# Properties

 Label 1840.2.a.n.1.2 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.79129 q^{3} -1.00000 q^{5} +1.79129 q^{7} +4.79129 q^{9} +O(q^{10})$$ $$q+2.79129 q^{3} -1.00000 q^{5} +1.79129 q^{7} +4.79129 q^{9} +0.791288 q^{11} +5.79129 q^{13} -2.79129 q^{15} +0.791288 q^{17} -5.79129 q^{19} +5.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} +7.58258 q^{29} +3.37386 q^{31} +2.20871 q^{33} -1.79129 q^{35} -4.00000 q^{37} +16.1652 q^{39} -6.79129 q^{41} -11.1652 q^{43} -4.79129 q^{45} +4.41742 q^{47} -3.79129 q^{49} +2.20871 q^{51} +6.00000 q^{53} -0.791288 q^{55} -16.1652 q^{57} +13.5826 q^{59} +10.3739 q^{61} +8.58258 q^{63} -5.79129 q^{65} -11.1652 q^{67} -2.79129 q^{69} -8.37386 q^{71} +12.7477 q^{73} +2.79129 q^{75} +1.41742 q^{77} -8.00000 q^{79} -0.417424 q^{81} +6.00000 q^{83} -0.791288 q^{85} +21.1652 q^{87} +15.1652 q^{89} +10.3739 q^{91} +9.41742 q^{93} +5.79129 q^{95} -7.95644 q^{97} +3.79129 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 2q^{5} - q^{7} + 5q^{9} + O(q^{10})$$ $$2q + q^{3} - 2q^{5} - q^{7} + 5q^{9} - 3q^{11} + 7q^{13} - q^{15} - 3q^{17} - 7q^{19} + 10q^{21} - 2q^{23} + 2q^{25} + 10q^{27} + 6q^{29} - 7q^{31} + 9q^{33} + q^{35} - 8q^{37} + 14q^{39} - 9q^{41} - 4q^{43} - 5q^{45} + 18q^{47} - 3q^{49} + 9q^{51} + 12q^{53} + 3q^{55} - 14q^{57} + 18q^{59} + 7q^{61} + 8q^{63} - 7q^{65} - 4q^{67} - q^{69} - 3q^{71} - 2q^{73} + q^{75} + 12q^{77} - 16q^{79} - 10q^{81} + 12q^{83} + 3q^{85} + 24q^{87} + 12q^{89} + 7q^{91} + 28q^{93} + 7q^{95} + 7q^{97} + 3q^{99} + O(q^{100})$$

## 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
$$3$$ 2.79129 1.61155 0.805775 0.592221i $$-0.201749\pi$$
0.805775 + 0.592221i $$0.201749\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.79129 0.677043 0.338522 0.940959i $$-0.390073\pi$$
0.338522 + 0.940959i $$0.390073\pi$$
$$8$$ 0 0
$$9$$ 4.79129 1.59710
$$10$$ 0 0
$$11$$ 0.791288 0.238582 0.119291 0.992859i $$-0.461938\pi$$
0.119291 + 0.992859i $$0.461938\pi$$
$$12$$ 0 0
$$13$$ 5.79129 1.60621 0.803107 0.595835i $$-0.203179\pi$$
0.803107 + 0.595835i $$0.203179\pi$$
$$14$$ 0 0
$$15$$ −2.79129 −0.720707
$$16$$ 0 0
$$17$$ 0.791288 0.191915 0.0959577 0.995385i $$-0.469409\pi$$
0.0959577 + 0.995385i $$0.469409\pi$$
$$18$$ 0 0
$$19$$ −5.79129 −1.32861 −0.664306 0.747460i $$-0.731273\pi$$
−0.664306 + 0.747460i $$0.731273\pi$$
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ 7.58258 1.40805 0.704024 0.710176i $$-0.251385\pi$$
0.704024 + 0.710176i $$0.251385\pi$$
$$30$$ 0 0
$$31$$ 3.37386 0.605964 0.302982 0.952996i $$-0.402018\pi$$
0.302982 + 0.952996i $$0.402018\pi$$
$$32$$ 0 0
$$33$$ 2.20871 0.384487
$$34$$ 0 0
$$35$$ −1.79129 −0.302783
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ 16.1652 2.58850
$$40$$ 0 0
$$41$$ −6.79129 −1.06062 −0.530310 0.847804i $$-0.677925\pi$$
−0.530310 + 0.847804i $$0.677925\pi$$
$$42$$ 0 0
$$43$$ −11.1652 −1.70267 −0.851335 0.524623i $$-0.824206\pi$$
−0.851335 + 0.524623i $$0.824206\pi$$
$$44$$ 0 0
$$45$$ −4.79129 −0.714243
$$46$$ 0 0
$$47$$ 4.41742 0.644348 0.322174 0.946681i $$-0.395586\pi$$
0.322174 + 0.946681i $$0.395586\pi$$
$$48$$ 0 0
$$49$$ −3.79129 −0.541613
$$50$$ 0 0
$$51$$ 2.20871 0.309282
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −0.791288 −0.106697
$$56$$ 0 0
$$57$$ −16.1652 −2.14113
$$58$$ 0 0
$$59$$ 13.5826 1.76830 0.884150 0.467202i $$-0.154738\pi$$
0.884150 + 0.467202i $$0.154738\pi$$
$$60$$ 0 0
$$61$$ 10.3739 1.32824 0.664119 0.747627i $$-0.268807\pi$$
0.664119 + 0.747627i $$0.268807\pi$$
$$62$$ 0 0
$$63$$ 8.58258 1.08130
$$64$$ 0 0
$$65$$ −5.79129 −0.718321
$$66$$ 0 0
$$67$$ −11.1652 −1.36404 −0.682020 0.731333i $$-0.738898\pi$$
−0.682020 + 0.731333i $$0.738898\pi$$
$$68$$ 0 0
$$69$$ −2.79129 −0.336032
$$70$$ 0 0
$$71$$ −8.37386 −0.993795 −0.496897 0.867809i $$-0.665527\pi$$
−0.496897 + 0.867809i $$0.665527\pi$$
$$72$$ 0 0
$$73$$ 12.7477 1.49201 0.746004 0.665941i $$-0.231970\pi$$
0.746004 + 0.665941i $$0.231970\pi$$
$$74$$ 0 0
$$75$$ 2.79129 0.322310
$$76$$ 0 0
$$77$$ 1.41742 0.161530
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −0.417424 −0.0463805
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ −0.791288 −0.0858272
$$86$$ 0 0
$$87$$ 21.1652 2.26914
$$88$$ 0 0
$$89$$ 15.1652 1.60750 0.803751 0.594965i $$-0.202834\pi$$
0.803751 + 0.594965i $$0.202834\pi$$
$$90$$ 0 0
$$91$$ 10.3739 1.08748
$$92$$ 0 0
$$93$$ 9.41742 0.976541
$$94$$ 0 0
$$95$$ 5.79129 0.594174
$$96$$ 0 0
$$97$$ −7.95644 −0.807854 −0.403927 0.914791i $$-0.632355\pi$$
−0.403927 + 0.914791i $$0.632355\pi$$
$$98$$ 0 0
$$99$$ 3.79129 0.381039
$$100$$ 0 0
$$101$$ 4.41742 0.439550 0.219775 0.975551i $$-0.429468\pi$$
0.219775 + 0.975551i $$0.429468\pi$$
$$102$$ 0 0
$$103$$ 6.37386 0.628035 0.314018 0.949417i $$-0.398325\pi$$
0.314018 + 0.949417i $$0.398325\pi$$
$$104$$ 0 0
$$105$$ −5.00000 −0.487950
$$106$$ 0 0
$$107$$ −4.41742 −0.427049 −0.213524 0.976938i $$-0.568494\pi$$
−0.213524 + 0.976938i $$0.568494\pi$$
$$108$$ 0 0
$$109$$ −3.37386 −0.323158 −0.161579 0.986860i $$-0.551659\pi$$
−0.161579 + 0.986860i $$0.551659\pi$$
$$110$$ 0 0
$$111$$ −11.1652 −1.05975
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 27.7477 2.56528
$$118$$ 0 0
$$119$$ 1.41742 0.129935
$$120$$ 0 0
$$121$$ −10.3739 −0.943079
$$122$$ 0 0
$$123$$ −18.9564 −1.70924
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −12.7477 −1.13118 −0.565589 0.824687i $$-0.691351\pi$$
−0.565589 + 0.824687i $$0.691351\pi$$
$$128$$ 0 0
$$129$$ −31.1652 −2.74394
$$130$$ 0 0
$$131$$ 9.16515 0.800763 0.400381 0.916349i $$-0.368878\pi$$
0.400381 + 0.916349i $$0.368878\pi$$
$$132$$ 0 0
$$133$$ −10.3739 −0.899528
$$134$$ 0 0
$$135$$ −5.00000 −0.430331
$$136$$ 0 0
$$137$$ 3.79129 0.323912 0.161956 0.986798i $$-0.448220\pi$$
0.161956 + 0.986798i $$0.448220\pi$$
$$138$$ 0 0
$$139$$ −12.7477 −1.08125 −0.540624 0.841264i $$-0.681812\pi$$
−0.540624 + 0.841264i $$0.681812\pi$$
$$140$$ 0 0
$$141$$ 12.3303 1.03840
$$142$$ 0 0
$$143$$ 4.58258 0.383214
$$144$$ 0 0
$$145$$ −7.58258 −0.629699
$$146$$ 0 0
$$147$$ −10.5826 −0.872836
$$148$$ 0 0
$$149$$ 8.20871 0.672484 0.336242 0.941776i $$-0.390844\pi$$
0.336242 + 0.941776i $$0.390844\pi$$
$$150$$ 0 0
$$151$$ 10.7913 0.878183 0.439091 0.898442i $$-0.355300\pi$$
0.439091 + 0.898442i $$0.355300\pi$$
$$152$$ 0 0
$$153$$ 3.79129 0.306507
$$154$$ 0 0
$$155$$ −3.37386 −0.270995
$$156$$ 0 0
$$157$$ −14.7477 −1.17700 −0.588498 0.808498i $$-0.700281\pi$$
−0.588498 + 0.808498i $$0.700281\pi$$
$$158$$ 0 0
$$159$$ 16.7477 1.32818
$$160$$ 0 0
$$161$$ −1.79129 −0.141173
$$162$$ 0 0
$$163$$ −8.62614 −0.675651 −0.337826 0.941209i $$-0.609691\pi$$
−0.337826 + 0.941209i $$0.609691\pi$$
$$164$$ 0 0
$$165$$ −2.20871 −0.171948
$$166$$ 0 0
$$167$$ −18.3303 −1.41844 −0.709221 0.704987i $$-0.750953\pi$$
−0.709221 + 0.704987i $$0.750953\pi$$
$$168$$ 0 0
$$169$$ 20.5390 1.57992
$$170$$ 0 0
$$171$$ −27.7477 −2.12192
$$172$$ 0 0
$$173$$ −18.7913 −1.42868 −0.714338 0.699801i $$-0.753272\pi$$
−0.714338 + 0.699801i $$0.753272\pi$$
$$174$$ 0 0
$$175$$ 1.79129 0.135409
$$176$$ 0 0
$$177$$ 37.9129 2.84971
$$178$$ 0 0
$$179$$ −10.7477 −0.803323 −0.401661 0.915788i $$-0.631567\pi$$
−0.401661 + 0.915788i $$0.631567\pi$$
$$180$$ 0 0
$$181$$ −18.5390 −1.37799 −0.688997 0.724764i $$-0.741949\pi$$
−0.688997 + 0.724764i $$0.741949\pi$$
$$182$$ 0 0
$$183$$ 28.9564 2.14052
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ 0.626136 0.0457876
$$188$$ 0 0
$$189$$ 8.95644 0.651485
$$190$$ 0 0
$$191$$ −25.5826 −1.85109 −0.925545 0.378637i $$-0.876393\pi$$
−0.925545 + 0.378637i $$0.876393\pi$$
$$192$$ 0 0
$$193$$ −20.7477 −1.49345 −0.746727 0.665131i $$-0.768376\pi$$
−0.746727 + 0.665131i $$0.768376\pi$$
$$194$$ 0 0
$$195$$ −16.1652 −1.15761
$$196$$ 0 0
$$197$$ −11.5390 −0.822121 −0.411060 0.911608i $$-0.634841\pi$$
−0.411060 + 0.911608i $$0.634841\pi$$
$$198$$ 0 0
$$199$$ 16.3303 1.15762 0.578812 0.815461i $$-0.303516\pi$$
0.578812 + 0.815461i $$0.303516\pi$$
$$200$$ 0 0
$$201$$ −31.1652 −2.19822
$$202$$ 0 0
$$203$$ 13.5826 0.953310
$$204$$ 0 0
$$205$$ 6.79129 0.474324
$$206$$ 0 0
$$207$$ −4.79129 −0.333018
$$208$$ 0 0
$$209$$ −4.58258 −0.316983
$$210$$ 0 0
$$211$$ 10.0000 0.688428 0.344214 0.938891i $$-0.388145\pi$$
0.344214 + 0.938891i $$0.388145\pi$$
$$212$$ 0 0
$$213$$ −23.3739 −1.60155
$$214$$ 0 0
$$215$$ 11.1652 0.761457
$$216$$ 0 0
$$217$$ 6.04356 0.410264
$$218$$ 0 0
$$219$$ 35.5826 2.40445
$$220$$ 0 0
$$221$$ 4.58258 0.308257
$$222$$ 0 0
$$223$$ 7.16515 0.479814 0.239907 0.970796i $$-0.422883\pi$$
0.239907 + 0.970796i $$0.422883\pi$$
$$224$$ 0 0
$$225$$ 4.79129 0.319419
$$226$$ 0 0
$$227$$ 22.7477 1.50982 0.754910 0.655829i $$-0.227681\pi$$
0.754910 + 0.655829i $$0.227681\pi$$
$$228$$ 0 0
$$229$$ 20.3303 1.34346 0.671732 0.740794i $$-0.265551\pi$$
0.671732 + 0.740794i $$0.265551\pi$$
$$230$$ 0 0
$$231$$ 3.95644 0.260315
$$232$$ 0 0
$$233$$ −1.58258 −0.103678 −0.0518390 0.998655i $$-0.516508\pi$$
−0.0518390 + 0.998655i $$0.516508\pi$$
$$234$$ 0 0
$$235$$ −4.41742 −0.288161
$$236$$ 0 0
$$237$$ −22.3303 −1.45051
$$238$$ 0 0
$$239$$ 15.1652 0.980952 0.490476 0.871455i $$-0.336823\pi$$
0.490476 + 0.871455i $$0.336823\pi$$
$$240$$ 0 0
$$241$$ −28.0000 −1.80364 −0.901819 0.432113i $$-0.857768\pi$$
−0.901819 + 0.432113i $$0.857768\pi$$
$$242$$ 0 0
$$243$$ −16.1652 −1.03699
$$244$$ 0 0
$$245$$ 3.79129 0.242216
$$246$$ 0 0
$$247$$ −33.5390 −2.13404
$$248$$ 0 0
$$249$$ 16.7477 1.06134
$$250$$ 0 0
$$251$$ −26.2087 −1.65428 −0.827140 0.561996i $$-0.810033\pi$$
−0.827140 + 0.561996i $$0.810033\pi$$
$$252$$ 0 0
$$253$$ −0.791288 −0.0497478
$$254$$ 0 0
$$255$$ −2.20871 −0.138315
$$256$$ 0 0
$$257$$ −4.74773 −0.296155 −0.148078 0.988976i $$-0.547309\pi$$
−0.148078 + 0.988976i $$0.547309\pi$$
$$258$$ 0 0
$$259$$ −7.16515 −0.445221
$$260$$ 0 0
$$261$$ 36.3303 2.24879
$$262$$ 0 0
$$263$$ −11.2087 −0.691159 −0.345579 0.938390i $$-0.612318\pi$$
−0.345579 + 0.938390i $$0.612318\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 42.3303 2.59057
$$268$$ 0 0
$$269$$ −10.7477 −0.655300 −0.327650 0.944799i $$-0.606257\pi$$
−0.327650 + 0.944799i $$0.606257\pi$$
$$270$$ 0 0
$$271$$ −18.1216 −1.10081 −0.550404 0.834898i $$-0.685526\pi$$
−0.550404 + 0.834898i $$0.685526\pi$$
$$272$$ 0 0
$$273$$ 28.9564 1.75252
$$274$$ 0 0
$$275$$ 0.791288 0.0477165
$$276$$ 0 0
$$277$$ 17.1652 1.03135 0.515677 0.856783i $$-0.327540\pi$$
0.515677 + 0.856783i $$0.327540\pi$$
$$278$$ 0 0
$$279$$ 16.1652 0.967782
$$280$$ 0 0
$$281$$ 10.7477 0.641156 0.320578 0.947222i $$-0.396123\pi$$
0.320578 + 0.947222i $$0.396123\pi$$
$$282$$ 0 0
$$283$$ −8.33030 −0.495185 −0.247593 0.968864i $$-0.579639\pi$$
−0.247593 + 0.968864i $$0.579639\pi$$
$$284$$ 0 0
$$285$$ 16.1652 0.957541
$$286$$ 0 0
$$287$$ −12.1652 −0.718086
$$288$$ 0 0
$$289$$ −16.3739 −0.963168
$$290$$ 0 0
$$291$$ −22.2087 −1.30190
$$292$$ 0 0
$$293$$ 27.4955 1.60630 0.803151 0.595776i $$-0.203155\pi$$
0.803151 + 0.595776i $$0.203155\pi$$
$$294$$ 0 0
$$295$$ −13.5826 −0.790808
$$296$$ 0 0
$$297$$ 3.95644 0.229576
$$298$$ 0 0
$$299$$ −5.79129 −0.334919
$$300$$ 0 0
$$301$$ −20.0000 −1.15278
$$302$$ 0 0
$$303$$ 12.3303 0.708357
$$304$$ 0 0
$$305$$ −10.3739 −0.594006
$$306$$ 0 0
$$307$$ 15.5390 0.886858 0.443429 0.896309i $$-0.353762\pi$$
0.443429 + 0.896309i $$0.353762\pi$$
$$308$$ 0 0
$$309$$ 17.7913 1.01211
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −4.62614 −0.261485 −0.130742 0.991416i $$-0.541736\pi$$
−0.130742 + 0.991416i $$0.541736\pi$$
$$314$$ 0 0
$$315$$ −8.58258 −0.483573
$$316$$ 0 0
$$317$$ 9.79129 0.549934 0.274967 0.961454i $$-0.411333\pi$$
0.274967 + 0.961454i $$0.411333\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ −12.3303 −0.688210
$$322$$ 0 0
$$323$$ −4.58258 −0.254981
$$324$$ 0 0
$$325$$ 5.79129 0.321243
$$326$$ 0 0
$$327$$ −9.41742 −0.520785
$$328$$ 0 0
$$329$$ 7.91288 0.436251
$$330$$ 0 0
$$331$$ 20.7477 1.14040 0.570199 0.821507i $$-0.306866\pi$$
0.570199 + 0.821507i $$0.306866\pi$$
$$332$$ 0 0
$$333$$ −19.1652 −1.05024
$$334$$ 0 0
$$335$$ 11.1652 0.610017
$$336$$ 0 0
$$337$$ −12.2087 −0.665051 −0.332525 0.943094i $$-0.607901\pi$$
−0.332525 + 0.943094i $$0.607901\pi$$
$$338$$ 0 0
$$339$$ 16.7477 0.909612
$$340$$ 0 0
$$341$$ 2.66970 0.144572
$$342$$ 0 0
$$343$$ −19.3303 −1.04374
$$344$$ 0 0
$$345$$ 2.79129 0.150278
$$346$$ 0 0
$$347$$ −5.20871 −0.279618 −0.139809 0.990178i $$-0.544649\pi$$
−0.139809 + 0.990178i $$0.544649\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 28.9564 1.54558
$$352$$ 0 0
$$353$$ 3.16515 0.168464 0.0842320 0.996446i $$-0.473156\pi$$
0.0842320 + 0.996446i $$0.473156\pi$$
$$354$$ 0 0
$$355$$ 8.37386 0.444439
$$356$$ 0 0
$$357$$ 3.95644 0.209397
$$358$$ 0 0
$$359$$ −9.16515 −0.483718 −0.241859 0.970311i $$-0.577757\pi$$
−0.241859 + 0.970311i $$0.577757\pi$$
$$360$$ 0 0
$$361$$ 14.5390 0.765211
$$362$$ 0 0
$$363$$ −28.9564 −1.51982
$$364$$ 0 0
$$365$$ −12.7477 −0.667247
$$366$$ 0 0
$$367$$ 19.1652 1.00041 0.500206 0.865906i $$-0.333257\pi$$
0.500206 + 0.865906i $$0.333257\pi$$
$$368$$ 0 0
$$369$$ −32.5390 −1.69391
$$370$$ 0 0
$$371$$ 10.7477 0.557994
$$372$$ 0 0
$$373$$ 12.7477 0.660052 0.330026 0.943972i $$-0.392942\pi$$
0.330026 + 0.943972i $$0.392942\pi$$
$$374$$ 0 0
$$375$$ −2.79129 −0.144141
$$376$$ 0 0
$$377$$ 43.9129 2.26163
$$378$$ 0 0
$$379$$ 6.37386 0.327403 0.163702 0.986510i $$-0.447657\pi$$
0.163702 + 0.986510i $$0.447657\pi$$
$$380$$ 0 0
$$381$$ −35.5826 −1.82295
$$382$$ 0 0
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ −1.41742 −0.0722386
$$386$$ 0 0
$$387$$ −53.4955 −2.71933
$$388$$ 0 0
$$389$$ −20.7042 −1.04974 −0.524871 0.851182i $$-0.675887\pi$$
−0.524871 + 0.851182i $$0.675887\pi$$
$$390$$ 0 0
$$391$$ −0.791288 −0.0400171
$$392$$ 0 0
$$393$$ 25.5826 1.29047
$$394$$ 0 0
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ −15.5390 −0.779881 −0.389940 0.920840i $$-0.627504\pi$$
−0.389940 + 0.920840i $$0.627504\pi$$
$$398$$ 0 0
$$399$$ −28.9564 −1.44964
$$400$$ 0 0
$$401$$ −4.74773 −0.237090 −0.118545 0.992949i $$-0.537823\pi$$
−0.118545 + 0.992949i $$0.537823\pi$$
$$402$$ 0 0
$$403$$ 19.5390 0.973308
$$404$$ 0 0
$$405$$ 0.417424 0.0207420
$$406$$ 0 0
$$407$$ −3.16515 −0.156891
$$408$$ 0 0
$$409$$ −18.2087 −0.900363 −0.450181 0.892937i $$-0.648641\pi$$
−0.450181 + 0.892937i $$0.648641\pi$$
$$410$$ 0 0
$$411$$ 10.5826 0.522000
$$412$$ 0 0
$$413$$ 24.3303 1.19722
$$414$$ 0 0
$$415$$ −6.00000 −0.294528
$$416$$ 0 0
$$417$$ −35.5826 −1.74249
$$418$$ 0 0
$$419$$ −20.8348 −1.01785 −0.508924 0.860811i $$-0.669957\pi$$
−0.508924 + 0.860811i $$0.669957\pi$$
$$420$$ 0 0
$$421$$ 18.1216 0.883192 0.441596 0.897214i $$-0.354412\pi$$
0.441596 + 0.897214i $$0.354412\pi$$
$$422$$ 0 0
$$423$$ 21.1652 1.02908
$$424$$ 0 0
$$425$$ 0.791288 0.0383831
$$426$$ 0 0
$$427$$ 18.5826 0.899274
$$428$$ 0 0
$$429$$ 12.7913 0.617569
$$430$$ 0 0
$$431$$ −25.9129 −1.24818 −0.624090 0.781353i $$-0.714530\pi$$
−0.624090 + 0.781353i $$0.714530\pi$$
$$432$$ 0 0
$$433$$ −30.5390 −1.46761 −0.733806 0.679359i $$-0.762258\pi$$
−0.733806 + 0.679359i $$0.762258\pi$$
$$434$$ 0 0
$$435$$ −21.1652 −1.01479
$$436$$ 0 0
$$437$$ 5.79129 0.277035
$$438$$ 0 0
$$439$$ 6.53901 0.312090 0.156045 0.987750i $$-0.450125\pi$$
0.156045 + 0.987750i $$0.450125\pi$$
$$440$$ 0 0
$$441$$ −18.1652 −0.865007
$$442$$ 0 0
$$443$$ 39.7913 1.89054 0.945271 0.326288i $$-0.105798\pi$$
0.945271 + 0.326288i $$0.105798\pi$$
$$444$$ 0 0
$$445$$ −15.1652 −0.718897
$$446$$ 0 0
$$447$$ 22.9129 1.08374
$$448$$ 0 0
$$449$$ −16.1216 −0.760825 −0.380412 0.924817i $$-0.624218\pi$$
−0.380412 + 0.924817i $$0.624218\pi$$
$$450$$ 0 0
$$451$$ −5.37386 −0.253045
$$452$$ 0 0
$$453$$ 30.1216 1.41524
$$454$$ 0 0
$$455$$ −10.3739 −0.486334
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 0 0
$$459$$ 3.95644 0.184671
$$460$$ 0 0
$$461$$ −28.7477 −1.33892 −0.669458 0.742850i $$-0.733473\pi$$
−0.669458 + 0.742850i $$0.733473\pi$$
$$462$$ 0 0
$$463$$ 10.0000 0.464739 0.232370 0.972628i $$-0.425352\pi$$
0.232370 + 0.972628i $$0.425352\pi$$
$$464$$ 0 0
$$465$$ −9.41742 −0.436723
$$466$$ 0 0
$$467$$ 19.9129 0.921458 0.460729 0.887541i $$-0.347588\pi$$
0.460729 + 0.887541i $$0.347588\pi$$
$$468$$ 0 0
$$469$$ −20.0000 −0.923514
$$470$$ 0 0
$$471$$ −41.1652 −1.89679
$$472$$ 0 0
$$473$$ −8.83485 −0.406227
$$474$$ 0 0
$$475$$ −5.79129 −0.265723
$$476$$ 0 0
$$477$$ 28.7477 1.31627
$$478$$ 0 0
$$479$$ −15.4955 −0.708005 −0.354003 0.935244i $$-0.615180\pi$$
−0.354003 + 0.935244i $$0.615180\pi$$
$$480$$ 0 0
$$481$$ −23.1652 −1.05624
$$482$$ 0 0
$$483$$ −5.00000 −0.227508
$$484$$ 0 0
$$485$$ 7.95644 0.361283
$$486$$ 0 0
$$487$$ −6.41742 −0.290801 −0.145401 0.989373i $$-0.546447\pi$$
−0.145401 + 0.989373i $$0.546447\pi$$
$$488$$ 0 0
$$489$$ −24.0780 −1.08885
$$490$$ 0 0
$$491$$ −10.7477 −0.485038 −0.242519 0.970147i $$-0.577974\pi$$
−0.242519 + 0.970147i $$0.577974\pi$$
$$492$$ 0 0
$$493$$ 6.00000 0.270226
$$494$$ 0 0
$$495$$ −3.79129 −0.170406
$$496$$ 0 0
$$497$$ −15.0000 −0.672842
$$498$$ 0 0
$$499$$ −4.83485 −0.216438 −0.108219 0.994127i $$-0.534515\pi$$
−0.108219 + 0.994127i $$0.534515\pi$$
$$500$$ 0 0
$$501$$ −51.1652 −2.28589
$$502$$ 0 0
$$503$$ −14.2087 −0.633535 −0.316768 0.948503i $$-0.602598\pi$$
−0.316768 + 0.948503i $$0.602598\pi$$
$$504$$ 0 0
$$505$$ −4.41742 −0.196573
$$506$$ 0 0
$$507$$ 57.3303 2.54613
$$508$$ 0 0
$$509$$ 34.7477 1.54017 0.770083 0.637944i $$-0.220215\pi$$
0.770083 + 0.637944i $$0.220215\pi$$
$$510$$ 0 0
$$511$$ 22.8348 1.01015
$$512$$ 0 0
$$513$$ −28.9564 −1.27846
$$514$$ 0 0
$$515$$ −6.37386 −0.280866
$$516$$ 0 0
$$517$$ 3.49545 0.153730
$$518$$ 0 0
$$519$$ −52.4519 −2.30238
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ −17.1652 −0.750580 −0.375290 0.926908i $$-0.622457\pi$$
−0.375290 + 0.926908i $$0.622457\pi$$
$$524$$ 0 0
$$525$$ 5.00000 0.218218
$$526$$ 0 0
$$527$$ 2.66970 0.116294
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 65.0780 2.82415
$$532$$ 0 0
$$533$$ −39.3303 −1.70358
$$534$$ 0 0
$$535$$ 4.41742 0.190982
$$536$$ 0 0
$$537$$ −30.0000 −1.29460
$$538$$ 0 0
$$539$$ −3.00000 −0.129219
$$540$$ 0 0
$$541$$ 1.66970 0.0717859 0.0358929 0.999356i $$-0.488572\pi$$
0.0358929 + 0.999356i $$0.488572\pi$$
$$542$$ 0 0
$$543$$ −51.7477 −2.22071
$$544$$ 0 0
$$545$$ 3.37386 0.144520
$$546$$ 0 0
$$547$$ 26.1216 1.11688 0.558439 0.829545i $$-0.311400\pi$$
0.558439 + 0.829545i $$0.311400\pi$$
$$548$$ 0 0
$$549$$ 49.7042 2.12132
$$550$$ 0 0
$$551$$ −43.9129 −1.87075
$$552$$ 0 0
$$553$$ −14.3303 −0.609386
$$554$$ 0 0
$$555$$ 11.1652 0.473934
$$556$$ 0 0
$$557$$ −6.33030 −0.268224 −0.134112 0.990966i $$-0.542818\pi$$
−0.134112 + 0.990966i $$0.542818\pi$$
$$558$$ 0 0
$$559$$ −64.6606 −2.73485
$$560$$ 0 0
$$561$$ 1.74773 0.0737891
$$562$$ 0 0
$$563$$ 15.1652 0.639135 0.319567 0.947564i $$-0.396462\pi$$
0.319567 + 0.947564i $$0.396462\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ −0.747727 −0.0314016
$$568$$ 0 0
$$569$$ −39.4955 −1.65574 −0.827868 0.560923i $$-0.810446\pi$$
−0.827868 + 0.560923i $$0.810446\pi$$
$$570$$ 0 0
$$571$$ 11.1216 0.465424 0.232712 0.972546i $$-0.425240\pi$$
0.232712 + 0.972546i $$0.425240\pi$$
$$572$$ 0 0
$$573$$ −71.4083 −2.98313
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 41.1652 1.71373 0.856864 0.515543i $$-0.172410\pi$$
0.856864 + 0.515543i $$0.172410\pi$$
$$578$$ 0 0
$$579$$ −57.9129 −2.40678
$$580$$ 0 0
$$581$$ 10.7477 0.445891
$$582$$ 0 0
$$583$$ 4.74773 0.196631
$$584$$ 0 0
$$585$$ −27.7477 −1.14723
$$586$$ 0 0
$$587$$ 30.7913 1.27089 0.635446 0.772145i $$-0.280816\pi$$
0.635446 + 0.772145i $$0.280816\pi$$
$$588$$ 0 0
$$589$$ −19.5390 −0.805091
$$590$$ 0 0
$$591$$ −32.2087 −1.32489
$$592$$ 0 0
$$593$$ −31.9129 −1.31050 −0.655252 0.755410i $$-0.727438\pi$$
−0.655252 + 0.755410i $$0.727438\pi$$
$$594$$ 0 0
$$595$$ −1.41742 −0.0581087
$$596$$ 0 0
$$597$$ 45.5826 1.86557
$$598$$ 0 0
$$599$$ −1.12159 −0.0458270 −0.0229135 0.999737i $$-0.507294\pi$$
−0.0229135 + 0.999737i $$0.507294\pi$$
$$600$$ 0 0
$$601$$ −18.2087 −0.742749 −0.371374 0.928483i $$-0.621113\pi$$
−0.371374 + 0.928483i $$0.621113\pi$$
$$602$$ 0 0
$$603$$ −53.4955 −2.17850
$$604$$ 0 0
$$605$$ 10.3739 0.421758
$$606$$ 0 0
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 0 0
$$609$$ 37.9129 1.53631
$$610$$ 0 0
$$611$$ 25.5826 1.03496
$$612$$ 0 0
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ 0 0
$$615$$ 18.9564 0.764397
$$616$$ 0 0
$$617$$ −23.8693 −0.960943 −0.480471 0.877010i $$-0.659534\pi$$
−0.480471 + 0.877010i $$0.659534\pi$$
$$618$$ 0 0
$$619$$ 1.79129 0.0719979 0.0359990 0.999352i $$-0.488539\pi$$
0.0359990 + 0.999352i $$0.488539\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ 0 0
$$623$$ 27.1652 1.08835
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −12.7913 −0.510835
$$628$$ 0 0
$$629$$ −3.16515 −0.126203
$$630$$ 0 0
$$631$$ −27.9129 −1.11119 −0.555597 0.831452i $$-0.687510\pi$$
−0.555597 + 0.831452i $$0.687510\pi$$
$$632$$ 0 0
$$633$$ 27.9129 1.10944
$$634$$ 0 0
$$635$$ 12.7477 0.505878
$$636$$ 0 0
$$637$$ −21.9564 −0.869946
$$638$$ 0 0
$$639$$ −40.1216 −1.58719
$$640$$ 0 0
$$641$$ 15.1652 0.598987 0.299494 0.954098i $$-0.403182\pi$$
0.299494 + 0.954098i $$0.403182\pi$$
$$642$$ 0 0
$$643$$ −6.74773 −0.266104 −0.133052 0.991109i $$-0.542478\pi$$
−0.133052 + 0.991109i $$0.542478\pi$$
$$644$$ 0 0
$$645$$ 31.1652 1.22713
$$646$$ 0 0
$$647$$ −21.1652 −0.832088 −0.416044 0.909344i $$-0.636584\pi$$
−0.416044 + 0.909344i $$0.636584\pi$$
$$648$$ 0 0
$$649$$ 10.7477 0.421885
$$650$$ 0 0
$$651$$ 16.8693 0.661161
$$652$$ 0 0
$$653$$ 3.46099 0.135439 0.0677194 0.997704i $$-0.478428\pi$$
0.0677194 + 0.997704i $$0.478428\pi$$
$$654$$ 0 0
$$655$$ −9.16515 −0.358112
$$656$$ 0 0
$$657$$ 61.0780 2.38288
$$658$$ 0 0
$$659$$ 8.83485 0.344157 0.172078 0.985083i $$-0.444952\pi$$
0.172078 + 0.985083i $$0.444952\pi$$
$$660$$ 0 0
$$661$$ −25.6261 −0.996741 −0.498371 0.866964i $$-0.666068\pi$$
−0.498371 + 0.866964i $$0.666068\pi$$
$$662$$ 0 0
$$663$$ 12.7913 0.496772
$$664$$ 0 0
$$665$$ 10.3739 0.402281
$$666$$ 0 0
$$667$$ −7.58258 −0.293599
$$668$$ 0 0
$$669$$ 20.0000 0.773245
$$670$$ 0 0
$$671$$ 8.20871 0.316894
$$672$$ 0 0
$$673$$ 38.0000 1.46479 0.732396 0.680879i $$-0.238402\pi$$
0.732396 + 0.680879i $$0.238402\pi$$
$$674$$ 0 0
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ 42.6606 1.63958 0.819790 0.572664i $$-0.194090\pi$$
0.819790 + 0.572664i $$0.194090\pi$$
$$678$$ 0 0
$$679$$ −14.2523 −0.546952
$$680$$ 0 0
$$681$$ 63.4955 2.43315
$$682$$ 0 0
$$683$$ 11.3739 0.435209 0.217604 0.976037i $$-0.430176\pi$$
0.217604 + 0.976037i $$0.430176\pi$$
$$684$$ 0 0
$$685$$ −3.79129 −0.144858
$$686$$ 0 0
$$687$$ 56.7477 2.16506
$$688$$ 0 0
$$689$$ 34.7477 1.32378
$$690$$ 0 0
$$691$$ −42.7477 −1.62620 −0.813100 0.582124i $$-0.802222\pi$$
−0.813100 + 0.582124i $$0.802222\pi$$
$$692$$ 0 0
$$693$$ 6.79129 0.257980
$$694$$ 0 0
$$695$$ 12.7477 0.483549
$$696$$ 0 0
$$697$$ −5.37386 −0.203550
$$698$$ 0 0
$$699$$ −4.41742 −0.167082
$$700$$ 0 0
$$701$$ 23.3739 0.882819 0.441409 0.897306i $$-0.354479\pi$$
0.441409 + 0.897306i $$0.354479\pi$$
$$702$$ 0 0
$$703$$ 23.1652 0.873690
$$704$$ 0 0
$$705$$ −12.3303 −0.464386
$$706$$ 0 0
$$707$$ 7.91288 0.297594
$$708$$ 0 0
$$709$$ 2.46099 0.0924242 0.0462121 0.998932i $$-0.485285\pi$$
0.0462121 + 0.998932i $$0.485285\pi$$
$$710$$ 0 0
$$711$$ −38.3303 −1.43750
$$712$$ 0 0
$$713$$ −3.37386 −0.126352
$$714$$ 0 0
$$715$$ −4.58258 −0.171379
$$716$$ 0 0
$$717$$ 42.3303 1.58085
$$718$$ 0 0
$$719$$ −2.53901 −0.0946893 −0.0473446 0.998879i $$-0.515076\pi$$
−0.0473446 + 0.998879i $$0.515076\pi$$
$$720$$ 0 0
$$721$$ 11.4174 0.425207
$$722$$ 0 0
$$723$$ −78.1561 −2.90666
$$724$$ 0 0
$$725$$ 7.58258 0.281610
$$726$$ 0 0
$$727$$ −39.1216 −1.45094 −0.725470 0.688254i $$-0.758377\pi$$
−0.725470 + 0.688254i $$0.758377\pi$$
$$728$$ 0 0
$$729$$ −43.8693 −1.62479
$$730$$ 0 0
$$731$$ −8.83485 −0.326769
$$732$$ 0 0
$$733$$ 26.0000 0.960332 0.480166 0.877178i $$-0.340576\pi$$
0.480166 + 0.877178i $$0.340576\pi$$
$$734$$ 0 0
$$735$$ 10.5826 0.390344
$$736$$ 0 0
$$737$$ −8.83485 −0.325436
$$738$$ 0 0
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ 0 0
$$741$$ −93.6170 −3.43911
$$742$$ 0 0
$$743$$ 12.9564 0.475326 0.237663 0.971348i $$-0.423619\pi$$
0.237663 + 0.971348i $$0.423619\pi$$
$$744$$ 0 0
$$745$$ −8.20871 −0.300744
$$746$$ 0 0
$$747$$ 28.7477 1.05182
$$748$$ 0 0
$$749$$ −7.91288 −0.289130
$$750$$ 0 0
$$751$$ 8.74773 0.319209 0.159605 0.987181i $$-0.448978\pi$$
0.159605 + 0.987181i $$0.448978\pi$$
$$752$$ 0 0
$$753$$ −73.1561 −2.66596
$$754$$ 0 0
$$755$$ −10.7913 −0.392735
$$756$$ 0 0
$$757$$ −10.3303 −0.375461 −0.187731 0.982221i $$-0.560113\pi$$
−0.187731 + 0.982221i $$0.560113\pi$$
$$758$$ 0 0
$$759$$ −2.20871 −0.0801712
$$760$$ 0 0
$$761$$ −11.0436 −0.400329 −0.200164 0.979762i $$-0.564148\pi$$
−0.200164 + 0.979762i $$0.564148\pi$$
$$762$$ 0 0
$$763$$ −6.04356 −0.218792
$$764$$ 0 0
$$765$$ −3.79129 −0.137074
$$766$$ 0 0
$$767$$ 78.6606 2.84027
$$768$$ 0 0
$$769$$ −40.3303 −1.45435 −0.727174 0.686453i $$-0.759167\pi$$
−0.727174 + 0.686453i $$0.759167\pi$$
$$770$$ 0 0
$$771$$ −13.2523 −0.477269
$$772$$ 0 0
$$773$$ 33.4955 1.20475 0.602374 0.798214i $$-0.294222\pi$$
0.602374 + 0.798214i $$0.294222\pi$$
$$774$$ 0 0
$$775$$ 3.37386 0.121193
$$776$$ 0 0
$$777$$ −20.0000 −0.717496
$$778$$ 0 0
$$779$$ 39.3303 1.40915
$$780$$ 0 0
$$781$$ −6.62614 −0.237102
$$782$$ 0 0
$$783$$ 37.9129 1.35490
$$784$$ 0 0
$$785$$ 14.7477 0.526369
$$786$$ 0 0
$$787$$ 17.5826 0.626751 0.313376 0.949629i $$-0.398540\pi$$
0.313376 + 0.949629i $$0.398540\pi$$
$$788$$ 0 0
$$789$$ −31.2867 −1.11384
$$790$$ 0 0
$$791$$ 10.7477 0.382145
$$792$$ 0 0
$$793$$ 60.0780 2.13343
$$794$$ 0 0
$$795$$ −16.7477 −0.593981
$$796$$ 0 0
$$797$$ −4.08712 −0.144773 −0.0723866 0.997377i $$-0.523062\pi$$
−0.0723866 + 0.997377i $$0.523062\pi$$
$$798$$ 0 0
$$799$$ 3.49545 0.123660
$$800$$ 0 0
$$801$$ 72.6606 2.56734
$$802$$ 0 0
$$803$$ 10.0871 0.355967
$$804$$ 0 0
$$805$$ 1.79129 0.0631346
$$806$$ 0 0
$$807$$ −30.0000 −1.05605
$$808$$ 0 0
$$809$$ 33.9564 1.19384 0.596922 0.802299i $$-0.296390\pi$$
0.596922 + 0.802299i $$0.296390\pi$$
$$810$$ 0 0
$$811$$ 2.08712 0.0732887 0.0366444 0.999328i $$-0.488333\pi$$
0.0366444 + 0.999328i $$0.488333\pi$$
$$812$$ 0 0
$$813$$ −50.5826 −1.77401
$$814$$ 0 0
$$815$$ 8.62614 0.302160
$$816$$ 0 0
$$817$$ 64.6606 2.26219
$$818$$ 0 0
$$819$$ 49.7042 1.73680
$$820$$ 0 0
$$821$$ 21.1652 0.738669 0.369334 0.929297i $$-0.379586\pi$$
0.369334 + 0.929297i $$0.379586\pi$$
$$822$$ 0 0
$$823$$ −22.8348 −0.795973 −0.397986 0.917391i $$-0.630291\pi$$
−0.397986 + 0.917391i $$0.630291\pi$$
$$824$$ 0 0
$$825$$ 2.20871 0.0768975
$$826$$ 0 0
$$827$$ 23.0780 0.802502 0.401251 0.915968i $$-0.368576\pi$$
0.401251 + 0.915968i $$0.368576\pi$$
$$828$$ 0 0
$$829$$ 23.4955 0.816031 0.408015 0.912975i $$-0.366221\pi$$
0.408015 + 0.912975i $$0.366221\pi$$
$$830$$ 0 0
$$831$$ 47.9129 1.66208
$$832$$ 0 0
$$833$$ −3.00000 −0.103944
$$834$$ 0 0
$$835$$ 18.3303 0.634346
$$836$$ 0 0
$$837$$ 16.8693 0.583089
$$838$$ 0 0
$$839$$ 31.5826 1.09035 0.545176 0.838322i $$-0.316463\pi$$
0.545176 + 0.838322i $$0.316463\pi$$
$$840$$ 0 0
$$841$$ 28.4955 0.982602
$$842$$ 0 0
$$843$$ 30.0000 1.03325
$$844$$ 0 0
$$845$$ −20.5390 −0.706564
$$846$$ 0 0
$$847$$ −18.5826 −0.638505
$$848$$ 0 0
$$849$$ −23.2523 −0.798016
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ 0 0
$$853$$ 40.5390 1.38803 0.694015 0.719961i $$-0.255840\pi$$
0.694015 + 0.719961i $$0.255840\pi$$
$$854$$ 0 0
$$855$$ 27.7477 0.948952
$$856$$ 0 0
$$857$$ −9.16515 −0.313076 −0.156538 0.987672i $$-0.550033\pi$$
−0.156538 + 0.987672i $$0.550033\pi$$
$$858$$ 0 0
$$859$$ 26.7477 0.912621 0.456310 0.889821i $$-0.349171\pi$$
0.456310 + 0.889821i $$0.349171\pi$$
$$860$$ 0 0
$$861$$ −33.9564 −1.15723
$$862$$ 0 0
$$863$$ 22.4174 0.763098 0.381549 0.924349i $$-0.375391\pi$$
0.381549 + 0.924349i $$0.375391\pi$$
$$864$$ 0 0
$$865$$ 18.7913 0.638923
$$866$$ 0 0
$$867$$ −45.7042 −1.55219
$$868$$ 0 0
$$869$$ −6.33030 −0.214741
$$870$$ 0 0
$$871$$ −64.6606 −2.19094
$$872$$ 0 0
$$873$$ −38.1216 −1.29022
$$874$$ 0 0
$$875$$ −1.79129 −0.0605566
$$876$$ 0 0
$$877$$ −42.7042 −1.44202 −0.721009 0.692926i $$-0.756321\pi$$
−0.721009 + 0.692926i $$0.756321\pi$$
$$878$$ 0 0
$$879$$ 76.7477 2.58864
$$880$$ 0 0
$$881$$ 30.3303 1.02185 0.510927 0.859624i $$-0.329302\pi$$
0.510927 + 0.859624i $$0.329302\pi$$
$$882$$ 0 0
$$883$$ 34.9564 1.17638 0.588189 0.808724i $$-0.299841\pi$$
0.588189 + 0.808724i $$0.299841\pi$$
$$884$$ 0 0
$$885$$ −37.9129 −1.27443
$$886$$ 0 0
$$887$$ −15.1652 −0.509196 −0.254598 0.967047i $$-0.581943\pi$$
−0.254598 + 0.967047i $$0.581943\pi$$
$$888$$ 0 0
$$889$$ −22.8348 −0.765856
$$890$$ 0 0
$$891$$ −0.330303 −0.0110656
$$892$$ 0 0
$$893$$ −25.5826 −0.856088
$$894$$ 0 0
$$895$$ 10.7477 0.359257
$$896$$ 0 0
$$897$$ −16.1652 −0.539739
$$898$$ 0 0
$$899$$ 25.5826 0.853227
$$900$$ 0 0
$$901$$ 4.74773 0.158170
$$902$$ 0 0
$$903$$ −55.8258 −1.85776
$$904$$ 0 0
$$905$$ 18.5390 0.616258
$$906$$ 0 0
$$907$$ −6.74773 −0.224055 −0.112027 0.993705i $$-0.535734\pi$$
−0.112027 + 0.993705i $$0.535734\pi$$
$$908$$ 0 0
$$909$$ 21.1652 0.702004
$$910$$ 0 0
$$911$$ −4.41742 −0.146356 −0.0731779 0.997319i $$-0.523314\pi$$
−0.0731779 + 0.997319i $$0.523314\pi$$
$$912$$ 0 0
$$913$$ 4.74773 0.157127
$$914$$ 0 0
$$915$$ −28.9564 −0.957270
$$916$$ 0 0
$$917$$ 16.4174 0.542151
$$918$$ 0 0
$$919$$ 55.1652 1.81973 0.909865 0.414904i $$-0.136185\pi$$
0.909865 + 0.414904i $$0.136185\pi$$
$$920$$ 0 0
$$921$$ 43.3739 1.42922
$$922$$ 0 0
$$923$$ −48.4955 −1.59625
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 0 0
$$927$$ 30.5390 1.00303
$$928$$ 0 0
$$929$$ 15.4955 0.508389 0.254195 0.967153i $$-0.418190\pi$$
0.254195 + 0.967153i $$0.418190\pi$$
$$930$$ 0 0
$$931$$ 21.9564 0.719593
$$932$$ 0 0
$$933$$ −33.4955 −1.09659
$$934$$ 0 0
$$935$$ −0.626136 −0.0204769
$$936$$ 0 0
$$937$$ 44.6261 1.45787 0.728936 0.684582i $$-0.240015\pi$$
0.728936 + 0.684582i $$0.240015\pi$$
$$938$$ 0 0
$$939$$ −12.9129 −0.421396
$$940$$ 0 0
$$941$$ 32.0436 1.04459 0.522295 0.852765i $$-0.325076\pi$$
0.522295 + 0.852765i $$0.325076\pi$$
$$942$$ 0 0
$$943$$ 6.79129 0.221155
$$944$$ 0 0
$$945$$ −8.95644 −0.291353
$$946$$ 0 0
$$947$$ −2.53901 −0.0825069 −0.0412534 0.999149i $$-0.513135\pi$$
−0.0412534 + 0.999149i $$0.513135\pi$$
$$948$$ 0 0
$$949$$ 73.8258 2.39649
$$950$$ 0 0
$$951$$ 27.3303 0.886246
$$952$$ 0 0
$$953$$ 5.53901 0.179426 0.0897131 0.995968i $$-0.471405\pi$$
0.0897131 + 0.995968i $$0.471405\pi$$
$$954$$ 0 0
$$955$$ 25.5826 0.827833
$$956$$ 0 0
$$957$$ 16.7477 0.541377
$$958$$ 0 0
$$959$$ 6.79129 0.219302
$$960$$ 0 0
$$961$$ −19.6170 −0.632808
$$962$$ 0 0
$$963$$ −21.1652 −0.682037
$$964$$ 0 0
$$965$$ 20.7477 0.667893
$$966$$ 0 0
$$967$$ 32.7477 1.05310 0.526548 0.850145i $$-0.323486\pi$$
0.526548 + 0.850145i $$0.323486\pi$$
$$968$$ 0 0
$$969$$ −12.7913 −0.410915
$$970$$ 0 0
$$971$$ −15.9564 −0.512067 −0.256033 0.966668i $$-0.582416\pi$$
−0.256033 + 0.966668i $$0.582416\pi$$
$$972$$ 0 0
$$973$$ −22.8348 −0.732052
$$974$$ 0 0
$$975$$ 16.1652 0.517699
$$976$$ 0 0
$$977$$ −34.1216 −1.09165 −0.545823 0.837900i $$-0.683783\pi$$
−0.545823 + 0.837900i $$0.683783\pi$$
$$978$$ 0 0
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ −16.1652 −0.516114
$$982$$ 0 0
$$983$$ 14.3739 0.458455 0.229228 0.973373i $$-0.426380\pi$$
0.229228 + 0.973373i $$0.426380\pi$$
$$984$$ 0 0
$$985$$ 11.5390 0.367664
$$986$$ 0 0
$$987$$ 22.0871 0.703041
$$988$$ 0 0
$$989$$ 11.1652 0.355031
$$990$$ 0 0
$$991$$ 33.2087 1.05491 0.527455 0.849583i $$-0.323146\pi$$
0.527455 + 0.849583i $$0.323146\pi$$
$$992$$ 0 0
$$993$$ 57.9129 1.83781
$$994$$ 0 0
$$995$$ −16.3303 −0.517705
$$996$$ 0 0
$$997$$ −43.4955 −1.37751 −0.688757 0.724992i $$-0.741843\pi$$
−0.688757 + 0.724992i $$0.741843\pi$$
$$998$$ 0 0
$$999$$ −20.0000 −0.632772
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 1840.2.a.n.1.2 2
4.3 odd 2 230.2.a.a.1.1 2
5.4 even 2 9200.2.a.bs.1.1 2
8.3 odd 2 7360.2.a.bq.1.2 2
8.5 even 2 7360.2.a.bk.1.1 2
12.11 even 2 2070.2.a.x.1.1 2
20.3 even 4 1150.2.b.g.599.3 4
20.7 even 4 1150.2.b.g.599.2 4
20.19 odd 2 1150.2.a.o.1.2 2
92.91 even 2 5290.2.a.e.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.1 2 4.3 odd 2
1150.2.a.o.1.2 2 20.19 odd 2
1150.2.b.g.599.2 4 20.7 even 4
1150.2.b.g.599.3 4 20.3 even 4
1840.2.a.n.1.2 2 1.1 even 1 trivial
2070.2.a.x.1.1 2 12.11 even 2
5290.2.a.e.1.1 2 92.91 even 2
7360.2.a.bk.1.1 2 8.5 even 2
7360.2.a.bq.1.2 2 8.3 odd 2
9200.2.a.bs.1.1 2 5.4 even 2