# Properties

 Label 1150.2.a.o.1.2 Level $1150$ Weight $2$ Character 1150.1 Self dual yes Analytic conductor $9.183$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.18279623245$$ 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 $$\chi$$ $$=$$ 1150.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.79129 q^{3} +1.00000 q^{4} +2.79129 q^{6} +1.79129 q^{7} +1.00000 q^{8} +4.79129 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.79129 q^{3} +1.00000 q^{4} +2.79129 q^{6} +1.79129 q^{7} +1.00000 q^{8} +4.79129 q^{9} -0.791288 q^{11} +2.79129 q^{12} -5.79129 q^{13} +1.79129 q^{14} +1.00000 q^{16} -0.791288 q^{17} +4.79129 q^{18} +5.79129 q^{19} +5.00000 q^{21} -0.791288 q^{22} -1.00000 q^{23} +2.79129 q^{24} -5.79129 q^{26} +5.00000 q^{27} +1.79129 q^{28} +7.58258 q^{29} -3.37386 q^{31} +1.00000 q^{32} -2.20871 q^{33} -0.791288 q^{34} +4.79129 q^{36} +4.00000 q^{37} +5.79129 q^{38} -16.1652 q^{39} -6.79129 q^{41} +5.00000 q^{42} -11.1652 q^{43} -0.791288 q^{44} -1.00000 q^{46} +4.41742 q^{47} +2.79129 q^{48} -3.79129 q^{49} -2.20871 q^{51} -5.79129 q^{52} -6.00000 q^{53} +5.00000 q^{54} +1.79129 q^{56} +16.1652 q^{57} +7.58258 q^{58} -13.5826 q^{59} +10.3739 q^{61} -3.37386 q^{62} +8.58258 q^{63} +1.00000 q^{64} -2.20871 q^{66} -11.1652 q^{67} -0.791288 q^{68} -2.79129 q^{69} +8.37386 q^{71} +4.79129 q^{72} -12.7477 q^{73} +4.00000 q^{74} +5.79129 q^{76} -1.41742 q^{77} -16.1652 q^{78} +8.00000 q^{79} -0.417424 q^{81} -6.79129 q^{82} +6.00000 q^{83} +5.00000 q^{84} -11.1652 q^{86} +21.1652 q^{87} -0.791288 q^{88} +15.1652 q^{89} -10.3739 q^{91} -1.00000 q^{92} -9.41742 q^{93} +4.41742 q^{94} +2.79129 q^{96} +7.95644 q^{97} -3.79129 q^{98} -3.79129 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + q^{3} + 2q^{4} + q^{6} - q^{7} + 2q^{8} + 5q^{9} + O(q^{10})$$ $$2q + 2q^{2} + q^{3} + 2q^{4} + q^{6} - q^{7} + 2q^{8} + 5q^{9} + 3q^{11} + q^{12} - 7q^{13} - q^{14} + 2q^{16} + 3q^{17} + 5q^{18} + 7q^{19} + 10q^{21} + 3q^{22} - 2q^{23} + q^{24} - 7q^{26} + 10q^{27} - q^{28} + 6q^{29} + 7q^{31} + 2q^{32} - 9q^{33} + 3q^{34} + 5q^{36} + 8q^{37} + 7q^{38} - 14q^{39} - 9q^{41} + 10q^{42} - 4q^{43} + 3q^{44} - 2q^{46} + 18q^{47} + q^{48} - 3q^{49} - 9q^{51} - 7q^{52} - 12q^{53} + 10q^{54} - q^{56} + 14q^{57} + 6q^{58} - 18q^{59} + 7q^{61} + 7q^{62} + 8q^{63} + 2q^{64} - 9q^{66} - 4q^{67} + 3q^{68} - q^{69} + 3q^{71} + 5q^{72} + 2q^{73} + 8q^{74} + 7q^{76} - 12q^{77} - 14q^{78} + 16q^{79} - 10q^{81} - 9q^{82} + 12q^{83} + 10q^{84} - 4q^{86} + 24q^{87} + 3q^{88} + 12q^{89} - 7q^{91} - 2q^{92} - 28q^{93} + 18q^{94} + q^{96} - 7q^{97} - 3q^{98} - 3q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 2.79129 1.61155 0.805775 0.592221i $$-0.201749\pi$$
0.805775 + 0.592221i $$0.201749\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.79129 1.13954
$$7$$ 1.79129 0.677043 0.338522 0.940959i $$-0.390073\pi$$
0.338522 + 0.940959i $$0.390073\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 4.79129 1.59710
$$10$$ 0 0
$$11$$ −0.791288 −0.238582 −0.119291 0.992859i $$-0.538062\pi$$
−0.119291 + 0.992859i $$0.538062\pi$$
$$12$$ 2.79129 0.805775
$$13$$ −5.79129 −1.60621 −0.803107 0.595835i $$-0.796821\pi$$
−0.803107 + 0.595835i $$0.796821\pi$$
$$14$$ 1.79129 0.478742
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −0.791288 −0.191915 −0.0959577 0.995385i $$-0.530591\pi$$
−0.0959577 + 0.995385i $$0.530591\pi$$
$$18$$ 4.79129 1.12932
$$19$$ 5.79129 1.32861 0.664306 0.747460i $$-0.268727\pi$$
0.664306 + 0.747460i $$0.268727\pi$$
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ −0.791288 −0.168703
$$23$$ −1.00000 −0.208514
$$24$$ 2.79129 0.569769
$$25$$ 0 0
$$26$$ −5.79129 −1.13576
$$27$$ 5.00000 0.962250
$$28$$ 1.79129 0.338522
$$29$$ 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 $$-0.597982\pi$$
−0.302982 + 0.952996i $$0.597982\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.20871 −0.384487
$$34$$ −0.791288 −0.135705
$$35$$ 0 0
$$36$$ 4.79129 0.798548
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 5.79129 0.939471
$$39$$ −16.1652 −2.58850
$$40$$ 0 0
$$41$$ −6.79129 −1.06062 −0.530310 0.847804i $$-0.677925\pi$$
−0.530310 + 0.847804i $$0.677925\pi$$
$$42$$ 5.00000 0.771517
$$43$$ −11.1652 −1.70267 −0.851335 0.524623i $$-0.824206\pi$$
−0.851335 + 0.524623i $$0.824206\pi$$
$$44$$ −0.791288 −0.119291
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 4.41742 0.644348 0.322174 0.946681i $$-0.395586\pi$$
0.322174 + 0.946681i $$0.395586\pi$$
$$48$$ 2.79129 0.402888
$$49$$ −3.79129 −0.541613
$$50$$ 0 0
$$51$$ −2.20871 −0.309282
$$52$$ −5.79129 −0.803107
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 0 0
$$56$$ 1.79129 0.239371
$$57$$ 16.1652 2.14113
$$58$$ 7.58258 0.995641
$$59$$ −13.5826 −1.76830 −0.884150 0.467202i $$-0.845262\pi$$
−0.884150 + 0.467202i $$0.845262\pi$$
$$60$$ 0 0
$$61$$ 10.3739 1.32824 0.664119 0.747627i $$-0.268807\pi$$
0.664119 + 0.747627i $$0.268807\pi$$
$$62$$ −3.37386 −0.428481
$$63$$ 8.58258 1.08130
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.20871 −0.271874
$$67$$ −11.1652 −1.36404 −0.682020 0.731333i $$-0.738898\pi$$
−0.682020 + 0.731333i $$0.738898\pi$$
$$68$$ −0.791288 −0.0959577
$$69$$ −2.79129 −0.336032
$$70$$ 0 0
$$71$$ 8.37386 0.993795 0.496897 0.867809i $$-0.334473\pi$$
0.496897 + 0.867809i $$0.334473\pi$$
$$72$$ 4.79129 0.564659
$$73$$ −12.7477 −1.49201 −0.746004 0.665941i $$-0.768030\pi$$
−0.746004 + 0.665941i $$0.768030\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 5.79129 0.664306
$$77$$ −1.41742 −0.161530
$$78$$ −16.1652 −1.83034
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ −0.417424 −0.0463805
$$82$$ −6.79129 −0.749972
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 5.00000 0.545545
$$85$$ 0 0
$$86$$ −11.1652 −1.20397
$$87$$ 21.1652 2.26914
$$88$$ −0.791288 −0.0843516
$$89$$ 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
$$92$$ −1.00000 −0.104257
$$93$$ −9.41742 −0.976541
$$94$$ 4.41742 0.455623
$$95$$ 0 0
$$96$$ 2.79129 0.284885
$$97$$ 7.95644 0.807854 0.403927 0.914791i $$-0.367645\pi$$
0.403927 + 0.914791i $$0.367645\pi$$
$$98$$ −3.79129 −0.382978
$$99$$ −3.79129 −0.381039
$$100$$ 0 0
$$101$$ 4.41742 0.439550 0.219775 0.975551i $$-0.429468\pi$$
0.219775 + 0.975551i $$0.429468\pi$$
$$102$$ −2.20871 −0.218695
$$103$$ 6.37386 0.628035 0.314018 0.949417i $$-0.398325\pi$$
0.314018 + 0.949417i $$0.398325\pi$$
$$104$$ −5.79129 −0.567882
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −4.41742 −0.427049 −0.213524 0.976938i $$-0.568494\pi$$
−0.213524 + 0.976938i $$0.568494\pi$$
$$108$$ 5.00000 0.481125
$$109$$ −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
$$112$$ 1.79129 0.169261
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 16.1652 1.51401
$$115$$ 0 0
$$116$$ 7.58258 0.704024
$$117$$ −27.7477 −2.56528
$$118$$ −13.5826 −1.25038
$$119$$ −1.41742 −0.129935
$$120$$ 0 0
$$121$$ −10.3739 −0.943079
$$122$$ 10.3739 0.939205
$$123$$ −18.9564 −1.70924
$$124$$ −3.37386 −0.302982
$$125$$ 0 0
$$126$$ 8.58258 0.764597
$$127$$ −12.7477 −1.13118 −0.565589 0.824687i $$-0.691351\pi$$
−0.565589 + 0.824687i $$0.691351\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −31.1652 −2.74394
$$130$$ 0 0
$$131$$ −9.16515 −0.800763 −0.400381 0.916349i $$-0.631122\pi$$
−0.400381 + 0.916349i $$0.631122\pi$$
$$132$$ −2.20871 −0.192244
$$133$$ 10.3739 0.899528
$$134$$ −11.1652 −0.964522
$$135$$ 0 0
$$136$$ −0.791288 −0.0678524
$$137$$ −3.79129 −0.323912 −0.161956 0.986798i $$-0.551780\pi$$
−0.161956 + 0.986798i $$0.551780\pi$$
$$138$$ −2.79129 −0.237610
$$139$$ 12.7477 1.08125 0.540624 0.841264i $$-0.318188\pi$$
0.540624 + 0.841264i $$0.318188\pi$$
$$140$$ 0 0
$$141$$ 12.3303 1.03840
$$142$$ 8.37386 0.702719
$$143$$ 4.58258 0.383214
$$144$$ 4.79129 0.399274
$$145$$ 0 0
$$146$$ −12.7477 −1.05501
$$147$$ −10.5826 −0.872836
$$148$$ 4.00000 0.328798
$$149$$ 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 $$-0.644700\pi$$
−0.439091 + 0.898442i $$0.644700\pi$$
$$152$$ 5.79129 0.469735
$$153$$ −3.79129 −0.306507
$$154$$ −1.41742 −0.114219
$$155$$ 0 0
$$156$$ −16.1652 −1.29425
$$157$$ 14.7477 1.17700 0.588498 0.808498i $$-0.299719\pi$$
0.588498 + 0.808498i $$0.299719\pi$$
$$158$$ 8.00000 0.636446
$$159$$ −16.7477 −1.32818
$$160$$ 0 0
$$161$$ −1.79129 −0.141173
$$162$$ −0.417424 −0.0327960
$$163$$ −8.62614 −0.675651 −0.337826 0.941209i $$-0.609691\pi$$
−0.337826 + 0.941209i $$0.609691\pi$$
$$164$$ −6.79129 −0.530310
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ −18.3303 −1.41844 −0.709221 0.704987i $$-0.750953\pi$$
−0.709221 + 0.704987i $$0.750953\pi$$
$$168$$ 5.00000 0.385758
$$169$$ 20.5390 1.57992
$$170$$ 0 0
$$171$$ 27.7477 2.12192
$$172$$ −11.1652 −0.851335
$$173$$ 18.7913 1.42868 0.714338 0.699801i $$-0.246728\pi$$
0.714338 + 0.699801i $$0.246728\pi$$
$$174$$ 21.1652 1.60453
$$175$$ 0 0
$$176$$ −0.791288 −0.0596456
$$177$$ −37.9129 −2.84971
$$178$$ 15.1652 1.13668
$$179$$ 10.7477 0.803323 0.401661 0.915788i $$-0.368433\pi$$
0.401661 + 0.915788i $$0.368433\pi$$
$$180$$ 0 0
$$181$$ −18.5390 −1.37799 −0.688997 0.724764i $$-0.741949\pi$$
−0.688997 + 0.724764i $$0.741949\pi$$
$$182$$ −10.3739 −0.768962
$$183$$ 28.9564 2.14052
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ −9.41742 −0.690519
$$187$$ 0.626136 0.0457876
$$188$$ 4.41742 0.322174
$$189$$ 8.95644 0.651485
$$190$$ 0 0
$$191$$ 25.5826 1.85109 0.925545 0.378637i $$-0.123607\pi$$
0.925545 + 0.378637i $$0.123607\pi$$
$$192$$ 2.79129 0.201444
$$193$$ 20.7477 1.49345 0.746727 0.665131i $$-0.231624\pi$$
0.746727 + 0.665131i $$0.231624\pi$$
$$194$$ 7.95644 0.571239
$$195$$ 0 0
$$196$$ −3.79129 −0.270806
$$197$$ 11.5390 0.822121 0.411060 0.911608i $$-0.365159\pi$$
0.411060 + 0.911608i $$0.365159\pi$$
$$198$$ −3.79129 −0.269435
$$199$$ −16.3303 −1.15762 −0.578812 0.815461i $$-0.696484\pi$$
−0.578812 + 0.815461i $$0.696484\pi$$
$$200$$ 0 0
$$201$$ −31.1652 −2.19822
$$202$$ 4.41742 0.310809
$$203$$ 13.5826 0.953310
$$204$$ −2.20871 −0.154641
$$205$$ 0 0
$$206$$ 6.37386 0.444088
$$207$$ −4.79129 −0.333018
$$208$$ −5.79129 −0.401554
$$209$$ −4.58258 −0.316983
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 23.3739 1.60155
$$214$$ −4.41742 −0.301969
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ −6.04356 −0.410264
$$218$$ −3.37386 −0.228507
$$219$$ −35.5826 −2.40445
$$220$$ 0 0
$$221$$ 4.58258 0.308257
$$222$$ 11.1652 0.749356
$$223$$ 7.16515 0.479814 0.239907 0.970796i $$-0.422883\pi$$
0.239907 + 0.970796i $$0.422883\pi$$
$$224$$ 1.79129 0.119685
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 22.7477 1.50982 0.754910 0.655829i $$-0.227681\pi$$
0.754910 + 0.655829i $$0.227681\pi$$
$$228$$ 16.1652 1.07056
$$229$$ 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
$$232$$ 7.58258 0.497820
$$233$$ 1.58258 0.103678 0.0518390 0.998655i $$-0.483492\pi$$
0.0518390 + 0.998655i $$0.483492\pi$$
$$234$$ −27.7477 −1.81393
$$235$$ 0 0
$$236$$ −13.5826 −0.884150
$$237$$ 22.3303 1.45051
$$238$$ −1.41742 −0.0918780
$$239$$ −15.1652 −0.980952 −0.490476 0.871455i $$-0.663177\pi$$
−0.490476 + 0.871455i $$0.663177\pi$$
$$240$$ 0 0
$$241$$ −28.0000 −1.80364 −0.901819 0.432113i $$-0.857768\pi$$
−0.901819 + 0.432113i $$0.857768\pi$$
$$242$$ −10.3739 −0.666857
$$243$$ −16.1652 −1.03699
$$244$$ 10.3739 0.664119
$$245$$ 0 0
$$246$$ −18.9564 −1.20862
$$247$$ −33.5390 −2.13404
$$248$$ −3.37386 −0.214241
$$249$$ 16.7477 1.06134
$$250$$ 0 0
$$251$$ 26.2087 1.65428 0.827140 0.561996i $$-0.189967\pi$$
0.827140 + 0.561996i $$0.189967\pi$$
$$252$$ 8.58258 0.540651
$$253$$ 0.791288 0.0497478
$$254$$ −12.7477 −0.799864
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 4.74773 0.296155 0.148078 0.988976i $$-0.452691\pi$$
0.148078 + 0.988976i $$0.452691\pi$$
$$258$$ −31.1652 −1.94026
$$259$$ 7.16515 0.445221
$$260$$ 0 0
$$261$$ 36.3303 2.24879
$$262$$ −9.16515 −0.566225
$$263$$ −11.2087 −0.691159 −0.345579 0.938390i $$-0.612318\pi$$
−0.345579 + 0.938390i $$0.612318\pi$$
$$264$$ −2.20871 −0.135937
$$265$$ 0 0
$$266$$ 10.3739 0.636062
$$267$$ 42.3303 2.59057
$$268$$ −11.1652 −0.682020
$$269$$ −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 $$-0.314474\pi$$
0.550404 + 0.834898i $$0.314474\pi$$
$$272$$ −0.791288 −0.0479789
$$273$$ −28.9564 −1.75252
$$274$$ −3.79129 −0.229040
$$275$$ 0 0
$$276$$ −2.79129 −0.168016
$$277$$ −17.1652 −1.03135 −0.515677 0.856783i $$-0.672460\pi$$
−0.515677 + 0.856783i $$0.672460\pi$$
$$278$$ 12.7477 0.764558
$$279$$ −16.1652 −0.967782
$$280$$ 0 0
$$281$$ 10.7477 0.641156 0.320578 0.947222i $$-0.396123\pi$$
0.320578 + 0.947222i $$0.396123\pi$$
$$282$$ 12.3303 0.734259
$$283$$ −8.33030 −0.495185 −0.247593 0.968864i $$-0.579639\pi$$
−0.247593 + 0.968864i $$0.579639\pi$$
$$284$$ 8.37386 0.496897
$$285$$ 0 0
$$286$$ 4.58258 0.270973
$$287$$ −12.1652 −0.718086
$$288$$ 4.79129 0.282329
$$289$$ −16.3739 −0.963168
$$290$$ 0 0
$$291$$ 22.2087 1.30190
$$292$$ −12.7477 −0.746004
$$293$$ −27.4955 −1.60630 −0.803151 0.595776i $$-0.796845\pi$$
−0.803151 + 0.595776i $$0.796845\pi$$
$$294$$ −10.5826 −0.617188
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ −3.95644 −0.229576
$$298$$ 8.20871 0.475518
$$299$$ 5.79129 0.334919
$$300$$ 0 0
$$301$$ −20.0000 −1.15278
$$302$$ −10.7913 −0.620969
$$303$$ 12.3303 0.708357
$$304$$ 5.79129 0.332153
$$305$$ 0 0
$$306$$ −3.79129 −0.216734
$$307$$ 15.5390 0.886858 0.443429 0.896309i $$-0.353762\pi$$
0.443429 + 0.896309i $$0.353762\pi$$
$$308$$ −1.41742 −0.0807652
$$309$$ 17.7913 1.01211
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ −16.1652 −0.915171
$$313$$ 4.62614 0.261485 0.130742 0.991416i $$-0.458264\pi$$
0.130742 + 0.991416i $$0.458264\pi$$
$$314$$ 14.7477 0.832262
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ −9.79129 −0.549934 −0.274967 0.961454i $$-0.588667\pi$$
−0.274967 + 0.961454i $$0.588667\pi$$
$$318$$ −16.7477 −0.939166
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ −12.3303 −0.688210
$$322$$ −1.79129 −0.0998246
$$323$$ −4.58258 −0.254981
$$324$$ −0.417424 −0.0231902
$$325$$ 0 0
$$326$$ −8.62614 −0.477758
$$327$$ −9.41742 −0.520785
$$328$$ −6.79129 −0.374986
$$329$$ 7.91288 0.436251
$$330$$ 0 0
$$331$$ −20.7477 −1.14040 −0.570199 0.821507i $$-0.693134\pi$$
−0.570199 + 0.821507i $$0.693134\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 19.1652 1.05024
$$334$$ −18.3303 −1.00299
$$335$$ 0 0
$$336$$ 5.00000 0.272772
$$337$$ 12.2087 0.665051 0.332525 0.943094i $$-0.392099\pi$$
0.332525 + 0.943094i $$0.392099\pi$$
$$338$$ 20.5390 1.11718
$$339$$ −16.7477 −0.909612
$$340$$ 0 0
$$341$$ 2.66970 0.144572
$$342$$ 27.7477 1.50043
$$343$$ −19.3303 −1.04374
$$344$$ −11.1652 −0.601985
$$345$$ 0 0
$$346$$ 18.7913 1.01023
$$347$$ −5.20871 −0.279618 −0.139809 0.990178i $$-0.544649\pi$$
−0.139809 + 0.990178i $$0.544649\pi$$
$$348$$ 21.1652 1.13457
$$349$$ 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
$$352$$ −0.791288 −0.0421758
$$353$$ −3.16515 −0.168464 −0.0842320 0.996446i $$-0.526844\pi$$
−0.0842320 + 0.996446i $$0.526844\pi$$
$$354$$ −37.9129 −2.01505
$$355$$ 0 0
$$356$$ 15.1652 0.803751
$$357$$ −3.95644 −0.209397
$$358$$ 10.7477 0.568035
$$359$$ 9.16515 0.483718 0.241859 0.970311i $$-0.422243\pi$$
0.241859 + 0.970311i $$0.422243\pi$$
$$360$$ 0 0
$$361$$ 14.5390 0.765211
$$362$$ −18.5390 −0.974389
$$363$$ −28.9564 −1.51982
$$364$$ −10.3739 −0.543738
$$365$$ 0 0
$$366$$ 28.9564 1.51358
$$367$$ 19.1652 1.00041 0.500206 0.865906i $$-0.333257\pi$$
0.500206 + 0.865906i $$0.333257\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −32.5390 −1.69391
$$370$$ 0 0
$$371$$ −10.7477 −0.557994
$$372$$ −9.41742 −0.488271
$$373$$ −12.7477 −0.660052 −0.330026 0.943972i $$-0.607058\pi$$
−0.330026 + 0.943972i $$0.607058\pi$$
$$374$$ 0.626136 0.0323767
$$375$$ 0 0
$$376$$ 4.41742 0.227811
$$377$$ −43.9129 −2.26163
$$378$$ 8.95644 0.460670
$$379$$ −6.37386 −0.327403 −0.163702 0.986510i $$-0.552343\pi$$
−0.163702 + 0.986510i $$0.552343\pi$$
$$380$$ 0 0
$$381$$ −35.5826 −1.82295
$$382$$ 25.5826 1.30892
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 2.79129 0.142442
$$385$$ 0 0
$$386$$ 20.7477 1.05603
$$387$$ −53.4955 −2.71933
$$388$$ 7.95644 0.403927
$$389$$ −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
$$392$$ −3.79129 −0.191489
$$393$$ −25.5826 −1.29047
$$394$$ 11.5390 0.581327
$$395$$ 0 0
$$396$$ −3.79129 −0.190519
$$397$$ 15.5390 0.779881 0.389940 0.920840i $$-0.372496\pi$$
0.389940 + 0.920840i $$0.372496\pi$$
$$398$$ −16.3303 −0.818564
$$399$$ 28.9564 1.44964
$$400$$ 0 0
$$401$$ −4.74773 −0.237090 −0.118545 0.992949i $$-0.537823\pi$$
−0.118545 + 0.992949i $$0.537823\pi$$
$$402$$ −31.1652 −1.55438
$$403$$ 19.5390 0.973308
$$404$$ 4.41742 0.219775
$$405$$ 0 0
$$406$$ 13.5826 0.674092
$$407$$ −3.16515 −0.156891
$$408$$ −2.20871 −0.109348
$$409$$ −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
$$412$$ 6.37386 0.314018
$$413$$ −24.3303 −1.19722
$$414$$ −4.79129 −0.235479
$$415$$ 0 0
$$416$$ −5.79129 −0.283941
$$417$$ 35.5826 1.74249
$$418$$ −4.58258 −0.224141
$$419$$ 20.8348 1.01785 0.508924 0.860811i $$-0.330043\pi$$
0.508924 + 0.860811i $$0.330043\pi$$
$$420$$ 0 0
$$421$$ 18.1216 0.883192 0.441596 0.897214i $$-0.354412\pi$$
0.441596 + 0.897214i $$0.354412\pi$$
$$422$$ −10.0000 −0.486792
$$423$$ 21.1652 1.02908
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 23.3739 1.13247
$$427$$ 18.5826 0.899274
$$428$$ −4.41742 −0.213524
$$429$$ 12.7913 0.617569
$$430$$ 0 0
$$431$$ 25.9129 1.24818 0.624090 0.781353i $$-0.285470\pi$$
0.624090 + 0.781353i $$0.285470\pi$$
$$432$$ 5.00000 0.240563
$$433$$ 30.5390 1.46761 0.733806 0.679359i $$-0.237742\pi$$
0.733806 + 0.679359i $$0.237742\pi$$
$$434$$ −6.04356 −0.290100
$$435$$ 0 0
$$436$$ −3.37386 −0.161579
$$437$$ −5.79129 −0.277035
$$438$$ −35.5826 −1.70020
$$439$$ −6.53901 −0.312090 −0.156045 0.987750i $$-0.549875\pi$$
−0.156045 + 0.987750i $$0.549875\pi$$
$$440$$ 0 0
$$441$$ −18.1652 −0.865007
$$442$$ 4.58258 0.217971
$$443$$ 39.7913 1.89054 0.945271 0.326288i $$-0.105798\pi$$
0.945271 + 0.326288i $$0.105798\pi$$
$$444$$ 11.1652 0.529875
$$445$$ 0 0
$$446$$ 7.16515 0.339280
$$447$$ 22.9129 1.08374
$$448$$ 1.79129 0.0846304
$$449$$ −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
$$452$$ −6.00000 −0.282216
$$453$$ −30.1216 −1.41524
$$454$$ 22.7477 1.06760
$$455$$ 0 0
$$456$$ 16.1652 0.757003
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 20.3303 0.949973
$$459$$ −3.95644 −0.184671
$$460$$ 0 0
$$461$$ −28.7477 −1.33892 −0.669458 0.742850i $$-0.733473\pi$$
−0.669458 + 0.742850i $$0.733473\pi$$
$$462$$ −3.95644 −0.184070
$$463$$ 10.0000 0.464739 0.232370 0.972628i $$-0.425352\pi$$
0.232370 + 0.972628i $$0.425352\pi$$
$$464$$ 7.58258 0.352012
$$465$$ 0 0
$$466$$ 1.58258 0.0733114
$$467$$ 19.9129 0.921458 0.460729 0.887541i $$-0.347588\pi$$
0.460729 + 0.887541i $$0.347588\pi$$
$$468$$ −27.7477 −1.28264
$$469$$ −20.0000 −0.923514
$$470$$ 0 0
$$471$$ 41.1652 1.89679
$$472$$ −13.5826 −0.625189
$$473$$ 8.83485 0.406227
$$474$$ 22.3303 1.02566
$$475$$ 0 0
$$476$$ −1.41742 −0.0649675
$$477$$ −28.7477 −1.31627
$$478$$ −15.1652 −0.693638
$$479$$ 15.4955 0.708005 0.354003 0.935244i $$-0.384820\pi$$
0.354003 + 0.935244i $$0.384820\pi$$
$$480$$ 0 0
$$481$$ −23.1652 −1.05624
$$482$$ −28.0000 −1.27537
$$483$$ −5.00000 −0.227508
$$484$$ −10.3739 −0.471539
$$485$$ 0 0
$$486$$ −16.1652 −0.733266
$$487$$ −6.41742 −0.290801 −0.145401 0.989373i $$-0.546447\pi$$
−0.145401 + 0.989373i $$0.546447\pi$$
$$488$$ 10.3739 0.469603
$$489$$ −24.0780 −1.08885
$$490$$ 0 0
$$491$$ 10.7477 0.485038 0.242519 0.970147i $$-0.422026\pi$$
0.242519 + 0.970147i $$0.422026\pi$$
$$492$$ −18.9564 −0.854622
$$493$$ −6.00000 −0.270226
$$494$$ −33.5390 −1.50899
$$495$$ 0 0
$$496$$ −3.37386 −0.151491
$$497$$ 15.0000 0.672842
$$498$$ 16.7477 0.750484
$$499$$ 4.83485 0.216438 0.108219 0.994127i $$-0.465485\pi$$
0.108219 + 0.994127i $$0.465485\pi$$
$$500$$ 0 0
$$501$$ −51.1652 −2.28589
$$502$$ 26.2087 1.16975
$$503$$ −14.2087 −0.633535 −0.316768 0.948503i $$-0.602598\pi$$
−0.316768 + 0.948503i $$0.602598\pi$$
$$504$$ 8.58258 0.382298
$$505$$ 0 0
$$506$$ 0.791288 0.0351770
$$507$$ 57.3303 2.54613
$$508$$ −12.7477 −0.565589
$$509$$ 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
$$512$$ 1.00000 0.0441942
$$513$$ 28.9564 1.27846
$$514$$ 4.74773 0.209413
$$515$$ 0 0
$$516$$ −31.1652 −1.37197
$$517$$ −3.49545 −0.153730
$$518$$ 7.16515 0.314819
$$519$$ 52.4519 2.30238
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 36.3303 1.59013
$$523$$ −17.1652 −0.750580 −0.375290 0.926908i $$-0.622457\pi$$
−0.375290 + 0.926908i $$0.622457\pi$$
$$524$$ −9.16515 −0.400381
$$525$$ 0 0
$$526$$ −11.2087 −0.488723
$$527$$ 2.66970 0.116294
$$528$$ −2.20871 −0.0961219
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −65.0780 −2.82415
$$532$$ 10.3739 0.449764
$$533$$ 39.3303 1.70358
$$534$$ 42.3303 1.83181
$$535$$ 0 0
$$536$$ −11.1652 −0.482261
$$537$$ 30.0000 1.29460
$$538$$ −10.7477 −0.463367
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 1.66970 0.0717859 0.0358929 0.999356i $$-0.488572\pi$$
0.0358929 + 0.999356i $$0.488572\pi$$
$$542$$ 18.1216 0.778389
$$543$$ −51.7477 −2.22071
$$544$$ −0.791288 −0.0339262
$$545$$ 0 0
$$546$$ −28.9564 −1.23922
$$547$$ 26.1216 1.11688 0.558439 0.829545i $$-0.311400\pi$$
0.558439 + 0.829545i $$0.311400\pi$$
$$548$$ −3.79129 −0.161956
$$549$$ 49.7042 2.12132
$$550$$ 0 0
$$551$$ 43.9129 1.87075
$$552$$ −2.79129 −0.118805
$$553$$ 14.3303 0.609386
$$554$$ −17.1652 −0.729277
$$555$$ 0 0
$$556$$ 12.7477 0.540624
$$557$$ 6.33030 0.268224 0.134112 0.990966i $$-0.457182\pi$$
0.134112 + 0.990966i $$0.457182\pi$$
$$558$$ −16.1652 −0.684325
$$559$$ 64.6606 2.73485
$$560$$ 0 0
$$561$$ 1.74773 0.0737891
$$562$$ 10.7477 0.453366
$$563$$ 15.1652 0.639135 0.319567 0.947564i $$-0.396462\pi$$
0.319567 + 0.947564i $$0.396462\pi$$
$$564$$ 12.3303 0.519199
$$565$$ 0 0
$$566$$ −8.33030 −0.350149
$$567$$ −0.747727 −0.0314016
$$568$$ 8.37386 0.351360
$$569$$ −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 $$-0.574760\pi$$
−0.232712 + 0.972546i $$0.574760\pi$$
$$572$$ 4.58258 0.191607
$$573$$ 71.4083 2.98313
$$574$$ −12.1652 −0.507764
$$575$$ 0 0
$$576$$ 4.79129 0.199637
$$577$$ −41.1652 −1.71373 −0.856864 0.515543i $$-0.827590\pi$$
−0.856864 + 0.515543i $$0.827590\pi$$
$$578$$ −16.3739 −0.681063
$$579$$ 57.9129 2.40678
$$580$$ 0 0
$$581$$ 10.7477 0.445891
$$582$$ 22.2087 0.920581
$$583$$ 4.74773 0.196631
$$584$$ −12.7477 −0.527505
$$585$$ 0 0
$$586$$ −27.4955 −1.13583
$$587$$ 30.7913 1.27089 0.635446 0.772145i $$-0.280816\pi$$
0.635446 + 0.772145i $$0.280816\pi$$
$$588$$ −10.5826 −0.436418
$$589$$ −19.5390 −0.805091
$$590$$ 0 0
$$591$$ 32.2087 1.32489
$$592$$ 4.00000 0.164399
$$593$$ 31.9129 1.31050 0.655252 0.755410i $$-0.272562\pi$$
0.655252 + 0.755410i $$0.272562\pi$$
$$594$$ −3.95644 −0.162335
$$595$$ 0 0
$$596$$ 8.20871 0.336242
$$597$$ −45.5826 −1.86557
$$598$$ 5.79129 0.236823
$$599$$ 1.12159 0.0458270 0.0229135 0.999737i $$-0.492706\pi$$
0.0229135 + 0.999737i $$0.492706\pi$$
$$600$$ 0 0
$$601$$ −18.2087 −0.742749 −0.371374 0.928483i $$-0.621113\pi$$
−0.371374 + 0.928483i $$0.621113\pi$$
$$602$$ −20.0000 −0.815139
$$603$$ −53.4955 −2.17850
$$604$$ −10.7913 −0.439091
$$605$$ 0 0
$$606$$ 12.3303 0.500884
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 5.79129 0.234868
$$609$$ 37.9129 1.53631
$$610$$ 0 0
$$611$$ −25.5826 −1.03496
$$612$$ −3.79129 −0.153254
$$613$$ −14.0000 −0.565455 −0.282727 0.959200i $$-0.591239\pi$$
−0.282727 + 0.959200i $$0.591239\pi$$
$$614$$ 15.5390 0.627104
$$615$$ 0 0
$$616$$ −1.41742 −0.0571097
$$617$$ 23.8693 0.960943 0.480471 0.877010i $$-0.340466\pi$$
0.480471 + 0.877010i $$0.340466\pi$$
$$618$$ 17.7913 0.715671
$$619$$ −1.79129 −0.0719979 −0.0359990 0.999352i $$-0.511461\pi$$
−0.0359990 + 0.999352i $$0.511461\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ 12.0000 0.481156
$$623$$ 27.1652 1.08835
$$624$$ −16.1652 −0.647124
$$625$$ 0 0
$$626$$ 4.62614 0.184898
$$627$$ −12.7913 −0.510835
$$628$$ 14.7477 0.588498
$$629$$ −3.16515 −0.126203
$$630$$ 0 0
$$631$$ 27.9129 1.11119 0.555597 0.831452i $$-0.312490\pi$$
0.555597 + 0.831452i $$0.312490\pi$$
$$632$$ 8.00000 0.318223
$$633$$ −27.9129 −1.10944
$$634$$ −9.79129 −0.388862
$$635$$ 0 0
$$636$$ −16.7477 −0.664091
$$637$$ 21.9564 0.869946
$$638$$ −6.00000 −0.237542
$$639$$ 40.1216 1.58719
$$640$$ 0 0
$$641$$ 15.1652 0.598987 0.299494 0.954098i $$-0.403182\pi$$
0.299494 + 0.954098i $$0.403182\pi$$
$$642$$ −12.3303 −0.486638
$$643$$ −6.74773 −0.266104 −0.133052 0.991109i $$-0.542478\pi$$
−0.133052 + 0.991109i $$0.542478\pi$$
$$644$$ −1.79129 −0.0705866
$$645$$ 0 0
$$646$$ −4.58258 −0.180299
$$647$$ −21.1652 −0.832088 −0.416044 0.909344i $$-0.636584\pi$$
−0.416044 + 0.909344i $$0.636584\pi$$
$$648$$ −0.417424 −0.0163980
$$649$$ 10.7477 0.421885
$$650$$ 0 0
$$651$$ −16.8693 −0.661161
$$652$$ −8.62614 −0.337826
$$653$$ −3.46099 −0.135439 −0.0677194 0.997704i $$-0.521572\pi$$
−0.0677194 + 0.997704i $$0.521572\pi$$
$$654$$ −9.41742 −0.368250
$$655$$ 0 0
$$656$$ −6.79129 −0.265155
$$657$$ −61.0780 −2.38288
$$658$$ 7.91288 0.308476
$$659$$ −8.83485 −0.344157 −0.172078 0.985083i $$-0.555048\pi$$
−0.172078 + 0.985083i $$0.555048\pi$$
$$660$$ 0 0
$$661$$ −25.6261 −0.996741 −0.498371 0.866964i $$-0.666068\pi$$
−0.498371 + 0.866964i $$0.666068\pi$$
$$662$$ −20.7477 −0.806383
$$663$$ 12.7913 0.496772
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 19.1652 0.742635
$$667$$ −7.58258 −0.293599
$$668$$ −18.3303 −0.709221
$$669$$ 20.0000 0.773245
$$670$$ 0 0
$$671$$ −8.20871 −0.316894
$$672$$ 5.00000 0.192879
$$673$$ −38.0000 −1.46479 −0.732396 0.680879i $$-0.761598\pi$$
−0.732396 + 0.680879i $$0.761598\pi$$
$$674$$ 12.2087 0.470262
$$675$$ 0 0
$$676$$ 20.5390 0.789962
$$677$$ −42.6606 −1.63958 −0.819790 0.572664i $$-0.805910\pi$$
−0.819790 + 0.572664i $$0.805910\pi$$
$$678$$ −16.7477 −0.643193
$$679$$ 14.2523 0.546952
$$680$$ 0 0
$$681$$ 63.4955 2.43315
$$682$$ 2.66970 0.102228
$$683$$ 11.3739 0.435209 0.217604 0.976037i $$-0.430176\pi$$
0.217604 + 0.976037i $$0.430176\pi$$
$$684$$ 27.7477 1.06096
$$685$$ 0 0
$$686$$ −19.3303 −0.738034
$$687$$ 56.7477 2.16506
$$688$$ −11.1652 −0.425667
$$689$$ 34.7477 1.32378
$$690$$ 0 0
$$691$$ 42.7477 1.62620 0.813100 0.582124i $$-0.197778\pi$$
0.813100 + 0.582124i $$0.197778\pi$$
$$692$$ 18.7913 0.714338
$$693$$ −6.79129 −0.257980
$$694$$ −5.20871 −0.197720
$$695$$ 0 0
$$696$$ 21.1652 0.802263
$$697$$ 5.37386 0.203550
$$698$$ 26.0000 0.984115
$$699$$ 4.41742 0.167082
$$700$$ 0 0
$$701$$ 23.3739 0.882819 0.441409 0.897306i $$-0.354479\pi$$
0.441409 + 0.897306i $$0.354479\pi$$
$$702$$ −28.9564 −1.09289
$$703$$ 23.1652 0.873690
$$704$$ −0.791288 −0.0298228
$$705$$ 0 0
$$706$$ −3.16515 −0.119122
$$707$$ 7.91288 0.297594
$$708$$ −37.9129 −1.42485
$$709$$ 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
$$712$$ 15.1652 0.568338
$$713$$ 3.37386 0.126352
$$714$$ −3.95644 −0.148066
$$715$$ 0 0
$$716$$ 10.7477 0.401661
$$717$$ −42.3303 −1.58085
$$718$$ 9.16515 0.342040
$$719$$ 2.53901 0.0946893 0.0473446 0.998879i $$-0.484924\pi$$
0.0473446 + 0.998879i $$0.484924\pi$$
$$720$$ 0 0
$$721$$ 11.4174 0.425207
$$722$$ 14.5390 0.541086
$$723$$ −78.1561 −2.90666
$$724$$ −18.5390 −0.688997
$$725$$ 0 0
$$726$$ −28.9564 −1.07467
$$727$$ −39.1216 −1.45094 −0.725470 0.688254i $$-0.758377\pi$$
−0.725470 + 0.688254i $$0.758377\pi$$
$$728$$ −10.3739 −0.384481
$$729$$ −43.8693 −1.62479
$$730$$ 0 0
$$731$$ 8.83485 0.326769
$$732$$ 28.9564 1.07026
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ 19.1652 0.707399
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 8.83485 0.325436
$$738$$ −32.5390 −1.19778
$$739$$ 8.00000 0.294285 0.147142 0.989115i $$-0.452992\pi$$
0.147142 + 0.989115i $$0.452992\pi$$
$$740$$ 0 0
$$741$$ −93.6170 −3.43911
$$742$$ −10.7477 −0.394561
$$743$$ 12.9564 0.475326 0.237663 0.971348i $$-0.423619\pi$$
0.237663 + 0.971348i $$0.423619\pi$$
$$744$$ −9.41742 −0.345260
$$745$$ 0 0
$$746$$ −12.7477 −0.466727
$$747$$ 28.7477 1.05182
$$748$$ 0.626136 0.0228938
$$749$$ −7.91288 −0.289130
$$750$$ 0 0
$$751$$ −8.74773 −0.319209 −0.159605 0.987181i $$-0.551022\pi$$
−0.159605 + 0.987181i $$0.551022\pi$$
$$752$$ 4.41742 0.161087
$$753$$ 73.1561 2.66596
$$754$$ −43.9129 −1.59921
$$755$$ 0 0
$$756$$ 8.95644 0.325743
$$757$$ 10.3303 0.375461 0.187731 0.982221i $$-0.439887\pi$$
0.187731 + 0.982221i $$0.439887\pi$$
$$758$$ −6.37386 −0.231509
$$759$$ 2.20871 0.0801712
$$760$$ 0 0
$$761$$ −11.0436 −0.400329 −0.200164 0.979762i $$-0.564148\pi$$
−0.200164 + 0.979762i $$0.564148\pi$$
$$762$$ −35.5826 −1.28902
$$763$$ −6.04356 −0.218792
$$764$$ 25.5826 0.925545
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 78.6606 2.84027
$$768$$ 2.79129 0.100722
$$769$$ −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
$$772$$ 20.7477 0.746727
$$773$$ −33.4955 −1.20475 −0.602374 0.798214i $$-0.705778\pi$$
−0.602374 + 0.798214i $$0.705778\pi$$
$$774$$ −53.4955 −1.92285
$$775$$ 0 0
$$776$$ 7.95644 0.285620
$$777$$ 20.0000 0.717496
$$778$$ −20.7042 −0.742280
$$779$$ −39.3303 −1.40915
$$780$$ 0 0
$$781$$ −6.62614 −0.237102
$$782$$ 0.791288 0.0282964
$$783$$ 37.9129 1.35490
$$784$$ −3.79129 −0.135403
$$785$$ 0 0
$$786$$ −25.5826 −0.912500
$$787$$ 17.5826 0.626751 0.313376 0.949629i $$-0.398540\pi$$
0.313376 + 0.949629i $$0.398540\pi$$
$$788$$ 11.5390 0.411060
$$789$$ −31.2867 −1.11384
$$790$$ 0 0
$$791$$ −10.7477 −0.382145
$$792$$ −3.79129 −0.134718
$$793$$ −60.0780 −2.13343
$$794$$ 15.5390 0.551459
$$795$$ 0 0
$$796$$ −16.3303 −0.578812
$$797$$ 4.08712 0.144773 0.0723866 0.997377i $$-0.476938\pi$$
0.0723866 + 0.997377i $$0.476938\pi$$
$$798$$ 28.9564 1.02505
$$799$$ −3.49545 −0.123660
$$800$$ 0 0
$$801$$ 72.6606 2.56734
$$802$$ −4.74773 −0.167648
$$803$$ 10.0871 0.355967
$$804$$ −31.1652 −1.09911
$$805$$ 0 0
$$806$$ 19.5390 0.688232
$$807$$ −30.0000 −1.05605
$$808$$ 4.41742 0.155404
$$809$$ 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 $$-0.511667\pi$$
−0.0366444 + 0.999328i $$0.511667\pi$$
$$812$$ 13.5826 0.476655
$$813$$ 50.5826 1.77401
$$814$$ −3.16515 −0.110938
$$815$$ 0 0
$$816$$ −2.20871 −0.0773204
$$817$$ −64.6606 −2.26219
$$818$$ −18.2087 −0.636653
$$819$$ −49.7042 −1.73680
$$820$$ 0 0
$$821$$ 21.1652 0.738669 0.369334 0.929297i $$-0.379586\pi$$
0.369334 + 0.929297i $$0.379586\pi$$
$$822$$ −10.5826 −0.369110
$$823$$ −22.8348 −0.795973 −0.397986 0.917391i $$-0.630291\pi$$
−0.397986 + 0.917391i $$0.630291\pi$$
$$824$$ 6.37386 0.222044
$$825$$ 0 0
$$826$$ −24.3303 −0.846560
$$827$$ 23.0780 0.802502 0.401251 0.915968i $$-0.368576\pi$$
0.401251 + 0.915968i $$0.368576\pi$$
$$828$$ −4.79129 −0.166509
$$829$$ 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
$$832$$ −5.79129 −0.200777
$$833$$ 3.00000 0.103944
$$834$$ 35.5826 1.23212
$$835$$ 0 0
$$836$$ −4.58258 −0.158492
$$837$$ −16.8693 −0.583089
$$838$$ 20.8348 0.719728
$$839$$ −31.5826 −1.09035 −0.545176 0.838322i $$-0.683537\pi$$
−0.545176 + 0.838322i $$0.683537\pi$$
$$840$$ 0 0
$$841$$ 28.4955 0.982602
$$842$$ 18.1216 0.624511
$$843$$ 30.0000 1.03325
$$844$$ −10.0000 −0.344214
$$845$$ 0 0
$$846$$ 21.1652 0.727673
$$847$$ −18.5826 −0.638505
$$848$$ −6.00000 −0.206041
$$849$$ −23.2523 −0.798016
$$850$$ 0 0
$$851$$ −4.00000 −0.137118
$$852$$ 23.3739 0.800775
$$853$$ −40.5390 −1.38803 −0.694015 0.719961i $$-0.744160\pi$$
−0.694015 + 0.719961i $$0.744160\pi$$
$$854$$ 18.5826 0.635883
$$855$$ 0 0
$$856$$ −4.41742 −0.150984
$$857$$ 9.16515 0.313076 0.156538 0.987672i $$-0.449967\pi$$
0.156538 + 0.987672i $$0.449967\pi$$
$$858$$ 12.7913 0.436687
$$859$$ −26.7477 −0.912621 −0.456310 0.889821i $$-0.650829\pi$$
−0.456310 + 0.889821i $$0.650829\pi$$
$$860$$ 0 0
$$861$$ −33.9564 −1.15723
$$862$$ 25.9129 0.882596
$$863$$ 22.4174 0.763098 0.381549 0.924349i $$-0.375391\pi$$
0.381549 + 0.924349i $$0.375391\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ 30.5390 1.03776
$$867$$ −45.7042 −1.55219
$$868$$ −6.04356 −0.205132
$$869$$ −6.33030 −0.214741
$$870$$ 0 0
$$871$$ 64.6606 2.19094
$$872$$ −3.37386 −0.114253
$$873$$ 38.1216 1.29022
$$874$$ −5.79129 −0.195893
$$875$$ 0 0
$$876$$ −35.5826 −1.20222
$$877$$ 42.7042 1.44202 0.721009 0.692926i $$-0.243679\pi$$
0.721009 + 0.692926i $$0.243679\pi$$
$$878$$ −6.53901 −0.220681
$$879$$ −76.7477 −2.58864
$$880$$ 0 0
$$881$$ 30.3303 1.02185 0.510927 0.859624i $$-0.329302\pi$$
0.510927 + 0.859624i $$0.329302\pi$$
$$882$$ −18.1652 −0.611652
$$883$$ 34.9564 1.17638 0.588189 0.808724i $$-0.299841\pi$$
0.588189 + 0.808724i $$0.299841\pi$$
$$884$$ 4.58258 0.154129
$$885$$ 0 0
$$886$$ 39.7913 1.33681
$$887$$ −15.1652 −0.509196 −0.254598 0.967047i $$-0.581943\pi$$
−0.254598 + 0.967047i $$0.581943\pi$$
$$888$$ 11.1652 0.374678
$$889$$ −22.8348 −0.765856
$$890$$ 0 0
$$891$$ 0.330303 0.0110656
$$892$$ 7.16515 0.239907
$$893$$ 25.5826 0.856088
$$894$$ 22.9129 0.766321
$$895$$ 0 0
$$896$$ 1.79129 0.0598427
$$897$$ 16.1652 0.539739
$$898$$ −16.1216 −0.537984
$$899$$ −25.5826 −0.853227
$$900$$ 0 0
$$901$$ 4.74773 0.158170
$$902$$ 5.37386 0.178930
$$903$$ −55.8258 −1.85776
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −30.1216 −1.00072
$$907$$ −6.74773 −0.224055 −0.112027 0.993705i $$-0.535734\pi$$
−0.112027 + 0.993705i $$0.535734\pi$$
$$908$$ 22.7477 0.754910
$$909$$ 21.1652 0.702004
$$910$$ 0 0
$$911$$ 4.41742 0.146356 0.0731779 0.997319i $$-0.476686\pi$$
0.0731779 + 0.997319i $$0.476686\pi$$
$$912$$ 16.1652 0.535282
$$913$$ −4.74773 −0.157127
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ 20.3303 0.671732
$$917$$ −16.4174 −0.542151
$$918$$ −3.95644 −0.130582
$$919$$ −55.1652 −1.81973 −0.909865 0.414904i $$-0.863815\pi$$
−0.909865 + 0.414904i $$0.863815\pi$$
$$920$$ 0 0
$$921$$ 43.3739 1.42922
$$922$$ −28.7477 −0.946756
$$923$$ −48.4955 −1.59625
$$924$$ −3.95644 −0.130157
$$925$$ 0 0
$$926$$ 10.0000 0.328620
$$927$$ 30.5390 1.00303
$$928$$ 7.58258 0.248910
$$929$$ 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
$$932$$ 1.58258 0.0518390
$$933$$ 33.4955 1.09659
$$934$$ 19.9129 0.651569
$$935$$ 0 0
$$936$$ −27.7477 −0.906963
$$937$$ −44.6261 −1.45787 −0.728936 0.684582i $$-0.759985\pi$$
−0.728936 + 0.684582i $$0.759985\pi$$
$$938$$ −20.0000 −0.653023
$$939$$ 12.9129 0.421396
$$940$$ 0 0
$$941$$ 32.0436 1.04459 0.522295 0.852765i $$-0.325076\pi$$
0.522295 + 0.852765i $$0.325076\pi$$
$$942$$ 41.1652 1.34123
$$943$$ 6.79129 0.221155
$$944$$ −13.5826 −0.442075
$$945$$ 0 0
$$946$$ 8.83485 0.287246
$$947$$ −2.53901 −0.0825069 −0.0412534 0.999149i $$-0.513135\pi$$
−0.0412534 + 0.999149i $$0.513135\pi$$
$$948$$ 22.3303 0.725255
$$949$$ 73.8258 2.39649
$$950$$ 0 0
$$951$$ −27.3303 −0.886246
$$952$$ −1.41742 −0.0459390
$$953$$ −5.53901 −0.179426 −0.0897131 0.995968i $$-0.528595\pi$$
−0.0897131 + 0.995968i $$0.528595\pi$$
$$954$$ −28.7477 −0.930742
$$955$$ 0 0
$$956$$ −15.1652 −0.490476
$$957$$ −16.7477 −0.541377
$$958$$ 15.4955 0.500635
$$959$$ −6.79129 −0.219302
$$960$$ 0 0
$$961$$ −19.6170 −0.632808
$$962$$ −23.1652 −0.746874
$$963$$ −21.1652 −0.682037
$$964$$ −28.0000 −0.901819
$$965$$ 0 0
$$966$$ −5.00000 −0.160872
$$967$$ 32.7477 1.05310 0.526548 0.850145i $$-0.323486\pi$$
0.526548 + 0.850145i $$0.323486\pi$$
$$968$$ −10.3739 −0.333429
$$969$$ −12.7913 −0.410915
$$970$$ 0 0
$$971$$ 15.9564 0.512067 0.256033 0.966668i $$-0.417584\pi$$
0.256033 + 0.966668i $$0.417584\pi$$
$$972$$ −16.1652 −0.518497
$$973$$ 22.8348 0.732052
$$974$$ −6.41742 −0.205628
$$975$$ 0 0
$$976$$ 10.3739 0.332059
$$977$$ 34.1216 1.09165 0.545823 0.837900i $$-0.316217\pi$$
0.545823 + 0.837900i $$0.316217\pi$$
$$978$$ −24.0780 −0.769930
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ −16.1652 −0.516114
$$982$$ 10.7477 0.342974
$$983$$ 14.3739 0.458455 0.229228 0.973373i $$-0.426380\pi$$
0.229228 + 0.973373i $$0.426380\pi$$
$$984$$ −18.9564 −0.604309
$$985$$ 0 0
$$986$$ −6.00000 −0.191079
$$987$$ 22.0871 0.703041
$$988$$ −33.5390 −1.06702
$$989$$ 11.1652 0.355031
$$990$$ 0 0
$$991$$ −33.2087 −1.05491 −0.527455 0.849583i $$-0.676854\pi$$
−0.527455 + 0.849583i $$0.676854\pi$$
$$992$$ −3.37386 −0.107120
$$993$$ −57.9129 −1.83781
$$994$$ 15.0000 0.475771
$$995$$ 0 0
$$996$$ 16.7477 0.530672
$$997$$ 43.4955 1.37751 0.688757 0.724992i $$-0.258157\pi$$
0.688757 + 0.724992i $$0.258157\pi$$
$$998$$ 4.83485 0.153044
$$999$$ 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 1150.2.a.o.1.2 2
4.3 odd 2 9200.2.a.bs.1.1 2
5.2 odd 4 1150.2.b.g.599.3 4
5.3 odd 4 1150.2.b.g.599.2 4
5.4 even 2 230.2.a.a.1.1 2
15.14 odd 2 2070.2.a.x.1.1 2
20.19 odd 2 1840.2.a.n.1.2 2
40.19 odd 2 7360.2.a.bk.1.1 2
40.29 even 2 7360.2.a.bq.1.2 2
115.114 odd 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 5.4 even 2
1150.2.a.o.1.2 2 1.1 even 1 trivial
1150.2.b.g.599.2 4 5.3 odd 4
1150.2.b.g.599.3 4 5.2 odd 4
1840.2.a.n.1.2 2 20.19 odd 2
2070.2.a.x.1.1 2 15.14 odd 2
5290.2.a.e.1.1 2 115.114 odd 2
7360.2.a.bk.1.1 2 40.19 odd 2
7360.2.a.bq.1.2 2 40.29 even 2
9200.2.a.bs.1.1 2 4.3 odd 2