.999722i $$0.492498\pi$$
$$212$$ 0 0
$$213$$ −3.80776 −0.260904
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −30.9309 −2.09972
$$218$$ 0 0
$$219$$ −7.61553 −0.514610
$$220$$ 0 0
$$221$$ −17.3693 −1.16839
$$222$$ 0 0
$$223$$ 23.1231 1.54844 0.774219 0.632918i $$-0.218143\pi$$
0.774219 + 0.632918i $$0.218143\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ −6.49242 −0.429031 −0.214516 0.976721i $$-0.568817\pi$$
−0.214516 + 0.976721i $$0.568817\pi$$
$$230$$ 0 0
$$231$$ −5.56155 −0.365923
$$232$$ 0 0
$$233$$ −11.3153 −0.741293 −0.370646 0.928774i $$-0.620864\pi$$
−0.370646 + 0.928774i $$0.620864\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1.36932 0.0889467
$$238$$ 0 0
$$239$$ 0.876894 0.0567216 0.0283608 0.999598i $$-0.490971\pi$$
0.0283608 + 0.999598i $$0.490971\pi$$
$$240$$ 0 0
$$241$$ −28.7386 −1.85122 −0.925609 0.378481i $$-0.876447\pi$$
−0.925609 + 0.378481i $$0.876447\pi$$
$$242$$ 0 0
$$243$$ 5.75379 0.369106
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 7.61553 0.484564
$$248$$ 0 0
$$249$$ −15.6155 −0.989594
$$250$$ 0 0
$$251$$ 23.1231 1.45952 0.729759 0.683705i $$-0.239632\pi$$
0.729759 + 0.683705i $$0.239632\pi$$
$$252$$ 0 0
$$253$$ −7.12311 −0.447826
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 7.75379 0.483668 0.241834 0.970318i $$-0.422251\pi$$
0.241834 + 0.970318i $$0.422251\pi$$
$$258$$ 0 0
$$259$$ 34.9309 2.17050
$$260$$ 0 0
$$261$$ 0.246211 0.0152401
$$262$$ 0 0
$$263$$ −24.4384 −1.50694 −0.753470 0.657483i $$-0.771621\pi$$
−0.753470 + 0.657483i $$0.771621\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 15.3153 0.937284
$$268$$ 0 0
$$269$$ −22.8769 −1.39483 −0.697414 0.716668i $$-0.745666\pi$$
−0.697414 + 0.716668i $$0.745666\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 0 0
$$273$$ −17.3693 −1.05124
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −18.7386 −1.12590 −0.562948 0.826493i $$-0.690333\pi$$
−0.562948 + 0.826493i $$0.690333\pi$$
$$278$$ 0 0
$$279$$ −4.87689 −0.291972
$$280$$ 0 0
$$281$$ 18.4924 1.10317 0.551583 0.834120i $$-0.314024\pi$$
0.551583 + 0.834120i $$0.314024\pi$$
$$282$$ 0 0
$$283$$ 18.4924 1.09926 0.549630 0.835408i $$-0.314769\pi$$
0.549630 + 0.835408i $$0.314769\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 35.6155 2.10232
$$288$$ 0 0
$$289$$ 13.9309 0.819463
$$290$$ 0 0
$$291$$ −26.7386 −1.56745
$$292$$ 0 0
$$293$$ 11.1231 0.649819 0.324909 0.945745i $$-0.394666\pi$$
0.324909 + 0.945745i $$0.394666\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −5.56155 −0.322714
$$298$$ 0 0
$$299$$ −22.2462 −1.28653
$$300$$ 0 0
$$301$$ 18.2462 1.05169
$$302$$ 0 0
$$303$$ −25.3693 −1.45743
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −20.7386 −1.18362 −0.591808 0.806079i $$-0.701586\pi$$
−0.591808 + 0.806079i $$0.701586\pi$$
$$308$$ 0 0
$$309$$ 22.2462 1.26554
$$310$$ 0 0
$$311$$ 21.1771 1.20084 0.600421 0.799684i $$-0.295000\pi$$
0.600421 + 0.799684i $$0.295000\pi$$
$$312$$ 0 0
$$313$$ −16.2462 −0.918290 −0.459145 0.888361i $$-0.651844\pi$$
−0.459145 + 0.888361i $$0.651844\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.56155 0.200037 0.100018 0.994986i $$-0.468110\pi$$
0.100018 + 0.994986i $$0.468110\pi$$
$$318$$ 0 0
$$319$$ −0.438447 −0.0245483
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 0 0
$$323$$ −13.5616 −0.754585
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −3.12311 −0.172708
$$328$$ 0 0
$$329$$ −25.3693 −1.39866
$$330$$ 0 0
$$331$$ −2.24621 −0.123463 −0.0617315 0.998093i $$-0.519662\pi$$
−0.0617315 + 0.998093i $$0.519662\pi$$
$$332$$ 0 0
$$333$$ 5.50758 0.301813
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −6.93087 −0.377549 −0.188774 0.982021i $$-0.560451\pi$$
−0.188774 + 0.982021i $$0.560451\pi$$
$$338$$ 0 0
$$339$$ −1.75379 −0.0952527
$$340$$ 0 0
$$341$$ 8.68466 0.470301
$$342$$ 0 0
$$343$$ 4.68466 0.252948
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −24.2462 −1.30160 −0.650802 0.759247i $$-0.725567\pi$$
−0.650802 + 0.759247i $$0.725567\pi$$
$$348$$ 0 0
$$349$$ −24.7386 −1.32423 −0.662114 0.749403i $$-0.730341\pi$$
−0.662114 + 0.749403i $$0.730341\pi$$
$$350$$ 0 0
$$351$$ −17.3693 −0.927106
$$352$$ 0 0
$$353$$ −3.75379 −0.199794 −0.0998970 0.994998i $$-0.531851\pi$$
−0.0998970 + 0.994998i $$0.531851\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 30.9309 1.63704
$$358$$ 0 0
$$359$$ −8.49242 −0.448213 −0.224106 0.974565i $$-0.571946\pi$$
−0.224106 + 0.974565i $$0.571946\pi$$
$$360$$ 0 0
$$361$$ −13.0540 −0.687051
$$362$$ 0 0
$$363$$ 1.56155 0.0819603
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 0 0
$$369$$ 5.61553 0.292333
$$370$$ 0 0
$$371$$ −15.8078 −0.820698
$$372$$ 0 0
$$373$$ 29.3693 1.52069 0.760343 0.649522i $$-0.225031\pi$$
0.760343 + 0.649522i $$0.225031\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.36932 −0.0705234
$$378$$ 0 0
$$379$$ 27.6155 1.41851 0.709257 0.704950i $$-0.249030\pi$$
0.709257 + 0.704950i $$0.249030\pi$$
$$380$$ 0 0
$$381$$ 11.5076 0.589551
$$382$$ 0 0
$$383$$ 29.8617 1.52586 0.762932 0.646479i $$-0.223759\pi$$
0.762932 + 0.646479i $$0.223759\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.87689 0.146241
$$388$$ 0 0
$$389$$ −32.2462 −1.63495 −0.817474 0.575966i $$-0.804626\pi$$
−0.817474 + 0.575966i $$0.804626\pi$$
$$390$$ 0 0
$$391$$ 39.6155 2.00344
$$392$$ 0 0
$$393$$ 1.06913 0.0539305
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −5.50758 −0.276417 −0.138209 0.990403i $$-0.544134\pi$$
−0.138209 + 0.990403i $$0.544134\pi$$
$$398$$ 0 0
$$399$$ −13.5616 −0.678927
$$400$$ 0 0
$$401$$ −21.8078 −1.08903 −0.544514 0.838752i $$-0.683286\pi$$
−0.544514 + 0.838752i $$0.683286\pi$$
$$402$$ 0 0
$$403$$ 27.1231 1.35110
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −9.80776 −0.486153
$$408$$ 0 0
$$409$$ −17.1231 −0.846683 −0.423342 0.905970i $$-0.639143\pi$$
−0.423342 + 0.905970i $$0.639143\pi$$
$$410$$ 0 0
$$411$$ 30.2462 1.49194
$$412$$ 0 0
$$413$$ 47.6155 2.34301
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 25.7538 1.26117
$$418$$ 0 0
$$419$$ −5.75379 −0.281091 −0.140545 0.990074i $$-0.544886\pi$$
−0.140545 + 0.990074i $$0.544886\pi$$
$$420$$ 0 0
$$421$$ 21.1231 1.02948 0.514739 0.857347i $$-0.327889\pi$$
0.514739 + 0.857347i $$0.327889\pi$$
$$422$$ 0 0
$$423$$ −4.00000 −0.194487
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 12.6847 0.613854
$$428$$ 0 0
$$429$$ 4.87689 0.235459
$$430$$ 0 0
$$431$$ 20.4924 0.987085 0.493543 0.869722i $$-0.335702\pi$$
0.493543 + 0.869722i $$0.335702\pi$$
$$432$$ 0 0
$$433$$ 23.8617 1.14672 0.573361 0.819303i $$-0.305639\pi$$
0.573361 + 0.819303i $$0.305639\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −17.3693 −0.830887
$$438$$ 0 0
$$439$$ −24.9848 −1.19246 −0.596231 0.802813i $$-0.703336\pi$$
−0.596231 + 0.802813i $$0.703336\pi$$
$$440$$ 0 0
$$441$$ −3.19224 −0.152011
$$442$$ 0 0
$$443$$ −20.0000 −0.950229 −0.475114 0.879924i $$-0.657593\pi$$
−0.475114 + 0.879924i $$0.657593\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −2.05398 −0.0971497
$$448$$ 0 0
$$449$$ −28.7386 −1.35626 −0.678130 0.734942i $$-0.737209\pi$$
−0.678130 + 0.734942i $$0.737209\pi$$
$$450$$ 0 0
$$451$$ −10.0000 −0.470882
$$452$$ 0 0
$$453$$ −1.36932 −0.0643361
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.0540 1.78009 0.890045 0.455873i $$-0.150673\pi$$
0.890045 + 0.455873i $$0.150673\pi$$
$$458$$ 0 0
$$459$$ 30.9309 1.44373
$$460$$ 0 0
$$461$$ 28.9309 1.34744 0.673722 0.738984i $$-0.264694\pi$$
0.673722 + 0.738984i $$0.264694\pi$$
$$462$$ 0 0
$$463$$ 0.876894 0.0407527 0.0203764 0.999792i $$-0.493514\pi$$
0.0203764 + 0.999792i $$0.493514\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −8.68466 −0.401878 −0.200939 0.979604i $$-0.564399\pi$$
−0.200939 + 0.979604i $$0.564399\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 20.1922 0.930409
$$472$$ 0 0
$$473$$ −5.12311 −0.235561
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −2.49242 −0.114120
$$478$$ 0 0
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ −30.6307 −1.39664
$$482$$ 0 0
$$483$$ 39.6155 1.80257
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 34.2462 1.55184 0.775922 0.630829i $$-0.217285\pi$$
0.775922 + 0.630829i $$0.217285\pi$$
$$488$$ 0 0
$$489$$ −0.300187 −0.0135749
$$490$$ 0 0
$$491$$ −15.8078 −0.713394 −0.356697 0.934220i $$-0.616097\pi$$
−0.356697 + 0.934220i $$0.616097\pi$$
$$492$$ 0 0
$$493$$ 2.43845 0.109822
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.68466 0.389560
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −2.05398 −0.0917648
$$502$$ 0 0
$$503$$ −29.1231 −1.29854 −0.649268 0.760560i $$-0.724924\pi$$
−0.649268 + 0.760560i $$0.724924\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −5.06913 −0.225128
$$508$$ 0 0
$$509$$ −35.3693 −1.56772 −0.783859 0.620939i $$-0.786751\pi$$
−0.783859 + 0.620939i $$0.786751\pi$$
$$510$$ 0 0
$$511$$ 17.3693 0.768373
$$512$$ 0 0
$$513$$ −13.5616 −0.598757
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 7.12311 0.313274
$$518$$ 0 0
$$519$$ 33.3693 1.46475
$$520$$ 0 0
$$521$$ 9.50758 0.416535 0.208267 0.978072i $$-0.433218\pi$$
0.208267 + 0.978072i $$0.433218\pi$$
$$522$$ 0 0
$$523$$ −23.3693 −1.02187 −0.510934 0.859620i $$-0.670701\pi$$
−0.510934 + 0.859620i $$0.670701\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −48.3002 −2.10399
$$528$$ 0 0
$$529$$ 27.7386 1.20603
$$530$$ 0 0
$$531$$ 7.50758 0.325801
$$532$$ 0 0
$$533$$ −31.2311 −1.35277
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 13.2614 0.572270
$$538$$ 0 0
$$539$$ 5.68466 0.244856
$$540$$ 0 0
$$541$$ 11.5616 0.497070 0.248535 0.968623i $$-0.420051\pi$$
0.248535 + 0.968623i $$0.420051\pi$$
$$542$$ 0 0
$$543$$ 22.6307 0.971176
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 13.1231 0.561103 0.280552 0.959839i $$-0.409483\pi$$
0.280552 + 0.959839i $$0.409483\pi$$
$$548$$ 0 0
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −1.06913 −0.0455465
$$552$$ 0 0
$$553$$ −3.12311 −0.132808
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −7.12311 −0.301816 −0.150908 0.988548i $$-0.548220\pi$$
−0.150908 + 0.988548i $$0.548220\pi$$
$$558$$ 0 0
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ −8.68466 −0.366667
$$562$$ 0 0
$$563$$ 8.24621 0.347536 0.173768 0.984787i $$-0.444406\pi$$
0.173768 + 0.984787i $$0.444406\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 24.9309 1.04700
$$568$$ 0 0
$$569$$ −2.87689 −0.120606 −0.0603028 0.998180i $$-0.519207\pi$$
−0.0603028 + 0.998180i $$0.519207\pi$$
$$570$$ 0 0
$$571$$ 31.8078 1.33111 0.665557 0.746347i $$-0.268194\pi$$
0.665557 + 0.746347i $$0.268194\pi$$
$$572$$ 0 0
$$573$$ 2.73863 0.114408
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 30.4924 1.26942 0.634708 0.772752i $$-0.281120\pi$$
0.634708 + 0.772752i $$0.281120\pi$$
$$578$$ 0 0
$$579$$ −30.9309 −1.28544
$$580$$ 0 0
$$581$$ 35.6155 1.47758
$$582$$ 0 0
$$583$$ 4.43845 0.183822
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −21.5616 −0.889941 −0.444970 0.895545i $$-0.646786\pi$$
−0.444970 + 0.895545i $$0.646786\pi$$
$$588$$ 0 0
$$589$$ 21.1771 0.872586
$$590$$ 0 0
$$591$$ −28.4924 −1.17202
$$592$$ 0 0
$$593$$ 7.61553 0.312732 0.156366 0.987699i $$-0.450022\pi$$
0.156366 + 0.987699i $$0.450022\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −10.8229 −0.442953
$$598$$ 0 0
$$599$$ −15.3153 −0.625768 −0.312884 0.949791i $$-0.601295\pi$$
−0.312884 + 0.949791i $$0.601295\pi$$
$$600$$ 0 0
$$601$$ 37.2311 1.51869 0.759343 0.650690i $$-0.225520\pi$$
0.759343 + 0.650690i $$0.225520\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −34.6847 −1.40781 −0.703903 0.710296i $$-0.748561\pi$$
−0.703903 + 0.710296i $$0.748561\pi$$
$$608$$ 0 0
$$609$$ 2.43845 0.0988109
$$610$$ 0 0
$$611$$ 22.2462 0.899985
$$612$$ 0 0
$$613$$ −27.6155 −1.11538 −0.557690 0.830049i $$-0.688312\pi$$
−0.557690 + 0.830049i $$0.688312\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.87689 −0.115819 −0.0579097 0.998322i $$-0.518444\pi$$
−0.0579097 + 0.998322i $$0.518444\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 39.6155 1.58972
$$622$$ 0 0
$$623$$ −34.9309 −1.39948
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 3.80776 0.152067
$$628$$ 0 0
$$629$$ 54.5464 2.17491
$$630$$ 0 0
$$631$$ −2.43845 −0.0970730 −0.0485365 0.998821i $$-0.515456\pi$$
−0.0485365 + 0.998821i $$0.515456\pi$$
$$632$$ 0 0
$$633$$ 1.06913 0.0424941
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 17.7538 0.703431
$$638$$ 0 0
$$639$$ 1.36932 0.0541693
$$640$$ 0 0
$$641$$ 3.94602 0.155859 0.0779293 0.996959i $$-0.475169\pi$$
0.0779293 + 0.996959i $$0.475169\pi$$
$$642$$ 0 0
$$643$$ 19.3153 0.761723 0.380861 0.924632i $$-0.375628\pi$$
0.380861 + 0.924632i $$0.375628\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.36932 0.368346 0.184173 0.982894i $$-0.441039\pi$$
0.184173 + 0.982894i $$0.441039\pi$$
$$648$$ 0 0
$$649$$ −13.3693 −0.524792
$$650$$ 0 0
$$651$$ −48.3002 −1.89303
$$652$$ 0 0
$$653$$ −10.1922 −0.398853 −0.199427 0.979913i $$-0.563908\pi$$
−0.199427 + 0.979913i $$0.563908\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.73863 0.106844
$$658$$ 0 0
$$659$$ 5.06913 0.197465 0.0987326 0.995114i $$-0.468521\pi$$
0.0987326 + 0.995114i $$0.468521\pi$$
$$660$$ 0 0
$$661$$ −14.4924 −0.563690 −0.281845 0.959460i $$-0.590946\pi$$
−0.281845 + 0.959460i $$0.590946\pi$$
$$662$$ 0 0
$$663$$ −27.1231 −1.05337
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.12311 0.120927
$$668$$ 0 0
$$669$$ 36.1080 1.39601
$$670$$ 0 0
$$671$$ −3.56155 −0.137492
$$672$$ 0 0
$$673$$ 21.1771 0.816316 0.408158 0.912911i $$-0.366171\pi$$
0.408158 + 0.912911i $$0.366171\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.24621 0.240061 0.120031 0.992770i $$-0.461701\pi$$
0.120031 + 0.992770i $$0.461701\pi$$
$$678$$ 0 0
$$679$$ 60.9848 2.34038
$$680$$ 0 0
$$681$$ 28.1080 1.07710
$$682$$ 0 0
$$683$$ −3.80776 −0.145700 −0.0728500 0.997343i $$-0.523209\pi$$
−0.0728500 + 0.997343i $$0.523209\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −10.1383 −0.386799
$$688$$ 0 0
$$689$$ 13.8617 0.528090
$$690$$ 0 0
$$691$$ −35.6155 −1.35488 −0.677439 0.735579i $$-0.736910\pi$$
−0.677439 + 0.735579i $$0.736910\pi$$
$$692$$ 0 0
$$693$$ 2.00000 0.0759737
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 55.6155 2.10659
$$698$$ 0 0
$$699$$ −17.6695 −0.668322
$$700$$ 0 0
$$701$$ −36.5464 −1.38034 −0.690169 0.723648i $$-0.742464\pi$$
−0.690169 + 0.723648i $$0.742464\pi$$
$$702$$ 0 0
$$703$$ −23.9157 −0.901998
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 57.8617 2.17611
$$708$$ 0 0
$$709$$ 48.2462 1.81192 0.905962 0.423358i $$-0.139149\pi$$
0.905962 + 0.423358i $$0.139149\pi$$
$$710$$ 0 0
$$711$$ −0.492423 −0.0184673
$$712$$ 0 0
$$713$$ −61.8617 −2.31674
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 1.36932 0.0511381
$$718$$ 0 0
$$719$$ −42.0540 −1.56835 −0.784174 0.620541i $$-0.786913\pi$$
−0.784174 + 0.620541i $$0.786913\pi$$
$$720$$ 0 0
$$721$$ −50.7386 −1.88961
$$722$$ 0 0
$$723$$ −44.8769 −1.66899
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −23.6155 −0.875851 −0.437926 0.899011i $$-0.644287\pi$$
−0.437926 + 0.899011i $$0.644287\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ 28.4924 1.05383
$$732$$ 0 0
$$733$$ 4.00000 0.147743 0.0738717 0.997268i $$-0.476464\pi$$
0.0738717 + 0.997268i $$0.476464\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −46.7386 −1.71931 −0.859654 0.510876i $$-0.829321\pi$$
−0.859654 + 0.510876i $$0.829321\pi$$
$$740$$ 0 0
$$741$$ 11.8920 0.436865
$$742$$ 0 0
$$743$$ 28.4384 1.04331 0.521653 0.853158i $$-0.325316\pi$$
0.521653 + 0.853158i $$0.325316\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 5.61553 0.205461
$$748$$ 0 0
$$749$$ 18.2462 0.666702
$$750$$ 0 0
$$751$$ 18.4384 0.672828 0.336414 0.941714i $$-0.390786\pi$$
0.336414 + 0.941714i $$0.390786\pi$$
$$752$$ 0 0
$$753$$ 36.1080 1.31585
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 26.4924 0.962883 0.481442 0.876478i $$-0.340113\pi$$
0.481442 + 0.876478i $$0.340113\pi$$
$$758$$ 0 0
$$759$$ −11.1231 −0.403743
$$760$$ 0 0
$$761$$ 2.00000 0.0724999 0.0362500 0.999343i $$-0.488459\pi$$
0.0362500 + 0.999343i $$0.488459\pi$$
$$762$$ 0 0
$$763$$ 7.12311 0.257874
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −41.7538 −1.50764
$$768$$ 0 0
$$769$$ 30.9848 1.11734 0.558671 0.829389i $$-0.311311\pi$$
0.558671 + 0.829389i $$0.311311\pi$$
$$770$$ 0 0
$$771$$ 12.1080 0.436057
$$772$$ 0 0
$$773$$ −0.822919 −0.0295983 −0.0147992 0.999890i $$-0.504711\pi$$
−0.0147992 + 0.999890i $$0.504711\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 54.5464 1.95684
$$778$$ 0 0
$$779$$ −24.3845 −0.873664
$$780$$ 0 0
$$781$$ −2.43845 −0.0872545
$$782$$ 0 0
$$783$$ 2.43845 0.0871430
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 26.8769 0.958058 0.479029 0.877799i $$-0.340989\pi$$
0.479029 + 0.877799i $$0.340989\pi$$
$$788$$ 0 0
$$789$$ −38.1619 −1.35860
$$790$$ 0 0
$$791$$ 4.00000 0.142224
$$792$$ 0 0
$$793$$ −11.1231 −0.394993
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −12.7386 −0.451226 −0.225613 0.974217i $$-0.572438\pi$$
−0.225613 + 0.974217i $$0.572438\pi$$
$$798$$ 0 0
$$799$$ −39.6155 −1.40150
$$800$$ 0 0
$$801$$ −5.50758 −0.194601
$$802$$ 0 0
$$803$$ −4.87689 −0.172102
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −35.7235 −1.25753
$$808$$ 0 0
$$809$$ −52.7386 −1.85419 −0.927096 0.374824i $$-0.877703\pi$$
−0.927096 + 0.374824i $$0.877703\pi$$
$$810$$ 0 0
$$811$$ −46.4384 −1.63067 −0.815337 0.578986i $$-0.803448\pi$$
−0.815337 + 0.578986i $$0.803448\pi$$
$$812$$ 0 0
$$813$$ 18.7386 0.657193
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −12.4924 −0.437055
$$818$$ 0 0
$$819$$ 6.24621 0.218260
$$820$$ 0 0
$$821$$ 19.7538 0.689412 0.344706 0.938711i $$-0.387979\pi$$
0.344706 + 0.938711i $$0.387979\pi$$
$$822$$ 0 0
$$823$$ −18.7386 −0.653188 −0.326594 0.945165i $$-0.605901\pi$$
−0.326594 + 0.945165i $$0.605901\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −10.3845 −0.361103 −0.180552 0.983565i $$-0.557788\pi$$
−0.180552 + 0.983565i $$0.557788\pi$$
$$828$$ 0 0
$$829$$ 47.8617 1.66231 0.831153 0.556043i $$-0.187681\pi$$
0.831153 + 0.556043i $$0.187681\pi$$
$$830$$ 0 0
$$831$$ −29.2614 −1.01507
$$832$$ 0 0
$$833$$ −31.6155 −1.09541
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −48.3002 −1.66950
$$838$$ 0 0
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −28.8078 −0.993371
$$842$$ 0 0
$$843$$ 28.8769 0.994573
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −3.56155 −0.122376
$$848$$ 0 0
$$849$$ 28.8769 0.991052
$$850$$ 0 0
$$851$$ 69.8617 2.39483
$$852$$ 0 0
$$853$$ 40.1080 1.37327 0.686635 0.727002i $$-0.259087\pi$$
0.686635 + 0.727002i $$0.259087\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −50.0540 −1.70981 −0.854906 0.518784i $$-0.826385\pi$$
−0.854906 + 0.518784i $$0.826385\pi$$
$$858$$ 0 0
$$859$$ 21.3693 0.729112 0.364556 0.931182i $$-0.381221\pi$$
0.364556 + 0.931182i $$0.381221\pi$$
$$860$$ 0 0
$$861$$ 55.6155 1.89537
$$862$$ 0 0
$$863$$ −9.36932 −0.318935 −0.159468 0.987203i $$-0.550978\pi$$
−0.159468 + 0.987203i $$0.550978\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 21.7538 0.738797
$$868$$ 0 0
$$869$$ 0.876894 0.0297466
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 9.61553 0.325436
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −39.1231 −1.32109 −0.660547 0.750785i $$-0.729675\pi$$
−0.660547 + 0.750785i $$0.729675\pi$$
$$878$$ 0 0
$$879$$ 17.3693 0.585853
$$880$$ 0 0
$$881$$ −34.4924 −1.16208 −0.581040 0.813875i $$-0.697354\pi$$
−0.581040 + 0.813875i $$0.697354\pi$$
$$882$$ 0 0
$$883$$ 17.9460 0.603932 0.301966 0.953319i $$-0.402357\pi$$
0.301966 + 0.953319i $$0.402357\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 55.3693 1.85912 0.929560 0.368671i $$-0.120187\pi$$
0.929560 + 0.368671i $$0.120187\pi$$
$$888$$ 0 0
$$889$$ −26.2462 −0.880270
$$890$$ 0 0
$$891$$ −7.00000 −0.234509
$$892$$ 0 0
$$893$$ 17.3693 0.581242
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −34.7386 −1.15989
$$898$$ 0 0
$$899$$ −3.80776 −0.126996
$$900$$ 0 0
$$901$$ −24.6847 −0.822365
$$902$$ 0 0
$$903$$ 28.4924 0.948168
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −45.1771 −1.50008 −0.750040 0.661392i $$-0.769966\pi$$
−0.750040 + 0.661392i $$0.769966\pi$$
$$908$$ 0 0
$$909$$ 9.12311 0.302594
$$910$$ 0 0
$$911$$ −41.6695 −1.38057 −0.690286 0.723537i $$-0.742515\pi$$
−0.690286 + 0.723537i $$0.742515\pi$$
$$912$$ 0 0
$$913$$ −10.0000 −0.330952
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −2.43845 −0.0805246
$$918$$ 0 0
$$919$$ −9.75379 −0.321748 −0.160874 0.986975i $$-0.551431\pi$$
−0.160874 + 0.986975i $$0.551431\pi$$
$$920$$ 0 0
$$921$$ −32.3845 −1.06710
$$922$$ 0 0
$$923$$ −7.61553 −0.250668
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ 0.930870 0.0305408 0.0152704 0.999883i $$-0.495139\pi$$
0.0152704 + 0.999883i $$0.495139\pi$$
$$930$$ 0 0
$$931$$ 13.8617 0.454300
$$932$$ 0 0
$$933$$ 33.0691 1.08263
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 29.3693 0.959454 0.479727 0.877418i $$-0.340736\pi$$
0.479727 + 0.877418i $$0.340736\pi$$
$$938$$ 0 0
$$939$$ −25.3693 −0.827896
$$940$$ 0 0
$$941$$ −28.4384 −0.927067 −0.463533 0.886079i $$-0.653419\pi$$
−0.463533 + 0.886079i $$0.653419\pi$$
$$942$$ 0 0
$$943$$ 71.2311 2.31960
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −50.0540 −1.62654 −0.813268 0.581890i $$-0.802314\pi$$
−0.813268 + 0.581890i $$0.802314\pi$$
$$948$$ 0 0
$$949$$ −15.2311 −0.494421
$$950$$ 0 0
$$951$$ 5.56155 0.180346
$$952$$ 0 0
$$953$$ −37.6695 −1.22023 −0.610117 0.792311i $$-0.708878\pi$$
−0.610117 + 0.792311i $$0.708878\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −0.684658 −0.0221319
$$958$$ 0 0
$$959$$ −68.9848 −2.22764
$$960$$ 0 0
$$961$$ 44.4233 1.43301
$$962$$ 0 0
$$963$$ 2.87689 0.0927066
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.05398 0.258998 0.129499 0.991580i $$-0.458663\pi$$
0.129499 + 0.991580i $$0.458663\pi$$
$$968$$ 0 0
$$969$$ −21.1771 −0.680306
$$970$$ 0 0
$$971$$ 2.24621 0.0720843 0.0360422 0.999350i $$-0.488525\pi$$
0.0360422 + 0.999350i $$0.488525\pi$$
$$972$$ 0 0
$$973$$ −58.7386 −1.88307
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 51.3693 1.64345 0.821725 0.569884i $$-0.193012\pi$$
0.821725 + 0.569884i $$0.193012\pi$$
$$978$$ 0 0
$$979$$ 9.80776 0.313457
$$980$$ 0 0
$$981$$ 1.12311 0.0358580
$$982$$ 0 0
$$983$$ −20.8769 −0.665870 −0.332935 0.942950i $$-0.608039\pi$$
−0.332935 + 0.942950i $$0.608039\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −39.6155 −1.26098
$$988$$ 0 0
$$989$$ 36.4924 1.16039
$$990$$ 0 0
$$991$$ −52.4924 −1.66748 −0.833738 0.552160i $$-0.813804\pi$$
−0.833738 + 0.552160i $$0.813804\pi$$
$$992$$ 0 0
$$993$$ −3.50758 −0.111310
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −12.9848 −0.411234 −0.205617 0.978633i $$-0.565920\pi$$
−0.205617 + 0.978633i $$0.565920\pi$$
$$998$$ 0 0
$$999$$ 54.5464 1.72577
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 2200.2.a.m.1.2 2
4.3 odd 2 4400.2.a.br.1.1 2
5.2 odd 4 2200.2.b.h.1849.2 4
5.3 odd 4 2200.2.b.h.1849.3 4
5.4 even 2 440.2.a.f.1.1 2
15.14 odd 2 3960.2.a.be.1.2 2
20.3 even 4 4400.2.b.u.4049.2 4
20.7 even 4 4400.2.b.u.4049.3 4
20.19 odd 2 880.2.a.l.1.2 2
40.19 odd 2 3520.2.a.bs.1.1 2
40.29 even 2 3520.2.a.bl.1.2 2
55.54 odd 2 4840.2.a.n.1.1 2
60.59 even 2 7920.2.a.ca.1.1 2
220.219 even 2 9680.2.a.bl.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.f.1.1 2 5.4 even 2
880.2.a.l.1.2 2 20.19 odd 2
2200.2.a.m.1.2 2 1.1 even 1 trivial
2200.2.b.h.1849.2 4 5.2 odd 4
2200.2.b.h.1849.3 4 5.3 odd 4
3520.2.a.bl.1.2 2 40.29 even 2
3520.2.a.bs.1.1 2 40.19 odd 2
3960.2.a.be.1.2 2 15.14 odd 2
4400.2.a.br.1.1 2 4.3 odd 2
4400.2.b.u.4049.2 4 20.3 even 4
4400.2.b.u.4049.3 4 20.7 even 4
4840.2.a.n.1.1 2 55.54 odd 2
7920.2.a.ca.1.1 2 60.59 even 2
9680.2.a.bl.1.2 2 220.219 even 2