# Properties

 Label 7600.2.a.bf.1.1 Level $7600$ Weight $2$ Character 7600.1 Self dual yes Analytic conductor $60.686$ Analytic rank $0$ 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] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.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: $$60.6863055362$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 380) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 7600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.732051 q^{3} +2.00000 q^{7} -2.46410 q^{9} +O(q^{10})$$ $$q-0.732051 q^{3} +2.00000 q^{7} -2.46410 q^{9} -3.46410 q^{11} -0.732051 q^{13} -3.46410 q^{17} -1.00000 q^{19} -1.46410 q^{21} +3.46410 q^{23} +4.00000 q^{27} -3.46410 q^{29} -5.46410 q^{31} +2.53590 q^{33} -3.26795 q^{37} +0.535898 q^{39} -6.00000 q^{41} +8.92820 q^{43} -0.928203 q^{47} -3.00000 q^{49} +2.53590 q^{51} +7.26795 q^{53} +0.732051 q^{57} +6.92820 q^{59} -8.39230 q^{61} -4.92820 q^{63} +3.26795 q^{67} -2.53590 q^{69} +9.46410 q^{71} +7.46410 q^{73} -6.92820 q^{77} +10.9282 q^{79} +4.46410 q^{81} -3.46410 q^{83} +2.53590 q^{87} -8.53590 q^{89} -1.46410 q^{91} +4.00000 q^{93} -14.5885 q^{97} +8.53590 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 2 q^{13} - 2 q^{19} + 4 q^{21} + 8 q^{27} - 4 q^{31} + 12 q^{33} - 10 q^{37} + 8 q^{39} - 12 q^{41} + 4 q^{43} + 12 q^{47} - 6 q^{49} + 12 q^{51} + 18 q^{53} - 2 q^{57} + 4 q^{61} + 4 q^{63} + 10 q^{67} - 12 q^{69} + 12 q^{71} + 8 q^{73} + 8 q^{79} + 2 q^{81} + 12 q^{87} - 24 q^{89} + 4 q^{91} + 8 q^{93} + 2 q^{97} + 24 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 + 2 * q^13 - 2 * q^19 + 4 * q^21 + 8 * q^27 - 4 * q^31 + 12 * q^33 - 10 * q^37 + 8 * q^39 - 12 * q^41 + 4 * q^43 + 12 * q^47 - 6 * q^49 + 12 * q^51 + 18 * q^53 - 2 * q^57 + 4 * q^61 + 4 * q^63 + 10 * q^67 - 12 * q^69 + 12 * q^71 + 8 * q^73 + 8 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^89 + 4 * q^91 + 8 * q^93 + 2 * q^97 + 24 * 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
$$3$$ −0.732051 −0.422650 −0.211325 0.977416i $$-0.567778\pi$$
−0.211325 + 0.977416i $$0.567778\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ −2.46410 −0.821367
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ −0.732051 −0.203034 −0.101517 0.994834i $$-0.532370\pi$$
−0.101517 + 0.994834i $$0.532370\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.46410 −0.840168 −0.420084 0.907485i $$-0.637999\pi$$
−0.420084 + 0.907485i $$0.637999\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.46410 −0.319493
$$22$$ 0 0
$$23$$ 3.46410 0.722315 0.361158 0.932505i $$-0.382382\pi$$
0.361158 + 0.932505i $$0.382382\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ −3.46410 −0.643268 −0.321634 0.946864i $$-0.604232\pi$$
−0.321634 + 0.946864i $$0.604232\pi$$
$$30$$ 0 0
$$31$$ −5.46410 −0.981382 −0.490691 0.871334i $$-0.663256\pi$$
−0.490691 + 0.871334i $$0.663256\pi$$
$$32$$ 0 0
$$33$$ 2.53590 0.441443
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −3.26795 −0.537248 −0.268624 0.963245i $$-0.586569\pi$$
−0.268624 + 0.963245i $$0.586569\pi$$
$$38$$ 0 0
$$39$$ 0.535898 0.0858124
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 8.92820 1.36154 0.680769 0.732498i $$-0.261646\pi$$
0.680769 + 0.732498i $$0.261646\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −0.928203 −0.135392 −0.0676962 0.997706i $$-0.521565\pi$$
−0.0676962 + 0.997706i $$0.521565\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 2.53590 0.355097
$$52$$ 0 0
$$53$$ 7.26795 0.998330 0.499165 0.866507i $$-0.333640\pi$$
0.499165 + 0.866507i $$0.333640\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.732051 0.0969625
$$58$$ 0 0
$$59$$ 6.92820 0.901975 0.450988 0.892530i $$-0.351072\pi$$
0.450988 + 0.892530i $$0.351072\pi$$
$$60$$ 0 0
$$61$$ −8.39230 −1.07452 −0.537262 0.843415i $$-0.680541\pi$$
−0.537262 + 0.843415i $$0.680541\pi$$
$$62$$ 0 0
$$63$$ −4.92820 −0.620895
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.26795 0.399244 0.199622 0.979873i $$-0.436029\pi$$
0.199622 + 0.979873i $$0.436029\pi$$
$$68$$ 0 0
$$69$$ −2.53590 −0.305286
$$70$$ 0 0
$$71$$ 9.46410 1.12318 0.561591 0.827415i $$-0.310189\pi$$
0.561591 + 0.827415i $$0.310189\pi$$
$$72$$ 0 0
$$73$$ 7.46410 0.873607 0.436804 0.899557i $$-0.356111\pi$$
0.436804 + 0.899557i $$0.356111\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −6.92820 −0.789542
$$78$$ 0 0
$$79$$ 10.9282 1.22952 0.614759 0.788715i $$-0.289253\pi$$
0.614759 + 0.788715i $$0.289253\pi$$
$$80$$ 0 0
$$81$$ 4.46410 0.496011
$$82$$ 0 0
$$83$$ −3.46410 −0.380235 −0.190117 0.981761i $$-0.560887\pi$$
−0.190117 + 0.981761i $$0.560887\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.53590 0.271877
$$88$$ 0 0
$$89$$ −8.53590 −0.904803 −0.452402 0.891814i $$-0.649433\pi$$
−0.452402 + 0.891814i $$0.649433\pi$$
$$90$$ 0 0
$$91$$ −1.46410 −0.153480
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −14.5885 −1.48123 −0.740617 0.671928i $$-0.765467\pi$$
−0.740617 + 0.671928i $$0.765467\pi$$
$$98$$ 0 0
$$99$$ 8.53590 0.857890
$$100$$ 0 0
$$101$$ 4.39230 0.437051 0.218525 0.975831i $$-0.429875\pi$$
0.218525 + 0.975831i $$0.429875\pi$$
$$102$$ 0 0
$$103$$ −3.66025 −0.360656 −0.180328 0.983607i $$-0.557716\pi$$
−0.180328 + 0.983607i $$0.557716\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.6603 1.12724 0.563620 0.826034i $$-0.309408\pi$$
0.563620 + 0.826034i $$0.309408\pi$$
$$108$$ 0 0
$$109$$ 6.39230 0.612272 0.306136 0.951988i $$-0.400964\pi$$
0.306136 + 0.951988i $$0.400964\pi$$
$$110$$ 0 0
$$111$$ 2.39230 0.227068
$$112$$ 0 0
$$113$$ 18.5885 1.74865 0.874327 0.485336i $$-0.161303\pi$$
0.874327 + 0.485336i $$0.161303\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.80385 0.166766
$$118$$ 0 0
$$119$$ −6.92820 −0.635107
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 4.39230 0.396041
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −10.5885 −0.939574 −0.469787 0.882780i $$-0.655669\pi$$
−0.469787 + 0.882780i $$0.655669\pi$$
$$128$$ 0 0
$$129$$ −6.53590 −0.575454
$$130$$ 0 0
$$131$$ 5.07180 0.443125 0.221562 0.975146i $$-0.428884\pi$$
0.221562 + 0.975146i $$0.428884\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 17.3205 1.47979 0.739895 0.672722i $$-0.234875\pi$$
0.739895 + 0.672722i $$0.234875\pi$$
$$138$$ 0 0
$$139$$ −4.53590 −0.384730 −0.192365 0.981323i $$-0.561616\pi$$
−0.192365 + 0.981323i $$0.561616\pi$$
$$140$$ 0 0
$$141$$ 0.679492 0.0572235
$$142$$ 0 0
$$143$$ 2.53590 0.212062
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.19615 0.181136
$$148$$ 0 0
$$149$$ 16.3923 1.34291 0.671455 0.741045i $$-0.265670\pi$$
0.671455 + 0.741045i $$0.265670\pi$$
$$150$$ 0 0
$$151$$ −12.3923 −1.00847 −0.504236 0.863566i $$-0.668226\pi$$
−0.504236 + 0.863566i $$0.668226\pi$$
$$152$$ 0 0
$$153$$ 8.53590 0.690086
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −16.5359 −1.31971 −0.659854 0.751394i $$-0.729382\pi$$
−0.659854 + 0.751394i $$0.729382\pi$$
$$158$$ 0 0
$$159$$ −5.32051 −0.421944
$$160$$ 0 0
$$161$$ 6.92820 0.546019
$$162$$ 0 0
$$163$$ 20.9282 1.63922 0.819612 0.572919i $$-0.194189\pi$$
0.819612 + 0.572919i $$0.194189\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −7.26795 −0.562411 −0.281205 0.959648i $$-0.590734\pi$$
−0.281205 + 0.959648i $$0.590734\pi$$
$$168$$ 0 0
$$169$$ −12.4641 −0.958777
$$170$$ 0 0
$$171$$ 2.46410 0.188435
$$172$$ 0 0
$$173$$ −14.1962 −1.07931 −0.539657 0.841885i $$-0.681446\pi$$
−0.539657 + 0.841885i $$0.681446\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −5.07180 −0.381220
$$178$$ 0 0
$$179$$ 20.7846 1.55351 0.776757 0.629800i $$-0.216863\pi$$
0.776757 + 0.629800i $$0.216863\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 6.14359 0.454148
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ 0 0
$$189$$ 8.00000 0.581914
$$190$$ 0 0
$$191$$ 6.92820 0.501307 0.250654 0.968077i $$-0.419354\pi$$
0.250654 + 0.968077i $$0.419354\pi$$
$$192$$ 0 0
$$193$$ 18.1962 1.30979 0.654894 0.755721i $$-0.272713\pi$$
0.654894 + 0.755721i $$0.272713\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.9282 0.921096 0.460548 0.887635i $$-0.347653\pi$$
0.460548 + 0.887635i $$0.347653\pi$$
$$198$$ 0 0
$$199$$ −13.0718 −0.926635 −0.463318 0.886192i $$-0.653341\pi$$
−0.463318 + 0.886192i $$0.653341\pi$$
$$200$$ 0 0
$$201$$ −2.39230 −0.168740
$$202$$ 0 0
$$203$$ −6.92820 −0.486265
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −8.53590 −0.593286
$$208$$ 0 0
$$209$$ 3.46410 0.239617
$$210$$ 0 0
$$211$$ 15.3205 1.05471 0.527354 0.849646i $$-0.323184\pi$$
0.527354 + 0.849646i $$0.323184\pi$$
$$212$$ 0 0
$$213$$ −6.92820 −0.474713
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −10.9282 −0.741855
$$218$$ 0 0
$$219$$ −5.46410 −0.369230
$$220$$ 0 0
$$221$$ 2.53590 0.170583
$$222$$ 0 0
$$223$$ 7.66025 0.512969 0.256484 0.966548i $$-0.417436\pi$$
0.256484 + 0.966548i $$0.417436\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −2.19615 −0.145764 −0.0728819 0.997341i $$-0.523220\pi$$
−0.0728819 + 0.997341i $$0.523220\pi$$
$$228$$ 0 0
$$229$$ 10.5359 0.696232 0.348116 0.937452i $$-0.386822\pi$$
0.348116 + 0.937452i $$0.386822\pi$$
$$230$$ 0 0
$$231$$ 5.07180 0.333700
$$232$$ 0 0
$$233$$ −12.9282 −0.846955 −0.423477 0.905907i $$-0.639191\pi$$
−0.423477 + 0.905907i $$0.639191\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ −13.8564 −0.896296 −0.448148 0.893959i $$-0.647916\pi$$
−0.448148 + 0.893959i $$0.647916\pi$$
$$240$$ 0 0
$$241$$ 15.8564 1.02140 0.510700 0.859759i $$-0.329386\pi$$
0.510700 + 0.859759i $$0.329386\pi$$
$$242$$ 0 0
$$243$$ −15.2679 −0.979439
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.732051 0.0465793
$$248$$ 0 0
$$249$$ 2.53590 0.160706
$$250$$ 0 0
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −9.12436 −0.569162 −0.284581 0.958652i $$-0.591854\pi$$
−0.284581 + 0.958652i $$0.591854\pi$$
$$258$$ 0 0
$$259$$ −6.53590 −0.406121
$$260$$ 0 0
$$261$$ 8.53590 0.528359
$$262$$ 0 0
$$263$$ 15.4641 0.953557 0.476779 0.879023i $$-0.341804\pi$$
0.476779 + 0.879023i $$0.341804\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.24871 0.382415
$$268$$ 0 0
$$269$$ −10.3923 −0.633630 −0.316815 0.948487i $$-0.602613\pi$$
−0.316815 + 0.948487i $$0.602613\pi$$
$$270$$ 0 0
$$271$$ −18.3923 −1.11725 −0.558626 0.829419i $$-0.688671\pi$$
−0.558626 + 0.829419i $$0.688671\pi$$
$$272$$ 0 0
$$273$$ 1.07180 0.0648681
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.39230 0.143740 0.0718698 0.997414i $$-0.477103\pi$$
0.0718698 + 0.997414i $$0.477103\pi$$
$$278$$ 0 0
$$279$$ 13.4641 0.806075
$$280$$ 0 0
$$281$$ −12.9282 −0.771232 −0.385616 0.922659i $$-0.626011\pi$$
−0.385616 + 0.922659i $$0.626011\pi$$
$$282$$ 0 0
$$283$$ −2.39230 −0.142208 −0.0711039 0.997469i $$-0.522652\pi$$
−0.0711039 + 0.997469i $$0.522652\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ 0 0
$$291$$ 10.6795 0.626043
$$292$$ 0 0
$$293$$ −11.6603 −0.681199 −0.340600 0.940208i $$-0.610630\pi$$
−0.340600 + 0.940208i $$0.610630\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −13.8564 −0.804030
$$298$$ 0 0
$$299$$ −2.53590 −0.146655
$$300$$ 0 0
$$301$$ 17.8564 1.02923
$$302$$ 0 0
$$303$$ −3.21539 −0.184719
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −18.1962 −1.03851 −0.519255 0.854620i $$-0.673790\pi$$
−0.519255 + 0.854620i $$0.673790\pi$$
$$308$$ 0 0
$$309$$ 2.67949 0.152431
$$310$$ 0 0
$$311$$ 22.3923 1.26975 0.634876 0.772614i $$-0.281051\pi$$
0.634876 + 0.772614i $$0.281051\pi$$
$$312$$ 0 0
$$313$$ 30.7846 1.74005 0.870025 0.493008i $$-0.164103\pi$$
0.870025 + 0.493008i $$0.164103\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 21.1244 1.18646 0.593231 0.805032i $$-0.297852\pi$$
0.593231 + 0.805032i $$0.297852\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ −8.53590 −0.476427
$$322$$ 0 0
$$323$$ 3.46410 0.192748
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −4.67949 −0.258776
$$328$$ 0 0
$$329$$ −1.85641 −0.102347
$$330$$ 0 0
$$331$$ −26.2487 −1.44276 −0.721380 0.692540i $$-0.756492\pi$$
−0.721380 + 0.692540i $$0.756492\pi$$
$$332$$ 0 0
$$333$$ 8.05256 0.441278
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 20.7321 1.12935 0.564673 0.825314i $$-0.309002\pi$$
0.564673 + 0.825314i $$0.309002\pi$$
$$338$$ 0 0
$$339$$ −13.6077 −0.739069
$$340$$ 0 0
$$341$$ 18.9282 1.02502
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −31.1769 −1.67366 −0.836832 0.547459i $$-0.815595\pi$$
−0.836832 + 0.547459i $$0.815595\pi$$
$$348$$ 0 0
$$349$$ 7.07180 0.378545 0.189272 0.981925i $$-0.439387\pi$$
0.189272 + 0.981925i $$0.439387\pi$$
$$350$$ 0 0
$$351$$ −2.92820 −0.156296
$$352$$ 0 0
$$353$$ −22.3923 −1.19182 −0.595911 0.803050i $$-0.703209\pi$$
−0.595911 + 0.803050i $$0.703209\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 5.07180 0.268428
$$358$$ 0 0
$$359$$ 15.4641 0.816164 0.408082 0.912945i $$-0.366198\pi$$
0.408082 + 0.912945i $$0.366198\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −0.732051 −0.0384227
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −23.8564 −1.24529 −0.622647 0.782503i $$-0.713943\pi$$
−0.622647 + 0.782503i $$0.713943\pi$$
$$368$$ 0 0
$$369$$ 14.7846 0.769656
$$370$$ 0 0
$$371$$ 14.5359 0.754666
$$372$$ 0 0
$$373$$ −14.5885 −0.755362 −0.377681 0.925936i $$-0.623278\pi$$
−0.377681 + 0.925936i $$0.623278\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 2.53590 0.130605
$$378$$ 0 0
$$379$$ −1.07180 −0.0550545 −0.0275273 0.999621i $$-0.508763\pi$$
−0.0275273 + 0.999621i $$0.508763\pi$$
$$380$$ 0 0
$$381$$ 7.75129 0.397111
$$382$$ 0 0
$$383$$ 23.6603 1.20898 0.604491 0.796612i $$-0.293376\pi$$
0.604491 + 0.796612i $$0.293376\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −22.0000 −1.11832
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ −3.71281 −0.187287
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 12.5359 0.629159 0.314579 0.949231i $$-0.398137\pi$$
0.314579 + 0.949231i $$0.398137\pi$$
$$398$$ 0 0
$$399$$ 1.46410 0.0732968
$$400$$ 0 0
$$401$$ −2.78461 −0.139057 −0.0695284 0.997580i $$-0.522149\pi$$
−0.0695284 + 0.997580i $$0.522149\pi$$
$$402$$ 0 0
$$403$$ 4.00000 0.199254
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 11.3205 0.561137
$$408$$ 0 0
$$409$$ 6.39230 0.316079 0.158040 0.987433i $$-0.449483\pi$$
0.158040 + 0.987433i $$0.449483\pi$$
$$410$$ 0 0
$$411$$ −12.6795 −0.625433
$$412$$ 0 0
$$413$$ 13.8564 0.681829
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 3.32051 0.162606
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 0 0
$$423$$ 2.28719 0.111207
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −16.7846 −0.812264
$$428$$ 0 0
$$429$$ −1.85641 −0.0896281
$$430$$ 0 0
$$431$$ 26.5359 1.27819 0.639095 0.769128i $$-0.279309\pi$$
0.639095 + 0.769128i $$0.279309\pi$$
$$432$$ 0 0
$$433$$ 39.6603 1.90595 0.952975 0.303049i $$-0.0980045\pi$$
0.952975 + 0.303049i $$0.0980045\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −3.46410 −0.165710
$$438$$ 0 0
$$439$$ −23.7128 −1.13175 −0.565875 0.824491i $$-0.691462\pi$$
−0.565875 + 0.824491i $$0.691462\pi$$
$$440$$ 0 0
$$441$$ 7.39230 0.352015
$$442$$ 0 0
$$443$$ 34.3923 1.63403 0.817014 0.576618i $$-0.195628\pi$$
0.817014 + 0.576618i $$0.195628\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −12.0000 −0.567581
$$448$$ 0 0
$$449$$ 3.46410 0.163481 0.0817405 0.996654i $$-0.473952\pi$$
0.0817405 + 0.996654i $$0.473952\pi$$
$$450$$ 0 0
$$451$$ 20.7846 0.978709
$$452$$ 0 0
$$453$$ 9.07180 0.426230
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.78461 0.317371 0.158685 0.987329i $$-0.449274\pi$$
0.158685 + 0.987329i $$0.449274\pi$$
$$458$$ 0 0
$$459$$ −13.8564 −0.646762
$$460$$ 0 0
$$461$$ 33.7128 1.57016 0.785081 0.619393i $$-0.212621\pi$$
0.785081 + 0.619393i $$0.212621\pi$$
$$462$$ 0 0
$$463$$ 11.4641 0.532782 0.266391 0.963865i $$-0.414169\pi$$
0.266391 + 0.963865i $$0.414169\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 5.32051 0.246204 0.123102 0.992394i $$-0.460716\pi$$
0.123102 + 0.992394i $$0.460716\pi$$
$$468$$ 0 0
$$469$$ 6.53590 0.301800
$$470$$ 0 0
$$471$$ 12.1051 0.557774
$$472$$ 0 0
$$473$$ −30.9282 −1.42208
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −17.9090 −0.819995
$$478$$ 0 0
$$479$$ 1.60770 0.0734575 0.0367287 0.999325i $$-0.488306\pi$$
0.0367287 + 0.999325i $$0.488306\pi$$
$$480$$ 0 0
$$481$$ 2.39230 0.109080
$$482$$ 0 0
$$483$$ −5.07180 −0.230775
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −1.80385 −0.0817401 −0.0408701 0.999164i $$-0.513013\pi$$
−0.0408701 + 0.999164i $$0.513013\pi$$
$$488$$ 0 0
$$489$$ −15.3205 −0.692817
$$490$$ 0 0
$$491$$ −5.07180 −0.228887 −0.114443 0.993430i $$-0.536508\pi$$
−0.114443 + 0.993430i $$0.536508\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 18.9282 0.849046
$$498$$ 0 0
$$499$$ −16.5359 −0.740248 −0.370124 0.928982i $$-0.620685\pi$$
−0.370124 + 0.928982i $$0.620685\pi$$
$$500$$ 0 0
$$501$$ 5.32051 0.237703
$$502$$ 0 0
$$503$$ 8.53590 0.380597 0.190298 0.981726i $$-0.439054\pi$$
0.190298 + 0.981726i $$0.439054\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.12436 0.405227
$$508$$ 0 0
$$509$$ 29.3205 1.29961 0.649804 0.760102i $$-0.274851\pi$$
0.649804 + 0.760102i $$0.274851\pi$$
$$510$$ 0 0
$$511$$ 14.9282 0.660385
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.21539 0.141413
$$518$$ 0 0
$$519$$ 10.3923 0.456172
$$520$$ 0 0
$$521$$ 26.7846 1.17346 0.586728 0.809784i $$-0.300416\pi$$
0.586728 + 0.809784i $$0.300416\pi$$
$$522$$ 0 0
$$523$$ 35.3731 1.54676 0.773378 0.633945i $$-0.218565\pi$$
0.773378 + 0.633945i $$0.218565\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 18.9282 0.824525
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ −17.0718 −0.740853
$$532$$ 0 0
$$533$$ 4.39230 0.190252
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −15.2154 −0.656593
$$538$$ 0 0
$$539$$ 10.3923 0.447628
$$540$$ 0 0
$$541$$ 33.1769 1.42639 0.713193 0.700967i $$-0.247248\pi$$
0.713193 + 0.700967i $$0.247248\pi$$
$$542$$ 0 0
$$543$$ −10.2487 −0.439814
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 3.26795 0.139727 0.0698637 0.997557i $$-0.477744\pi$$
0.0698637 + 0.997557i $$0.477744\pi$$
$$548$$ 0 0
$$549$$ 20.6795 0.882579
$$550$$ 0 0
$$551$$ 3.46410 0.147576
$$552$$ 0 0
$$553$$ 21.8564 0.929429
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 29.3205 1.24235 0.621175 0.783672i $$-0.286656\pi$$
0.621175 + 0.783672i $$0.286656\pi$$
$$558$$ 0 0
$$559$$ −6.53590 −0.276439
$$560$$ 0 0
$$561$$ −8.78461 −0.370887
$$562$$ 0 0
$$563$$ 33.8038 1.42466 0.712331 0.701844i $$-0.247639\pi$$
0.712331 + 0.701844i $$0.247639\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 8.92820 0.374949
$$568$$ 0 0
$$569$$ 5.32051 0.223047 0.111524 0.993762i $$-0.464427\pi$$
0.111524 + 0.993762i $$0.464427\pi$$
$$570$$ 0 0
$$571$$ 2.39230 0.100115 0.0500574 0.998746i $$-0.484060\pi$$
0.0500574 + 0.998746i $$0.484060\pi$$
$$572$$ 0 0
$$573$$ −5.07180 −0.211877
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −34.7846 −1.44810 −0.724051 0.689746i $$-0.757722\pi$$
−0.724051 + 0.689746i $$0.757722\pi$$
$$578$$ 0 0
$$579$$ −13.3205 −0.553581
$$580$$ 0 0
$$581$$ −6.92820 −0.287430
$$582$$ 0 0
$$583$$ −25.1769 −1.04272
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −19.1769 −0.791516 −0.395758 0.918355i $$-0.629518\pi$$
−0.395758 + 0.918355i $$0.629518\pi$$
$$588$$ 0 0
$$589$$ 5.46410 0.225144
$$590$$ 0 0
$$591$$ −9.46410 −0.389301
$$592$$ 0 0
$$593$$ −4.14359 −0.170157 −0.0850785 0.996374i $$-0.527114\pi$$
−0.0850785 + 0.996374i $$0.527114\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 9.56922 0.391642
$$598$$ 0 0
$$599$$ −20.7846 −0.849236 −0.424618 0.905373i $$-0.639592\pi$$
−0.424618 + 0.905373i $$0.639592\pi$$
$$600$$ 0 0
$$601$$ −44.6410 −1.82095 −0.910473 0.413570i $$-0.864282\pi$$
−0.910473 + 0.413570i $$0.864282\pi$$
$$602$$ 0 0
$$603$$ −8.05256 −0.327926
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 25.9090 1.05161 0.525806 0.850604i $$-0.323764\pi$$
0.525806 + 0.850604i $$0.323764\pi$$
$$608$$ 0 0
$$609$$ 5.07180 0.205520
$$610$$ 0 0
$$611$$ 0.679492 0.0274893
$$612$$ 0 0
$$613$$ 14.3923 0.581300 0.290650 0.956829i $$-0.406129\pi$$
0.290650 + 0.956829i $$0.406129\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −22.3923 −0.901480 −0.450740 0.892655i $$-0.648840\pi$$
−0.450740 + 0.892655i $$0.648840\pi$$
$$618$$ 0 0
$$619$$ −18.3923 −0.739249 −0.369625 0.929181i $$-0.620514\pi$$
−0.369625 + 0.929181i $$0.620514\pi$$
$$620$$ 0 0
$$621$$ 13.8564 0.556038
$$622$$ 0 0
$$623$$ −17.0718 −0.683967
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −2.53590 −0.101274
$$628$$ 0 0
$$629$$ 11.3205 0.451378
$$630$$ 0 0
$$631$$ −4.53590 −0.180571 −0.0902856 0.995916i $$-0.528778\pi$$
−0.0902856 + 0.995916i $$0.528778\pi$$
$$632$$ 0 0
$$633$$ −11.2154 −0.445772
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.19615 0.0870147
$$638$$ 0 0
$$639$$ −23.3205 −0.922545
$$640$$ 0 0
$$641$$ −11.0718 −0.437310 −0.218655 0.975802i $$-0.570167\pi$$
−0.218655 + 0.975802i $$0.570167\pi$$
$$642$$ 0 0
$$643$$ 6.39230 0.252088 0.126044 0.992025i $$-0.459772\pi$$
0.126044 + 0.992025i $$0.459772\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8.53590 0.335581 0.167790 0.985823i $$-0.446337\pi$$
0.167790 + 0.985823i $$0.446337\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 8.00000 0.313545
$$652$$ 0 0
$$653$$ −20.5359 −0.803632 −0.401816 0.915720i $$-0.631621\pi$$
−0.401816 + 0.915720i $$0.631621\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −18.3923 −0.717552
$$658$$ 0 0
$$659$$ 32.7846 1.27711 0.638554 0.769577i $$-0.279533\pi$$
0.638554 + 0.769577i $$0.279533\pi$$
$$660$$ 0 0
$$661$$ 46.7846 1.81971 0.909855 0.414926i $$-0.136193\pi$$
0.909855 + 0.414926i $$0.136193\pi$$
$$662$$ 0 0
$$663$$ −1.85641 −0.0720969
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12.0000 −0.464642
$$668$$ 0 0
$$669$$ −5.60770 −0.216806
$$670$$ 0 0
$$671$$ 29.0718 1.12230
$$672$$ 0 0
$$673$$ 47.2679 1.82205 0.911023 0.412356i $$-0.135294\pi$$
0.911023 + 0.412356i $$0.135294\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 34.9808 1.34442 0.672210 0.740361i $$-0.265345\pi$$
0.672210 + 0.740361i $$0.265345\pi$$
$$678$$ 0 0
$$679$$ −29.1769 −1.11971
$$680$$ 0 0
$$681$$ 1.60770 0.0616070
$$682$$ 0 0
$$683$$ −0.339746 −0.0130000 −0.00650001 0.999979i $$-0.502069\pi$$
−0.00650001 + 0.999979i $$0.502069\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −7.71281 −0.294262
$$688$$ 0 0
$$689$$ −5.32051 −0.202695
$$690$$ 0 0
$$691$$ 11.1769 0.425190 0.212595 0.977140i $$-0.431809\pi$$
0.212595 + 0.977140i $$0.431809\pi$$
$$692$$ 0 0
$$693$$ 17.0718 0.648504
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 20.7846 0.787273
$$698$$ 0 0
$$699$$ 9.46410 0.357965
$$700$$ 0 0
$$701$$ −33.4641 −1.26392 −0.631961 0.775000i $$-0.717750\pi$$
−0.631961 + 0.775000i $$0.717750\pi$$
$$702$$ 0 0
$$703$$ 3.26795 0.123253
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.78461 0.330379
$$708$$ 0 0
$$709$$ 10.7846 0.405025 0.202512 0.979280i $$-0.435089\pi$$
0.202512 + 0.979280i $$0.435089\pi$$
$$710$$ 0 0
$$711$$ −26.9282 −1.00989
$$712$$ 0 0
$$713$$ −18.9282 −0.708867
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 10.1436 0.378819
$$718$$ 0 0
$$719$$ 17.3205 0.645946 0.322973 0.946408i $$-0.395318\pi$$
0.322973 + 0.946408i $$0.395318\pi$$
$$720$$ 0 0
$$721$$ −7.32051 −0.272630
$$722$$ 0 0
$$723$$ −11.6077 −0.431695
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 27.8564 1.03314 0.516568 0.856246i $$-0.327209\pi$$
0.516568 + 0.856246i $$0.327209\pi$$
$$728$$ 0 0
$$729$$ −2.21539 −0.0820515
$$730$$ 0 0
$$731$$ −30.9282 −1.14392
$$732$$ 0 0
$$733$$ 30.7846 1.13706 0.568528 0.822664i $$-0.307513\pi$$
0.568528 + 0.822664i $$0.307513\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −11.3205 −0.416996
$$738$$ 0 0
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ −0.535898 −0.0196867
$$742$$ 0 0
$$743$$ 45.1244 1.65545 0.827726 0.561132i $$-0.189634\pi$$
0.827726 + 0.561132i $$0.189634\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 8.53590 0.312312
$$748$$ 0 0
$$749$$ 23.3205 0.852113
$$750$$ 0 0
$$751$$ 18.5359 0.676385 0.338192 0.941077i $$-0.390185\pi$$
0.338192 + 0.941077i $$0.390185\pi$$
$$752$$ 0 0
$$753$$ −17.5692 −0.640258
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ 8.78461 0.318861
$$760$$ 0 0
$$761$$ 40.3923 1.46422 0.732110 0.681186i $$-0.238536\pi$$
0.732110 + 0.681186i $$0.238536\pi$$
$$762$$ 0 0
$$763$$ 12.7846 0.462834
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −5.07180 −0.183132
$$768$$ 0 0
$$769$$ −37.4641 −1.35099 −0.675495 0.737365i $$-0.736070\pi$$
−0.675495 + 0.737365i $$0.736070\pi$$
$$770$$ 0 0
$$771$$ 6.67949 0.240556
$$772$$ 0 0
$$773$$ 16.0526 0.577370 0.288685 0.957424i $$-0.406782\pi$$
0.288685 + 0.957424i $$0.406782\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 4.78461 0.171647
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ −32.7846 −1.17313
$$782$$ 0 0
$$783$$ −13.8564 −0.495188
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −1.80385 −0.0643002 −0.0321501 0.999483i $$-0.510235\pi$$
−0.0321501 + 0.999483i $$0.510235\pi$$
$$788$$ 0 0
$$789$$ −11.3205 −0.403021
$$790$$ 0 0
$$791$$ 37.1769 1.32186
$$792$$ 0 0
$$793$$ 6.14359 0.218165
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −11.6603 −0.413027 −0.206514 0.978444i $$-0.566212\pi$$
−0.206514 + 0.978444i $$0.566212\pi$$
$$798$$ 0 0
$$799$$ 3.21539 0.113752
$$800$$ 0 0
$$801$$ 21.0333 0.743176
$$802$$ 0 0
$$803$$ −25.8564 −0.912453
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 7.60770 0.267804
$$808$$ 0 0
$$809$$ −45.7128 −1.60718 −0.803588 0.595185i $$-0.797079\pi$$
−0.803588 + 0.595185i $$0.797079\pi$$
$$810$$ 0 0
$$811$$ −36.3923 −1.27791 −0.638953 0.769246i $$-0.720632\pi$$
−0.638953 + 0.769246i $$0.720632\pi$$
$$812$$ 0 0
$$813$$ 13.4641 0.472207
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.92820 −0.312358
$$818$$ 0 0
$$819$$ 3.60770 0.126063
$$820$$ 0 0
$$821$$ −43.8564 −1.53060 −0.765300 0.643674i $$-0.777409\pi$$
−0.765300 + 0.643674i $$0.777409\pi$$
$$822$$ 0 0
$$823$$ 43.5692 1.51873 0.759364 0.650666i $$-0.225510\pi$$
0.759364 + 0.650666i $$0.225510\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −12.3397 −0.429095 −0.214548 0.976714i $$-0.568828\pi$$
−0.214548 + 0.976714i $$0.568828\pi$$
$$828$$ 0 0
$$829$$ −40.2487 −1.39790 −0.698948 0.715173i $$-0.746348\pi$$
−0.698948 + 0.715173i $$0.746348\pi$$
$$830$$ 0 0
$$831$$ −1.75129 −0.0607515
$$832$$ 0 0
$$833$$ 10.3923 0.360072
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −21.8564 −0.755468
$$838$$ 0 0
$$839$$ −5.07180 −0.175098 −0.0875489 0.996160i $$-0.527903\pi$$
−0.0875489 + 0.996160i $$0.527903\pi$$
$$840$$ 0 0
$$841$$ −17.0000 −0.586207
$$842$$ 0 0
$$843$$ 9.46410 0.325961
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 0 0
$$849$$ 1.75129 0.0601041
$$850$$ 0 0
$$851$$ −11.3205 −0.388062
$$852$$ 0 0
$$853$$ −24.6410 −0.843692 −0.421846 0.906667i $$-0.638618\pi$$
−0.421846 + 0.906667i $$0.638618\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −14.1962 −0.484931 −0.242466 0.970160i $$-0.577956\pi$$
−0.242466 + 0.970160i $$0.577956\pi$$
$$858$$ 0 0
$$859$$ 7.71281 0.263158 0.131579 0.991306i $$-0.457995\pi$$
0.131579 + 0.991306i $$0.457995\pi$$
$$860$$ 0 0
$$861$$ 8.78461 0.299379
$$862$$ 0 0
$$863$$ −40.7321 −1.38654 −0.693268 0.720680i $$-0.743830\pi$$
−0.693268 + 0.720680i $$0.743830\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 3.66025 0.124309
$$868$$ 0 0
$$869$$ −37.8564 −1.28419
$$870$$ 0 0
$$871$$ −2.39230 −0.0810602
$$872$$ 0 0
$$873$$ 35.9474 1.21664
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 50.9808 1.72150 0.860749 0.509030i $$-0.169996\pi$$
0.860749 + 0.509030i $$0.169996\pi$$
$$878$$ 0 0
$$879$$ 8.53590 0.287909
$$880$$ 0 0
$$881$$ −35.3205 −1.18998 −0.594989 0.803734i $$-0.702844\pi$$
−0.594989 + 0.803734i $$0.702844\pi$$
$$882$$ 0 0
$$883$$ −22.0000 −0.740359 −0.370179 0.928960i $$-0.620704\pi$$
−0.370179 + 0.928960i $$0.620704\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 54.5885 1.83290 0.916451 0.400148i $$-0.131041\pi$$
0.916451 + 0.400148i $$0.131041\pi$$
$$888$$ 0 0
$$889$$ −21.1769 −0.710251
$$890$$ 0 0
$$891$$ −15.4641 −0.518067
$$892$$ 0 0
$$893$$ 0.928203 0.0310611
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.85641 0.0619836
$$898$$ 0 0
$$899$$ 18.9282 0.631291
$$900$$ 0 0
$$901$$ −25.1769 −0.838765
$$902$$ 0 0
$$903$$ −13.0718 −0.435002
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 38.5885 1.28131 0.640654 0.767829i $$-0.278663\pi$$
0.640654 + 0.767829i $$0.278663\pi$$
$$908$$ 0 0
$$909$$ −10.8231 −0.358979
$$910$$ 0 0
$$911$$ −21.4641 −0.711137 −0.355569 0.934650i $$-0.615713\pi$$
−0.355569 + 0.934650i $$0.615713\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 10.1436 0.334971
$$918$$ 0 0
$$919$$ 52.4974 1.73173 0.865865 0.500278i $$-0.166769\pi$$
0.865865 + 0.500278i $$0.166769\pi$$
$$920$$ 0 0
$$921$$ 13.3205 0.438926
$$922$$ 0 0
$$923$$ −6.92820 −0.228045
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 9.01924 0.296231
$$928$$ 0 0
$$929$$ 35.5692 1.16699 0.583494 0.812117i $$-0.301685\pi$$
0.583494 + 0.812117i $$0.301685\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ 0 0
$$933$$ −16.3923 −0.536660
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 11.1769 0.365134 0.182567 0.983193i $$-0.441559\pi$$
0.182567 + 0.983193i $$0.441559\pi$$
$$938$$ 0 0
$$939$$ −22.5359 −0.735431
$$940$$ 0 0
$$941$$ −16.6410 −0.542482 −0.271241 0.962512i $$-0.587434\pi$$
−0.271241 + 0.962512i $$0.587434\pi$$
$$942$$ 0 0
$$943$$ −20.7846 −0.676840
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 41.3205 1.34274 0.671368 0.741124i $$-0.265707\pi$$
0.671368 + 0.741124i $$0.265707\pi$$
$$948$$ 0 0
$$949$$ −5.46410 −0.177372
$$950$$ 0 0
$$951$$ −15.4641 −0.501458
$$952$$ 0 0
$$953$$ −29.4115 −0.952733 −0.476367 0.879247i $$-0.658047\pi$$
−0.476367 + 0.879247i $$0.658047\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −8.78461 −0.283966
$$958$$ 0 0
$$959$$ 34.6410 1.11862
$$960$$ 0 0
$$961$$ −1.14359 −0.0368901
$$962$$ 0 0
$$963$$ −28.7321 −0.925877
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −10.6795 −0.343429 −0.171715 0.985147i $$-0.554931\pi$$
−0.171715 + 0.985147i $$0.554931\pi$$
$$968$$ 0 0
$$969$$ −2.53590 −0.0814648
$$970$$ 0 0
$$971$$ 21.4641 0.688816 0.344408 0.938820i $$-0.388080\pi$$
0.344408 + 0.938820i $$0.388080\pi$$
$$972$$ 0 0
$$973$$ −9.07180 −0.290828
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −21.8038 −0.697567 −0.348783 0.937203i $$-0.613405\pi$$
−0.348783 + 0.937203i $$0.613405\pi$$
$$978$$ 0 0
$$979$$ 29.5692 0.945036
$$980$$ 0 0
$$981$$ −15.7513 −0.502900
$$982$$ 0 0
$$983$$ 56.4449 1.80031 0.900156 0.435568i $$-0.143452\pi$$
0.900156 + 0.435568i $$0.143452\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 1.35898 0.0432569
$$988$$ 0 0
$$989$$ 30.9282 0.983460
$$990$$ 0 0
$$991$$ −12.3923 −0.393655 −0.196827 0.980438i $$-0.563064\pi$$
−0.196827 + 0.980438i $$0.563064\pi$$
$$992$$ 0 0
$$993$$ 19.2154 0.609782
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −9.60770 −0.304279 −0.152139 0.988359i $$-0.548616\pi$$
−0.152139 + 0.988359i $$0.548616\pi$$
$$998$$ 0 0
$$999$$ −13.0718 −0.413573
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 7600.2.a.bf.1.1 2
4.3 odd 2 1900.2.a.d.1.2 2
5.4 even 2 1520.2.a.l.1.2 2
20.3 even 4 1900.2.c.e.1749.3 4
20.7 even 4 1900.2.c.e.1749.2 4
20.19 odd 2 380.2.a.d.1.1 2
40.19 odd 2 6080.2.a.z.1.2 2
40.29 even 2 6080.2.a.bj.1.1 2
60.59 even 2 3420.2.a.h.1.1 2
380.379 even 2 7220.2.a.h.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.1 2 20.19 odd 2
1520.2.a.l.1.2 2 5.4 even 2
1900.2.a.d.1.2 2 4.3 odd 2
1900.2.c.e.1749.2 4 20.7 even 4
1900.2.c.e.1749.3 4 20.3 even 4
3420.2.a.h.1.1 2 60.59 even 2
6080.2.a.z.1.2 2 40.19 odd 2
6080.2.a.bj.1.1 2 40.29 even 2
7220.2.a.h.1.2 2 380.379 even 2
7600.2.a.bf.1.1 2 1.1 even 1 trivial