# Properties

 Label 1150.2.b.g.599.4 Level $1150$ Weight $2$ Character 1150.599 Analytic conductor $9.183$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1150.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.18279623245$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 599.4 Root $$1.79129i$$ of defining polynomial Character $$\chi$$ $$=$$ 1150.599 Dual form 1150.2.b.g.599.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.79129i q^{3} -1.00000 q^{4} -1.79129 q^{6} -2.79129i q^{7} -1.00000i q^{8} -0.208712 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.79129i q^{3} -1.00000 q^{4} -1.79129 q^{6} -2.79129i q^{7} -1.00000i q^{8} -0.208712 q^{9} +3.79129 q^{11} -1.79129i q^{12} +1.20871i q^{13} +2.79129 q^{14} +1.00000 q^{16} +3.79129i q^{17} -0.208712i q^{18} -1.20871 q^{19} +5.00000 q^{21} +3.79129i q^{22} +1.00000i q^{23} +1.79129 q^{24} -1.20871 q^{26} +5.00000i q^{27} +2.79129i q^{28} +1.58258 q^{29} +10.3739 q^{31} +1.00000i q^{32} +6.79129i q^{33} -3.79129 q^{34} +0.208712 q^{36} +4.00000i q^{37} -1.20871i q^{38} -2.16515 q^{39} -2.20871 q^{41} +5.00000i q^{42} -7.16515i q^{43} -3.79129 q^{44} -1.00000 q^{46} +13.5826i q^{47} +1.79129i q^{48} -0.791288 q^{49} -6.79129 q^{51} -1.20871i q^{52} +6.00000i q^{53} -5.00000 q^{54} -2.79129 q^{56} -2.16515i q^{57} +1.58258i q^{58} +4.41742 q^{59} -3.37386 q^{61} +10.3739i q^{62} +0.582576i q^{63} -1.00000 q^{64} -6.79129 q^{66} +7.16515i q^{67} -3.79129i q^{68} -1.79129 q^{69} -5.37386 q^{71} +0.208712i q^{72} -14.7477i q^{73} -4.00000 q^{74} +1.20871 q^{76} -10.5826i q^{77} -2.16515i q^{78} -8.00000 q^{79} -9.58258 q^{81} -2.20871i q^{82} -6.00000i q^{83} -5.00000 q^{84} +7.16515 q^{86} +2.83485i q^{87} -3.79129i q^{88} +3.16515 q^{89} +3.37386 q^{91} -1.00000i q^{92} +18.5826i q^{93} -13.5826 q^{94} -1.79129 q^{96} -14.9564i q^{97} -0.791288i q^{98} -0.791288 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 2 q^{6} - 10 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} + 2 q^{6} - 10 q^{9} + 6 q^{11} + 2 q^{14} + 4 q^{16} - 14 q^{19} + 20 q^{21} - 2 q^{24} - 14 q^{26} - 12 q^{29} + 14 q^{31} - 6 q^{34} + 10 q^{36} + 28 q^{39} - 18 q^{41} - 6 q^{44} - 4 q^{46} + 6 q^{49} - 18 q^{51} - 20 q^{54} - 2 q^{56} + 36 q^{59} + 14 q^{61} - 4 q^{64} - 18 q^{66} + 2 q^{69} + 6 q^{71} - 16 q^{74} + 14 q^{76} - 32 q^{79} - 20 q^{81} - 20 q^{84} - 8 q^{86} - 24 q^{89} - 14 q^{91} - 36 q^{94} + 2 q^{96} + 6 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 1.79129i 1.03420i 0.855925 + 0.517100i $$0.172989\pi$$
−0.855925 + 0.517100i $$0.827011\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.79129 −0.731290
$$7$$ − 2.79129i − 1.05501i −0.849553 0.527504i $$-0.823128\pi$$
0.849553 0.527504i $$-0.176872\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −0.208712 −0.0695707
$$10$$ 0 0
$$11$$ 3.79129 1.14312 0.571558 0.820562i $$-0.306339\pi$$
0.571558 + 0.820562i $$0.306339\pi$$
$$12$$ − 1.79129i − 0.517100i
$$13$$ 1.20871i 0.335236i 0.985852 + 0.167618i $$0.0536076\pi$$
−0.985852 + 0.167618i $$0.946392\pi$$
$$14$$ 2.79129 0.746003
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.79129i 0.919522i 0.888043 + 0.459761i $$0.152065\pi$$
−0.888043 + 0.459761i $$0.847935\pi$$
$$18$$ − 0.208712i − 0.0491939i
$$19$$ −1.20871 −0.277298 −0.138649 0.990342i $$-0.544276\pi$$
−0.138649 + 0.990342i $$0.544276\pi$$
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ 3.79129i 0.808305i
$$23$$ 1.00000i 0.208514i
$$24$$ 1.79129 0.365645
$$25$$ 0 0
$$26$$ −1.20871 −0.237048
$$27$$ 5.00000i 0.962250i
$$28$$ 2.79129i 0.527504i
$$29$$ 1.58258 0.293877 0.146938 0.989146i $$-0.453058\pi$$
0.146938 + 0.989146i $$0.453058\pi$$
$$30$$ 0 0
$$31$$ 10.3739 1.86320 0.931600 0.363484i $$-0.118413\pi$$
0.931600 + 0.363484i $$0.118413\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 6.79129i 1.18221i
$$34$$ −3.79129 −0.650201
$$35$$ 0 0
$$36$$ 0.208712 0.0347854
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ − 1.20871i − 0.196079i
$$39$$ −2.16515 −0.346702
$$40$$ 0 0
$$41$$ −2.20871 −0.344943 −0.172471 0.985015i $$-0.555175\pi$$
−0.172471 + 0.985015i $$0.555175\pi$$
$$42$$ 5.00000i 0.771517i
$$43$$ − 7.16515i − 1.09268i −0.837565 0.546338i $$-0.816022\pi$$
0.837565 0.546338i $$-0.183978\pi$$
$$44$$ −3.79129 −0.571558
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 13.5826i 1.98122i 0.136710 + 0.990611i $$0.456347\pi$$
−0.136710 + 0.990611i $$0.543653\pi$$
$$48$$ 1.79129i 0.258550i
$$49$$ −0.791288 −0.113041
$$50$$ 0 0
$$51$$ −6.79129 −0.950971
$$52$$ − 1.20871i − 0.167618i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ −2.79129 −0.373002
$$57$$ − 2.16515i − 0.286781i
$$58$$ 1.58258i 0.207802i
$$59$$ 4.41742 0.575100 0.287550 0.957766i $$-0.407159\pi$$
0.287550 + 0.957766i $$0.407159\pi$$
$$60$$ 0 0
$$61$$ −3.37386 −0.431979 −0.215989 0.976396i $$-0.569298\pi$$
−0.215989 + 0.976396i $$0.569298\pi$$
$$62$$ 10.3739i 1.31748i
$$63$$ 0.582576i 0.0733976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −6.79129 −0.835950
$$67$$ 7.16515i 0.875363i 0.899130 + 0.437681i $$0.144200\pi$$
−0.899130 + 0.437681i $$0.855800\pi$$
$$68$$ − 3.79129i − 0.459761i
$$69$$ −1.79129 −0.215646
$$70$$ 0 0
$$71$$ −5.37386 −0.637760 −0.318880 0.947795i $$-0.603307\pi$$
−0.318880 + 0.947795i $$0.603307\pi$$
$$72$$ 0.208712i 0.0245970i
$$73$$ − 14.7477i − 1.72609i −0.505126 0.863045i $$-0.668554\pi$$
0.505126 0.863045i $$-0.331446\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ 1.20871 0.138649
$$77$$ − 10.5826i − 1.20600i
$$78$$ − 2.16515i − 0.245155i
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −9.58258 −1.06473
$$82$$ − 2.20871i − 0.243911i
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ −5.00000 −0.545545
$$85$$ 0 0
$$86$$ 7.16515 0.772638
$$87$$ 2.83485i 0.303928i
$$88$$ − 3.79129i − 0.404153i
$$89$$ 3.16515 0.335505 0.167753 0.985829i $$-0.446349\pi$$
0.167753 + 0.985829i $$0.446349\pi$$
$$90$$ 0 0
$$91$$ 3.37386 0.353677
$$92$$ − 1.00000i − 0.104257i
$$93$$ 18.5826i 1.92692i
$$94$$ −13.5826 −1.40094
$$95$$ 0 0
$$96$$ −1.79129 −0.182823
$$97$$ − 14.9564i − 1.51860i −0.650743 0.759298i $$-0.725542\pi$$
0.650743 0.759298i $$-0.274458\pi$$
$$98$$ − 0.791288i − 0.0799321i
$$99$$ −0.791288 −0.0795274
$$100$$ 0 0
$$101$$ 13.5826 1.35152 0.675758 0.737123i $$-0.263816\pi$$
0.675758 + 0.737123i $$0.263816\pi$$
$$102$$ − 6.79129i − 0.672438i
$$103$$ 7.37386i 0.726568i 0.931678 + 0.363284i $$0.118345\pi$$
−0.931678 + 0.363284i $$0.881655\pi$$
$$104$$ 1.20871 0.118524
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 13.5826i − 1.31308i −0.754292 0.656539i $$-0.772020\pi$$
0.754292 0.656539i $$-0.227980\pi$$
$$108$$ − 5.00000i − 0.481125i
$$109$$ −10.3739 −0.993636 −0.496818 0.867855i $$-0.665498\pi$$
−0.496818 + 0.867855i $$0.665498\pi$$
$$110$$ 0 0
$$111$$ −7.16515 −0.680086
$$112$$ − 2.79129i − 0.263752i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 2.16515 0.202785
$$115$$ 0 0
$$116$$ −1.58258 −0.146938
$$117$$ − 0.252273i − 0.0233226i
$$118$$ 4.41742i 0.406657i
$$119$$ 10.5826 0.970103
$$120$$ 0 0
$$121$$ 3.37386 0.306715
$$122$$ − 3.37386i − 0.305455i
$$123$$ − 3.95644i − 0.356740i
$$124$$ −10.3739 −0.931600
$$125$$ 0 0
$$126$$ −0.582576 −0.0519000
$$127$$ 14.7477i 1.30865i 0.756214 + 0.654325i $$0.227047\pi$$
−0.756214 + 0.654325i $$0.772953\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 12.8348 1.13005
$$130$$ 0 0
$$131$$ 9.16515 0.800763 0.400381 0.916349i $$-0.368878\pi$$
0.400381 + 0.916349i $$0.368878\pi$$
$$132$$ − 6.79129i − 0.591106i
$$133$$ 3.37386i 0.292551i
$$134$$ −7.16515 −0.618975
$$135$$ 0 0
$$136$$ 3.79129 0.325100
$$137$$ 0.791288i 0.0676043i 0.999429 + 0.0338021i $$0.0107616\pi$$
−0.999429 + 0.0338021i $$0.989238\pi$$
$$138$$ − 1.79129i − 0.152485i
$$139$$ 14.7477 1.25089 0.625443 0.780270i $$-0.284918\pi$$
0.625443 + 0.780270i $$0.284918\pi$$
$$140$$ 0 0
$$141$$ −24.3303 −2.04898
$$142$$ − 5.37386i − 0.450965i
$$143$$ 4.58258i 0.383214i
$$144$$ −0.208712 −0.0173927
$$145$$ 0 0
$$146$$ 14.7477 1.22053
$$147$$ − 1.41742i − 0.116907i
$$148$$ − 4.00000i − 0.328798i
$$149$$ −12.7913 −1.04790 −0.523952 0.851748i $$-0.675543\pi$$
−0.523952 + 0.851748i $$0.675543\pi$$
$$150$$ 0 0
$$151$$ −6.20871 −0.505258 −0.252629 0.967563i $$-0.581295\pi$$
−0.252629 + 0.967563i $$0.581295\pi$$
$$152$$ 1.20871i 0.0980395i
$$153$$ − 0.791288i − 0.0639718i
$$154$$ 10.5826 0.852768
$$155$$ 0 0
$$156$$ 2.16515 0.173351
$$157$$ − 12.7477i − 1.01738i −0.860950 0.508690i $$-0.830130\pi$$
0.860950 0.508690i $$-0.169870\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −10.7477 −0.852350
$$160$$ 0 0
$$161$$ 2.79129 0.219984
$$162$$ − 9.58258i − 0.752878i
$$163$$ 22.3739i 1.75246i 0.481897 + 0.876228i $$0.339948\pi$$
−0.481897 + 0.876228i $$0.660052\pi$$
$$164$$ 2.20871 0.172471
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 18.3303i 1.41844i 0.704987 + 0.709221i $$0.250953\pi$$
−0.704987 + 0.709221i $$0.749047\pi$$
$$168$$ − 5.00000i − 0.385758i
$$169$$ 11.5390 0.887617
$$170$$ 0 0
$$171$$ 0.252273 0.0192918
$$172$$ 7.16515i 0.546338i
$$173$$ − 14.2087i − 1.08027i −0.841579 0.540134i $$-0.818373\pi$$
0.841579 0.540134i $$-0.181627\pi$$
$$174$$ −2.83485 −0.214909
$$175$$ 0 0
$$176$$ 3.79129 0.285779
$$177$$ 7.91288i 0.594768i
$$178$$ 3.16515i 0.237238i
$$179$$ 16.7477 1.25178 0.625892 0.779910i $$-0.284735\pi$$
0.625892 + 0.779910i $$0.284735\pi$$
$$180$$ 0 0
$$181$$ 13.5390 1.00635 0.503174 0.864185i $$-0.332166\pi$$
0.503174 + 0.864185i $$0.332166\pi$$
$$182$$ 3.37386i 0.250087i
$$183$$ − 6.04356i − 0.446753i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ −18.5826 −1.36254
$$187$$ 14.3739i 1.05112i
$$188$$ − 13.5826i − 0.990611i
$$189$$ 13.9564 1.01518
$$190$$ 0 0
$$191$$ 16.4174 1.18792 0.593962 0.804493i $$-0.297563\pi$$
0.593962 + 0.804493i $$0.297563\pi$$
$$192$$ − 1.79129i − 0.129275i
$$193$$ 6.74773i 0.485712i 0.970062 + 0.242856i $$0.0780843\pi$$
−0.970062 + 0.242856i $$0.921916\pi$$
$$194$$ 14.9564 1.07381
$$195$$ 0 0
$$196$$ 0.791288 0.0565206
$$197$$ − 20.5390i − 1.46334i −0.681657 0.731672i $$-0.738740\pi$$
0.681657 0.731672i $$-0.261260\pi$$
$$198$$ − 0.791288i − 0.0562344i
$$199$$ −20.3303 −1.44118 −0.720588 0.693363i $$-0.756128\pi$$
−0.720588 + 0.693363i $$0.756128\pi$$
$$200$$ 0 0
$$201$$ −12.8348 −0.905300
$$202$$ 13.5826i 0.955667i
$$203$$ − 4.41742i − 0.310042i
$$204$$ 6.79129 0.475485
$$205$$ 0 0
$$206$$ −7.37386 −0.513761
$$207$$ − 0.208712i − 0.0145065i
$$208$$ 1.20871i 0.0838091i
$$209$$ −4.58258 −0.316983
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ − 9.62614i − 0.659572i
$$214$$ 13.5826 0.928486
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ − 28.9564i − 1.96569i
$$218$$ − 10.3739i − 0.702607i
$$219$$ 26.4174 1.78512
$$220$$ 0 0
$$221$$ −4.58258 −0.308257
$$222$$ − 7.16515i − 0.480893i
$$223$$ 11.1652i 0.747674i 0.927494 + 0.373837i $$0.121958\pi$$
−0.927494 + 0.373837i $$0.878042\pi$$
$$224$$ 2.79129 0.186501
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 4.74773i − 0.315118i −0.987510 0.157559i $$-0.949638\pi$$
0.987510 0.157559i $$-0.0503624\pi$$
$$228$$ 2.16515i 0.143391i
$$229$$ 16.3303 1.07914 0.539568 0.841942i $$-0.318587\pi$$
0.539568 + 0.841942i $$0.318587\pi$$
$$230$$ 0 0
$$231$$ 18.9564 1.24724
$$232$$ − 1.58258i − 0.103901i
$$233$$ 7.58258i 0.496751i 0.968664 + 0.248376i $$0.0798967\pi$$
−0.968664 + 0.248376i $$0.920103\pi$$
$$234$$ 0.252273 0.0164916
$$235$$ 0 0
$$236$$ −4.41742 −0.287550
$$237$$ − 14.3303i − 0.930853i
$$238$$ 10.5826i 0.685966i
$$239$$ −3.16515 −0.204737 −0.102368 0.994747i $$-0.532642\pi$$
−0.102368 + 0.994747i $$0.532642\pi$$
$$240$$ 0 0
$$241$$ −28.0000 −1.80364 −0.901819 0.432113i $$-0.857768\pi$$
−0.901819 + 0.432113i $$0.857768\pi$$
$$242$$ 3.37386i 0.216880i
$$243$$ − 2.16515i − 0.138895i
$$244$$ 3.37386 0.215989
$$245$$ 0 0
$$246$$ 3.95644 0.252253
$$247$$ − 1.46099i − 0.0929603i
$$248$$ − 10.3739i − 0.658741i
$$249$$ 10.7477 0.681110
$$250$$ 0 0
$$251$$ 30.7913 1.94353 0.971764 0.235953i $$-0.0758212\pi$$
0.971764 + 0.235953i $$0.0758212\pi$$
$$252$$ − 0.582576i − 0.0366988i
$$253$$ 3.79129i 0.238356i
$$254$$ −14.7477 −0.925355
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 22.7477i − 1.41896i −0.704723 0.709482i $$-0.748929\pi$$
0.704723 0.709482i $$-0.251071\pi$$
$$258$$ 12.8348i 0.799063i
$$259$$ 11.1652 0.693769
$$260$$ 0 0
$$261$$ −0.330303 −0.0204452
$$262$$ 9.16515i 0.566225i
$$263$$ 15.7913i 0.973733i 0.873477 + 0.486866i $$0.161860\pi$$
−0.873477 + 0.486866i $$0.838140\pi$$
$$264$$ 6.79129 0.417975
$$265$$ 0 0
$$266$$ −3.37386 −0.206865
$$267$$ 5.66970i 0.346980i
$$268$$ − 7.16515i − 0.437681i
$$269$$ −16.7477 −1.02113 −0.510563 0.859840i $$-0.670563\pi$$
−0.510563 + 0.859840i $$0.670563\pi$$
$$270$$ 0 0
$$271$$ −23.1216 −1.40454 −0.702268 0.711912i $$-0.747829\pi$$
−0.702268 + 0.711912i $$0.747829\pi$$
$$272$$ 3.79129i 0.229881i
$$273$$ 6.04356i 0.365773i
$$274$$ −0.791288 −0.0478034
$$275$$ 0 0
$$276$$ 1.79129 0.107823
$$277$$ 1.16515i 0.0700072i 0.999387 + 0.0350036i $$0.0111443\pi$$
−0.999387 + 0.0350036i $$0.988856\pi$$
$$278$$ 14.7477i 0.884510i
$$279$$ −2.16515 −0.129624
$$280$$ 0 0
$$281$$ −16.7477 −0.999086 −0.499543 0.866289i $$-0.666499\pi$$
−0.499543 + 0.866289i $$0.666499\pi$$
$$282$$ − 24.3303i − 1.44885i
$$283$$ − 28.3303i − 1.68406i −0.539429 0.842031i $$-0.681360\pi$$
0.539429 0.842031i $$-0.318640\pi$$
$$284$$ 5.37386 0.318880
$$285$$ 0 0
$$286$$ −4.58258 −0.270973
$$287$$ 6.16515i 0.363917i
$$288$$ − 0.208712i − 0.0122985i
$$289$$ 2.62614 0.154479
$$290$$ 0 0
$$291$$ 26.7913 1.57053
$$292$$ 14.7477i 0.863045i
$$293$$ − 27.4955i − 1.60630i −0.595776 0.803151i $$-0.703155\pi$$
0.595776 0.803151i $$-0.296845\pi$$
$$294$$ 1.41742 0.0826659
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ 18.9564i 1.09996i
$$298$$ − 12.7913i − 0.740979i
$$299$$ −1.20871 −0.0699016
$$300$$ 0 0
$$301$$ −20.0000 −1.15278
$$302$$ − 6.20871i − 0.357271i
$$303$$ 24.3303i 1.39774i
$$304$$ −1.20871 −0.0693244
$$305$$ 0 0
$$306$$ 0.791288 0.0452349
$$307$$ − 16.5390i − 0.943931i −0.881617 0.471966i $$-0.843545\pi$$
0.881617 0.471966i $$-0.156455\pi$$
$$308$$ 10.5826i 0.602998i
$$309$$ −13.2087 −0.751417
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 2.16515i 0.122578i
$$313$$ − 18.3739i − 1.03855i −0.854607 0.519276i $$-0.826202\pi$$
0.854607 0.519276i $$-0.173798\pi$$
$$314$$ 12.7477 0.719396
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ − 5.20871i − 0.292550i −0.989244 0.146275i $$-0.953271\pi$$
0.989244 0.146275i $$-0.0467285\pi$$
$$318$$ − 10.7477i − 0.602703i
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 24.3303 1.35799
$$322$$ 2.79129i 0.155552i
$$323$$ − 4.58258i − 0.254981i
$$324$$ 9.58258 0.532365
$$325$$ 0 0
$$326$$ −22.3739 −1.23917
$$327$$ − 18.5826i − 1.02762i
$$328$$ 2.20871i 0.121956i
$$329$$ 37.9129 2.09020
$$330$$ 0 0
$$331$$ 6.74773 0.370889 0.185444 0.982655i $$-0.440628\pi$$
0.185444 + 0.982655i $$0.440628\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ − 0.834849i − 0.0457494i
$$334$$ −18.3303 −1.00299
$$335$$ 0 0
$$336$$ 5.00000 0.272772
$$337$$ 16.7913i 0.914680i 0.889292 + 0.457340i $$0.151198\pi$$
−0.889292 + 0.457340i $$0.848802\pi$$
$$338$$ 11.5390i 0.627640i
$$339$$ −10.7477 −0.583736
$$340$$ 0 0
$$341$$ 39.3303 2.12986
$$342$$ 0.252273i 0.0136414i
$$343$$ − 17.3303i − 0.935748i
$$344$$ −7.16515 −0.386319
$$345$$ 0 0
$$346$$ 14.2087 0.763865
$$347$$ − 9.79129i − 0.525624i −0.964847 0.262812i $$-0.915350\pi$$
0.964847 0.262812i $$-0.0846499\pi$$
$$348$$ − 2.83485i − 0.151964i
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ −6.04356 −0.322581
$$352$$ 3.79129i 0.202076i
$$353$$ − 15.1652i − 0.807160i −0.914944 0.403580i $$-0.867766\pi$$
0.914944 0.403580i $$-0.132234\pi$$
$$354$$ −7.91288 −0.420565
$$355$$ 0 0
$$356$$ −3.16515 −0.167753
$$357$$ 18.9564i 1.00328i
$$358$$ 16.7477i 0.885145i
$$359$$ 9.16515 0.483718 0.241859 0.970311i $$-0.422243\pi$$
0.241859 + 0.970311i $$0.422243\pi$$
$$360$$ 0 0
$$361$$ −17.5390 −0.923106
$$362$$ 13.5390i 0.711595i
$$363$$ 6.04356i 0.317205i
$$364$$ −3.37386 −0.176838
$$365$$ 0 0
$$366$$ 6.04356 0.315902
$$367$$ 0.834849i 0.0435787i 0.999763 + 0.0217894i $$0.00693632\pi$$
−0.999763 + 0.0217894i $$0.993064\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 0.460985 0.0239979
$$370$$ 0 0
$$371$$ 16.7477 0.869499
$$372$$ − 18.5826i − 0.963462i
$$373$$ − 14.7477i − 0.763608i −0.924243 0.381804i $$-0.875303\pi$$
0.924243 0.381804i $$-0.124697\pi$$
$$374$$ −14.3739 −0.743255
$$375$$ 0 0
$$376$$ 13.5826 0.700468
$$377$$ 1.91288i 0.0985183i
$$378$$ 13.9564i 0.717842i
$$379$$ −7.37386 −0.378770 −0.189385 0.981903i $$-0.560649\pi$$
−0.189385 + 0.981903i $$0.560649\pi$$
$$380$$ 0 0
$$381$$ −26.4174 −1.35341
$$382$$ 16.4174i 0.839989i
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ 1.79129 0.0914113
$$385$$ 0 0
$$386$$ −6.74773 −0.343450
$$387$$ 1.49545i 0.0760182i
$$388$$ 14.9564i 0.759298i
$$389$$ −29.7042 −1.50606 −0.753031 0.657986i $$-0.771409\pi$$
−0.753031 + 0.657986i $$0.771409\pi$$
$$390$$ 0 0
$$391$$ −3.79129 −0.191734
$$392$$ 0.791288i 0.0399661i
$$393$$ 16.4174i 0.828150i
$$394$$ 20.5390 1.03474
$$395$$ 0 0
$$396$$ 0.791288 0.0397637
$$397$$ − 16.5390i − 0.830069i −0.909806 0.415035i $$-0.863769\pi$$
0.909806 0.415035i $$-0.136231\pi$$
$$398$$ − 20.3303i − 1.01907i
$$399$$ −6.04356 −0.302556
$$400$$ 0 0
$$401$$ 22.7477 1.13597 0.567984 0.823040i $$-0.307724\pi$$
0.567984 + 0.823040i $$0.307724\pi$$
$$402$$ − 12.8348i − 0.640144i
$$403$$ 12.5390i 0.624613i
$$404$$ −13.5826 −0.675758
$$405$$ 0 0
$$406$$ 4.41742 0.219233
$$407$$ 15.1652i 0.751709i
$$408$$ 6.79129i 0.336219i
$$409$$ 22.7913 1.12696 0.563478 0.826131i $$-0.309463\pi$$
0.563478 + 0.826131i $$0.309463\pi$$
$$410$$ 0 0
$$411$$ −1.41742 −0.0699164
$$412$$ − 7.37386i − 0.363284i
$$413$$ − 12.3303i − 0.606735i
$$414$$ 0.208712 0.0102576
$$415$$ 0 0
$$416$$ −1.20871 −0.0592620
$$417$$ 26.4174i 1.29367i
$$418$$ − 4.58258i − 0.224141i
$$419$$ −39.1652 −1.91334 −0.956671 0.291170i $$-0.905956\pi$$
−0.956671 + 0.291170i $$0.905956\pi$$
$$420$$ 0 0
$$421$$ −23.1216 −1.12688 −0.563439 0.826158i $$-0.690522\pi$$
−0.563439 + 0.826158i $$0.690522\pi$$
$$422$$ − 10.0000i − 0.486792i
$$423$$ − 2.83485i − 0.137835i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 9.62614 0.466388
$$427$$ 9.41742i 0.455741i
$$428$$ 13.5826i 0.656539i
$$429$$ −8.20871 −0.396320
$$430$$ 0 0
$$431$$ −19.9129 −0.959170 −0.479585 0.877496i $$-0.659213\pi$$
−0.479585 + 0.877496i $$0.659213\pi$$
$$432$$ 5.00000i 0.240563i
$$433$$ 1.53901i 0.0739603i 0.999316 + 0.0369802i $$0.0117738\pi$$
−0.999316 + 0.0369802i $$0.988226\pi$$
$$434$$ 28.9564 1.38995
$$435$$ 0 0
$$436$$ 10.3739 0.496818
$$437$$ − 1.20871i − 0.0578205i
$$438$$ 26.4174i 1.26227i
$$439$$ −25.5390 −1.21891 −0.609455 0.792820i $$-0.708612\pi$$
−0.609455 + 0.792820i $$0.708612\pi$$
$$440$$ 0 0
$$441$$ 0.165151 0.00786435
$$442$$ − 4.58258i − 0.217971i
$$443$$ − 35.2087i − 1.67282i −0.548107 0.836408i $$-0.684651\pi$$
0.548107 0.836408i $$-0.315349\pi$$
$$444$$ 7.16515 0.340043
$$445$$ 0 0
$$446$$ −11.1652 −0.528685
$$447$$ − 22.9129i − 1.08374i
$$448$$ 2.79129i 0.131876i
$$449$$ −25.1216 −1.18556 −0.592781 0.805364i $$-0.701970\pi$$
−0.592781 + 0.805364i $$0.701970\pi$$
$$450$$ 0 0
$$451$$ −8.37386 −0.394310
$$452$$ − 6.00000i − 0.282216i
$$453$$ − 11.1216i − 0.522538i
$$454$$ 4.74773 0.222822
$$455$$ 0 0
$$456$$ −2.16515 −0.101393
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ 16.3303i 0.763065i
$$459$$ −18.9564 −0.884811
$$460$$ 0 0
$$461$$ −1.25227 −0.0583242 −0.0291621 0.999575i $$-0.509284\pi$$
−0.0291621 + 0.999575i $$0.509284\pi$$
$$462$$ 18.9564i 0.881933i
$$463$$ − 10.0000i − 0.464739i −0.972628 0.232370i $$-0.925352\pi$$
0.972628 0.232370i $$-0.0746479\pi$$
$$464$$ 1.58258 0.0734692
$$465$$ 0 0
$$466$$ −7.58258 −0.351256
$$467$$ − 25.9129i − 1.19911i −0.800335 0.599553i $$-0.795345\pi$$
0.800335 0.599553i $$-0.204655\pi$$
$$468$$ 0.252273i 0.0116613i
$$469$$ 20.0000 0.923514
$$470$$ 0 0
$$471$$ 22.8348 1.05217
$$472$$ − 4.41742i − 0.203328i
$$473$$ − 27.1652i − 1.24905i
$$474$$ 14.3303 0.658213
$$475$$ 0 0
$$476$$ −10.5826 −0.485052
$$477$$ − 1.25227i − 0.0573376i
$$478$$ − 3.16515i − 0.144771i
$$479$$ 39.4955 1.80459 0.902297 0.431116i $$-0.141880\pi$$
0.902297 + 0.431116i $$0.141880\pi$$
$$480$$ 0 0
$$481$$ −4.83485 −0.220450
$$482$$ − 28.0000i − 1.27537i
$$483$$ 5.00000i 0.227508i
$$484$$ −3.37386 −0.153357
$$485$$ 0 0
$$486$$ 2.16515 0.0982133
$$487$$ − 15.5826i − 0.706114i −0.935602 0.353057i $$-0.885142\pi$$
0.935602 0.353057i $$-0.114858\pi$$
$$488$$ 3.37386i 0.152728i
$$489$$ −40.0780 −1.81239
$$490$$ 0 0
$$491$$ −16.7477 −0.755814 −0.377907 0.925843i $$-0.623356\pi$$
−0.377907 + 0.925843i $$0.623356\pi$$
$$492$$ 3.95644i 0.178370i
$$493$$ 6.00000i 0.270226i
$$494$$ 1.46099 0.0657328
$$495$$ 0 0
$$496$$ 10.3739 0.465800
$$497$$ 15.0000i 0.672842i
$$498$$ 10.7477i 0.481617i
$$499$$ −23.1652 −1.03701 −0.518507 0.855073i $$-0.673512\pi$$
−0.518507 + 0.855073i $$0.673512\pi$$
$$500$$ 0 0
$$501$$ −32.8348 −1.46695
$$502$$ 30.7913i 1.37428i
$$503$$ 18.7913i 0.837862i 0.908018 + 0.418931i $$0.137595\pi$$
−0.908018 + 0.418931i $$0.862405\pi$$
$$504$$ 0.582576 0.0259500
$$505$$ 0 0
$$506$$ −3.79129 −0.168543
$$507$$ 20.6697i 0.917973i
$$508$$ − 14.7477i − 0.654325i
$$509$$ −7.25227 −0.321451 −0.160726 0.986999i $$-0.551383\pi$$
−0.160726 + 0.986999i $$0.551383\pi$$
$$510$$ 0 0
$$511$$ −41.1652 −1.82104
$$512$$ 1.00000i 0.0441942i
$$513$$ − 6.04356i − 0.266830i
$$514$$ 22.7477 1.00336
$$515$$ 0 0
$$516$$ −12.8348 −0.565023
$$517$$ 51.4955i 2.26477i
$$518$$ 11.1652i 0.490569i
$$519$$ 25.4519 1.11721
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ − 0.330303i − 0.0144570i
$$523$$ − 1.16515i − 0.0509485i −0.999675 0.0254743i $$-0.991890\pi$$
0.999675 0.0254743i $$-0.00810958\pi$$
$$524$$ −9.16515 −0.400381
$$525$$ 0 0
$$526$$ −15.7913 −0.688533
$$527$$ 39.3303i 1.71325i
$$528$$ 6.79129i 0.295553i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −0.921970 −0.0400101
$$532$$ − 3.37386i − 0.146276i
$$533$$ − 2.66970i − 0.115637i
$$534$$ −5.66970 −0.245352
$$535$$ 0 0
$$536$$ 7.16515 0.309487
$$537$$ 30.0000i 1.29460i
$$538$$ − 16.7477i − 0.722046i
$$539$$ −3.00000 −0.129219
$$540$$ 0 0
$$541$$ 38.3303 1.64795 0.823974 0.566627i $$-0.191752\pi$$
0.823974 + 0.566627i $$0.191752\pi$$
$$542$$ − 23.1216i − 0.993157i
$$543$$ 24.2523i 1.04076i
$$544$$ −3.79129 −0.162550
$$545$$ 0 0
$$546$$ −6.04356 −0.258641
$$547$$ − 15.1216i − 0.646553i −0.946305 0.323276i $$-0.895216\pi$$
0.946305 0.323276i $$-0.104784\pi$$
$$548$$ − 0.791288i − 0.0338021i
$$549$$ 0.704166 0.0300531
$$550$$ 0 0
$$551$$ −1.91288 −0.0814914
$$552$$ 1.79129i 0.0762423i
$$553$$ 22.3303i 0.949581i
$$554$$ −1.16515 −0.0495025
$$555$$ 0 0
$$556$$ −14.7477 −0.625443
$$557$$ − 30.3303i − 1.28514i −0.766229 0.642568i $$-0.777869\pi$$
0.766229 0.642568i $$-0.222131\pi$$
$$558$$ − 2.16515i − 0.0916582i
$$559$$ 8.66061 0.366305
$$560$$ 0 0
$$561$$ −25.7477 −1.08707
$$562$$ − 16.7477i − 0.706460i
$$563$$ 3.16515i 0.133395i 0.997773 + 0.0666976i $$0.0212463\pi$$
−0.997773 + 0.0666976i $$0.978754\pi$$
$$564$$ 24.3303 1.02449
$$565$$ 0 0
$$566$$ 28.3303 1.19081
$$567$$ 26.7477i 1.12330i
$$568$$ 5.37386i 0.225482i
$$569$$ −15.4955 −0.649603 −0.324802 0.945782i $$-0.605298\pi$$
−0.324802 + 0.945782i $$0.605298\pi$$
$$570$$ 0 0
$$571$$ 30.1216 1.26055 0.630275 0.776372i $$-0.282942\pi$$
0.630275 + 0.776372i $$0.282942\pi$$
$$572$$ − 4.58258i − 0.191607i
$$573$$ 29.4083i 1.22855i
$$574$$ −6.16515 −0.257328
$$575$$ 0 0
$$576$$ 0.208712 0.00869634
$$577$$ − 22.8348i − 0.950627i −0.879816 0.475314i $$-0.842335\pi$$
0.879816 0.475314i $$-0.157665\pi$$
$$578$$ 2.62614i 0.109233i
$$579$$ −12.0871 −0.502324
$$580$$ 0 0
$$581$$ −16.7477 −0.694813
$$582$$ 26.7913i 1.11053i
$$583$$ 22.7477i 0.942115i
$$584$$ −14.7477 −0.610265
$$585$$ 0 0
$$586$$ 27.4955 1.13583
$$587$$ 26.2087i 1.08175i 0.841103 + 0.540875i $$0.181907\pi$$
−0.841103 + 0.540875i $$0.818093\pi$$
$$588$$ 1.41742i 0.0584536i
$$589$$ −12.5390 −0.516661
$$590$$ 0 0
$$591$$ 36.7913 1.51339
$$592$$ 4.00000i 0.164399i
$$593$$ 13.9129i 0.571333i 0.958329 + 0.285667i $$0.0922150\pi$$
−0.958329 + 0.285667i $$0.907785\pi$$
$$594$$ −18.9564 −0.777792
$$595$$ 0 0
$$596$$ 12.7913 0.523952
$$597$$ − 36.4174i − 1.49047i
$$598$$ − 1.20871i − 0.0494279i
$$599$$ 40.1216 1.63932 0.819662 0.572848i $$-0.194161\pi$$
0.819662 + 0.572848i $$0.194161\pi$$
$$600$$ 0 0
$$601$$ −22.7913 −0.929676 −0.464838 0.885396i $$-0.653887\pi$$
−0.464838 + 0.885396i $$0.653887\pi$$
$$602$$ − 20.0000i − 0.815139i
$$603$$ − 1.49545i − 0.0608996i
$$604$$ 6.20871 0.252629
$$605$$ 0 0
$$606$$ −24.3303 −0.988351
$$607$$ 28.0000i 1.13648i 0.822861 + 0.568242i $$0.192376\pi$$
−0.822861 + 0.568242i $$0.807624\pi$$
$$608$$ − 1.20871i − 0.0490198i
$$609$$ 7.91288 0.320646
$$610$$ 0 0
$$611$$ −16.4174 −0.664178
$$612$$ 0.791288i 0.0319859i
$$613$$ 14.0000i 0.565455i 0.959200 + 0.282727i $$0.0912392\pi$$
−0.959200 + 0.282727i $$0.908761\pi$$
$$614$$ 16.5390 0.667460
$$615$$ 0 0
$$616$$ −10.5826 −0.426384
$$617$$ − 44.8693i − 1.80637i −0.429251 0.903185i $$-0.641222\pi$$
0.429251 0.903185i $$-0.358778\pi$$
$$618$$ − 13.2087i − 0.531332i
$$619$$ −2.79129 −0.112191 −0.0560957 0.998425i $$-0.517865\pi$$
−0.0560957 + 0.998425i $$0.517865\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ 12.0000i 0.481156i
$$623$$ − 8.83485i − 0.353961i
$$624$$ −2.16515 −0.0866754
$$625$$ 0 0
$$626$$ 18.3739 0.734367
$$627$$ − 8.20871i − 0.327824i
$$628$$ 12.7477i 0.508690i
$$629$$ −15.1652 −0.604674
$$630$$ 0 0
$$631$$ −17.9129 −0.713100 −0.356550 0.934276i $$-0.616047\pi$$
−0.356550 + 0.934276i $$0.616047\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ − 17.9129i − 0.711973i
$$634$$ 5.20871 0.206864
$$635$$ 0 0
$$636$$ 10.7477 0.426175
$$637$$ − 0.956439i − 0.0378955i
$$638$$ 6.00000i 0.237542i
$$639$$ 1.12159 0.0443694
$$640$$ 0 0
$$641$$ −3.16515 −0.125016 −0.0625080 0.998044i $$-0.519910\pi$$
−0.0625080 + 0.998044i $$0.519910\pi$$
$$642$$ 24.3303i 0.960240i
$$643$$ − 20.7477i − 0.818210i −0.912487 0.409105i $$-0.865841\pi$$
0.912487 0.409105i $$-0.134159\pi$$
$$644$$ −2.79129 −0.109992
$$645$$ 0 0
$$646$$ 4.58258 0.180299
$$647$$ − 2.83485i − 0.111449i −0.998446 0.0557247i $$-0.982253\pi$$
0.998446 0.0557247i $$-0.0177469\pi$$
$$648$$ 9.58258i 0.376439i
$$649$$ 16.7477 0.657406
$$650$$ 0 0
$$651$$ 51.8693 2.03292
$$652$$ − 22.3739i − 0.876228i
$$653$$ 35.5390i 1.39075i 0.718647 + 0.695375i $$0.244762\pi$$
−0.718647 + 0.695375i $$0.755238\pi$$
$$654$$ 18.5826 0.726636
$$655$$ 0 0
$$656$$ −2.20871 −0.0862357
$$657$$ 3.07803i 0.120085i
$$658$$ 37.9129i 1.47800i
$$659$$ 27.1652 1.05820 0.529102 0.848558i $$-0.322529\pi$$
0.529102 + 0.848558i $$0.322529\pi$$
$$660$$ 0 0
$$661$$ −39.3739 −1.53147 −0.765733 0.643159i $$-0.777624\pi$$
−0.765733 + 0.643159i $$0.777624\pi$$
$$662$$ 6.74773i 0.262258i
$$663$$ − 8.20871i − 0.318800i
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 0.834849 0.0323497
$$667$$ 1.58258i 0.0612776i
$$668$$ − 18.3303i − 0.709221i
$$669$$ −20.0000 −0.773245
$$670$$ 0 0
$$671$$ −12.7913 −0.493802
$$672$$ 5.00000i 0.192879i
$$673$$ 38.0000i 1.46479i 0.680879 + 0.732396i $$0.261598\pi$$
−0.680879 + 0.732396i $$0.738402\pi$$
$$674$$ −16.7913 −0.646776
$$675$$ 0 0
$$676$$ −11.5390 −0.443808
$$677$$ 30.6606i 1.17838i 0.807993 + 0.589191i $$0.200554\pi$$
−0.807993 + 0.589191i $$0.799446\pi$$
$$678$$ − 10.7477i − 0.412764i
$$679$$ −41.7477 −1.60213
$$680$$ 0 0
$$681$$ 8.50455 0.325895
$$682$$ 39.3303i 1.50604i
$$683$$ 2.37386i 0.0908334i 0.998968 + 0.0454167i $$0.0144616\pi$$
−0.998968 + 0.0454167i $$0.985538\pi$$
$$684$$ −0.252273 −0.00964590
$$685$$ 0 0
$$686$$ 17.3303 0.661674
$$687$$ 29.2523i 1.11604i
$$688$$ − 7.16515i − 0.273169i
$$689$$ −7.25227 −0.276290
$$690$$ 0 0
$$691$$ 15.2523 0.580224 0.290112 0.956993i $$-0.406307\pi$$
0.290112 + 0.956993i $$0.406307\pi$$
$$692$$ 14.2087i 0.540134i
$$693$$ 2.20871i 0.0839020i
$$694$$ 9.79129 0.371672
$$695$$ 0 0
$$696$$ 2.83485 0.107455
$$697$$ − 8.37386i − 0.317183i
$$698$$ − 26.0000i − 0.984115i
$$699$$ −13.5826 −0.513740
$$700$$ 0 0
$$701$$ 9.62614 0.363574 0.181787 0.983338i $$-0.441812\pi$$
0.181787 + 0.983338i $$0.441812\pi$$
$$702$$ − 6.04356i − 0.228100i
$$703$$ − 4.83485i − 0.182350i
$$704$$ −3.79129 −0.142890
$$705$$ 0 0
$$706$$ 15.1652 0.570748
$$707$$ − 37.9129i − 1.42586i
$$708$$ − 7.91288i − 0.297384i
$$709$$ −34.5390 −1.29714 −0.648570 0.761155i $$-0.724633\pi$$
−0.648570 + 0.761155i $$0.724633\pi$$
$$710$$ 0 0
$$711$$ 1.66970 0.0626185
$$712$$ − 3.16515i − 0.118619i
$$713$$ 10.3739i 0.388504i
$$714$$ −18.9564 −0.709427
$$715$$ 0 0
$$716$$ −16.7477 −0.625892
$$717$$ − 5.66970i − 0.211739i
$$718$$ 9.16515i 0.342040i
$$719$$ 29.5390 1.10162 0.550810 0.834631i $$-0.314319\pi$$
0.550810 + 0.834631i $$0.314319\pi$$
$$720$$ 0 0
$$721$$ 20.5826 0.766535
$$722$$ − 17.5390i − 0.652735i
$$723$$ − 50.1561i − 1.86532i
$$724$$ −13.5390 −0.503174
$$725$$ 0 0
$$726$$ −6.04356 −0.224298
$$727$$ 2.12159i 0.0786854i 0.999226 + 0.0393427i $$0.0125264\pi$$
−0.999226 + 0.0393427i $$0.987474\pi$$
$$728$$ − 3.37386i − 0.125044i
$$729$$ −24.8693 −0.921086
$$730$$ 0 0
$$731$$ 27.1652 1.00474
$$732$$ 6.04356i 0.223376i
$$733$$ 26.0000i 0.960332i 0.877178 + 0.480166i $$0.159424\pi$$
−0.877178 + 0.480166i $$0.840576\pi$$
$$734$$ −0.834849 −0.0308148
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 27.1652i 1.00064i
$$738$$ 0.460985i 0.0169691i
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ 0 0
$$741$$ 2.61704 0.0961395
$$742$$ 16.7477i 0.614828i
$$743$$ 9.95644i 0.365266i 0.983181 + 0.182633i $$0.0584621\pi$$
−0.983181 + 0.182633i $$0.941538\pi$$
$$744$$ 18.5826 0.681270
$$745$$ 0 0
$$746$$ 14.7477 0.539953
$$747$$ 1.25227i 0.0458183i
$$748$$ − 14.3739i − 0.525561i
$$749$$ −37.9129 −1.38531
$$750$$ 0 0
$$751$$ 18.7477 0.684114 0.342057 0.939679i $$-0.388876\pi$$
0.342057 + 0.939679i $$0.388876\pi$$
$$752$$ 13.5826i 0.495306i
$$753$$ 55.1561i 2.01000i
$$754$$ −1.91288 −0.0696629
$$755$$ 0 0
$$756$$ −13.9564 −0.507591
$$757$$ − 26.3303i − 0.956991i −0.878090 0.478496i $$-0.841182\pi$$
0.878090 0.478496i $$-0.158818\pi$$
$$758$$ − 7.37386i − 0.267831i
$$759$$ −6.79129 −0.246508
$$760$$ 0 0
$$761$$ −33.9564 −1.23092 −0.615460 0.788168i $$-0.711030\pi$$
−0.615460 + 0.788168i $$0.711030\pi$$
$$762$$ − 26.4174i − 0.957002i
$$763$$ 28.9564i 1.04829i
$$764$$ −16.4174 −0.593962
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 5.33939i 0.192794i
$$768$$ 1.79129i 0.0646375i
$$769$$ 3.66970 0.132333 0.0661663 0.997809i $$-0.478923\pi$$
0.0661663 + 0.997809i $$0.478923\pi$$
$$770$$ 0 0
$$771$$ 40.7477 1.46749
$$772$$ − 6.74773i − 0.242856i
$$773$$ − 21.4955i − 0.773138i −0.922261 0.386569i $$-0.873660\pi$$
0.922261 0.386569i $$-0.126340\pi$$
$$774$$ −1.49545 −0.0537530
$$775$$ 0 0
$$776$$ −14.9564 −0.536905
$$777$$ 20.0000i 0.717496i
$$778$$ − 29.7042i − 1.06495i
$$779$$ 2.66970 0.0956518
$$780$$ 0 0
$$781$$ −20.3739 −0.729034
$$782$$ − 3.79129i − 0.135576i
$$783$$ 7.91288i 0.282783i
$$784$$ −0.791288 −0.0282603
$$785$$ 0 0
$$786$$ −16.4174 −0.585590
$$787$$ 8.41742i 0.300049i 0.988682 + 0.150024i $$0.0479352\pi$$
−0.988682 + 0.150024i $$0.952065\pi$$
$$788$$ 20.5390i 0.731672i
$$789$$ −28.2867 −1.00703
$$790$$ 0 0
$$791$$ 16.7477 0.595481
$$792$$ 0.791288i 0.0281172i
$$793$$ − 4.07803i − 0.144815i
$$794$$ 16.5390 0.586948
$$795$$ 0 0
$$796$$ 20.3303 0.720588
$$797$$ 49.9129i 1.76800i 0.467482 + 0.884002i $$0.345161\pi$$
−0.467482 + 0.884002i $$0.654839\pi$$
$$798$$ − 6.04356i − 0.213940i
$$799$$ −51.4955 −1.82178
$$800$$ 0 0
$$801$$ −0.660606 −0.0233413
$$802$$ 22.7477i 0.803250i
$$803$$ − 55.9129i − 1.97312i
$$804$$ 12.8348 0.452650
$$805$$ 0 0
$$806$$ −12.5390 −0.441668
$$807$$ − 30.0000i − 1.05605i
$$808$$ − 13.5826i − 0.477833i
$$809$$ −11.0436 −0.388271 −0.194135 0.980975i $$-0.562190\pi$$
−0.194135 + 0.980975i $$0.562190\pi$$
$$810$$ 0 0
$$811$$ −47.9129 −1.68245 −0.841224 0.540686i $$-0.818165\pi$$
−0.841224 + 0.540686i $$0.818165\pi$$
$$812$$ 4.41742i 0.155021i
$$813$$ − 41.4174i − 1.45257i
$$814$$ −15.1652 −0.531538
$$815$$ 0 0
$$816$$ −6.79129 −0.237743
$$817$$ 8.66061i 0.302996i
$$818$$ 22.7913i 0.796879i
$$819$$ −0.704166 −0.0246056
$$820$$ 0 0
$$821$$ 2.83485 0.0989369 0.0494684 0.998776i $$-0.484247\pi$$
0.0494684 + 0.998776i $$0.484247\pi$$
$$822$$ − 1.41742i − 0.0494383i
$$823$$ 41.1652i 1.43493i 0.696596 + 0.717463i $$0.254697\pi$$
−0.696596 + 0.717463i $$0.745303\pi$$
$$824$$ 7.37386 0.256881
$$825$$ 0 0
$$826$$ 12.3303 0.429026
$$827$$ − 41.0780i − 1.42842i −0.699930 0.714212i $$-0.746785\pi$$
0.699930 0.714212i $$-0.253215\pi$$
$$828$$ 0.208712i 0.00725325i
$$829$$ 31.4955 1.09388 0.546941 0.837171i $$-0.315792\pi$$
0.546941 + 0.837171i $$0.315792\pi$$
$$830$$ 0 0
$$831$$ −2.08712 −0.0724014
$$832$$ − 1.20871i − 0.0419046i
$$833$$ − 3.00000i − 0.103944i
$$834$$ −26.4174 −0.914761
$$835$$ 0 0
$$836$$ 4.58258 0.158492
$$837$$ 51.8693i 1.79287i
$$838$$ − 39.1652i − 1.35294i
$$839$$ 22.4174 0.773935 0.386968 0.922093i $$-0.373522\pi$$
0.386968 + 0.922093i $$0.373522\pi$$
$$840$$ 0 0
$$841$$ −26.4955 −0.913636
$$842$$ − 23.1216i − 0.796823i
$$843$$ − 30.0000i − 1.03325i
$$844$$ 10.0000 0.344214
$$845$$ 0 0
$$846$$ 2.83485 0.0974641
$$847$$ − 9.41742i − 0.323587i
$$848$$ 6.00000i 0.206041i
$$849$$ 50.7477 1.74166
$$850$$ 0 0
$$851$$ −4.00000 −0.137118
$$852$$ 9.62614i 0.329786i
$$853$$ 8.46099i 0.289699i 0.989454 + 0.144849i $$0.0462697\pi$$
−0.989454 + 0.144849i $$0.953730\pi$$
$$854$$ −9.41742 −0.322258
$$855$$ 0 0
$$856$$ −13.5826 −0.464243
$$857$$ − 9.16515i − 0.313076i −0.987672 0.156538i $$-0.949967\pi$$
0.987672 0.156538i $$-0.0500333\pi$$
$$858$$ − 8.20871i − 0.280241i
$$859$$ −0.747727 −0.0255121 −0.0127561 0.999919i $$-0.504060\pi$$
−0.0127561 + 0.999919i $$0.504060\pi$$
$$860$$ 0 0
$$861$$ −11.0436 −0.376364
$$862$$ − 19.9129i − 0.678235i
$$863$$ − 31.5826i − 1.07508i −0.843237 0.537542i $$-0.819353\pi$$
0.843237 0.537542i $$-0.180647\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −1.53901 −0.0522979
$$867$$ 4.70417i 0.159762i
$$868$$ 28.9564i 0.982846i
$$869$$ −30.3303 −1.02889
$$870$$ 0 0
$$871$$ −8.66061 −0.293453
$$872$$ 10.3739i 0.351303i
$$873$$ 3.12159i 0.105650i
$$874$$ 1.20871 0.0408853
$$875$$ 0 0
$$876$$ −26.4174 −0.892562
$$877$$ − 7.70417i − 0.260151i −0.991504 0.130076i $$-0.958478\pi$$
0.991504 0.130076i $$-0.0415220\pi$$
$$878$$ − 25.5390i − 0.861900i
$$879$$ 49.2523 1.66124
$$880$$ 0 0
$$881$$ −6.33030 −0.213273 −0.106637 0.994298i $$-0.534008\pi$$
−0.106637 + 0.994298i $$0.534008\pi$$
$$882$$ 0.165151i 0.00556094i
$$883$$ − 12.0436i − 0.405298i −0.979251 0.202649i $$-0.935045\pi$$
0.979251 0.202649i $$-0.0649551\pi$$
$$884$$ 4.58258 0.154129
$$885$$ 0 0
$$886$$ 35.2087 1.18286
$$887$$ 3.16515i 0.106275i 0.998587 + 0.0531377i $$0.0169222\pi$$
−0.998587 + 0.0531377i $$0.983078\pi$$
$$888$$ 7.16515i 0.240447i
$$889$$ 41.1652 1.38063
$$890$$ 0 0
$$891$$ −36.3303 −1.21711
$$892$$ − 11.1652i − 0.373837i
$$893$$ − 16.4174i − 0.549388i
$$894$$ 22.9129 0.766321
$$895$$ 0 0
$$896$$ −2.79129 −0.0932504
$$897$$ − 2.16515i − 0.0722923i
$$898$$ − 25.1216i − 0.838318i
$$899$$ 16.4174 0.547552
$$900$$ 0 0
$$901$$ −22.7477 −0.757837
$$902$$ − 8.37386i − 0.278819i
$$903$$ − 35.8258i − 1.19221i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 11.1216 0.369490
$$907$$ 20.7477i 0.688917i 0.938802 + 0.344458i $$0.111937\pi$$
−0.938802 + 0.344458i $$0.888063\pi$$
$$908$$ 4.74773i 0.157559i
$$909$$ −2.83485 −0.0940260
$$910$$ 0 0
$$911$$ 13.5826 0.450011 0.225005 0.974358i $$-0.427760\pi$$
0.225005 + 0.974358i $$0.427760\pi$$
$$912$$ − 2.16515i − 0.0716953i
$$913$$ − 22.7477i − 0.752840i
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −16.3303 −0.539568
$$917$$ − 25.5826i − 0.844811i
$$918$$ − 18.9564i − 0.625656i
$$919$$ 36.8348 1.21507 0.607535 0.794293i $$-0.292159\pi$$
0.607535 + 0.794293i $$0.292159\pi$$
$$920$$ 0 0
$$921$$ 29.6261 0.976214
$$922$$ − 1.25227i − 0.0412414i
$$923$$ − 6.49545i − 0.213800i
$$924$$ −18.9564 −0.623621
$$925$$ 0 0
$$926$$ 10.0000 0.328620
$$927$$ − 1.53901i − 0.0505479i
$$928$$ 1.58258i 0.0519506i
$$929$$ 39.4955 1.29580 0.647902 0.761724i $$-0.275647\pi$$
0.647902 + 0.761724i $$0.275647\pi$$
$$930$$ 0 0
$$931$$ 0.956439 0.0313460
$$932$$ − 7.58258i − 0.248376i
$$933$$ 21.4955i 0.703730i
$$934$$ 25.9129 0.847895
$$935$$ 0 0
$$936$$ −0.252273 −0.00824580
$$937$$ − 58.3739i − 1.90699i −0.301407 0.953495i $$-0.597456\pi$$
0.301407 0.953495i $$-0.402544\pi$$
$$938$$ 20.0000i 0.653023i
$$939$$ 32.9129 1.07407
$$940$$ 0 0
$$941$$ 54.9564 1.79153 0.895764 0.444529i $$-0.146629\pi$$
0.895764 + 0.444529i $$0.146629\pi$$
$$942$$ 22.8348i 0.744000i
$$943$$ − 2.20871i − 0.0719256i
$$944$$ 4.41742 0.143775
$$945$$ 0 0
$$946$$ 27.1652 0.883215
$$947$$ 29.5390i 0.959889i 0.877299 + 0.479945i $$0.159343\pi$$
−0.877299 + 0.479945i $$0.840657\pi$$
$$948$$ 14.3303i 0.465427i
$$949$$ 17.8258 0.578649
$$950$$ 0 0
$$951$$ 9.33030 0.302556
$$952$$ − 10.5826i − 0.342983i
$$953$$ − 26.5390i − 0.859683i −0.902904 0.429842i $$-0.858569\pi$$
0.902904 0.429842i $$-0.141431\pi$$
$$954$$ 1.25227 0.0405438
$$955$$ 0 0
$$956$$ 3.16515 0.102368
$$957$$ 10.7477i 0.347425i
$$958$$ 39.4955i 1.27604i
$$959$$ 2.20871 0.0713230
$$960$$ 0 0
$$961$$ 76.6170 2.47152
$$962$$ − 4.83485i − 0.155882i
$$963$$ 2.83485i 0.0913517i
$$964$$ 28.0000 0.901819
$$965$$ 0 0
$$966$$ −5.00000 −0.160872
$$967$$ 5.25227i 0.168902i 0.996428 + 0.0844509i $$0.0269136\pi$$
−0.996428 + 0.0844509i $$0.973086\pi$$
$$968$$ − 3.37386i − 0.108440i
$$969$$ 8.20871 0.263702
$$970$$ 0 0
$$971$$ −6.95644 −0.223243 −0.111621 0.993751i $$-0.535604\pi$$
−0.111621 + 0.993751i $$0.535604\pi$$
$$972$$ 2.16515i 0.0694473i
$$973$$ − 41.1652i − 1.31969i
$$974$$ 15.5826 0.499298
$$975$$ 0 0
$$976$$ −3.37386 −0.107995
$$977$$ − 7.12159i − 0.227840i −0.993490 0.113920i $$-0.963659\pi$$
0.993490 0.113920i $$-0.0363407\pi$$
$$978$$ − 40.0780i − 1.28155i
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ 2.16515 0.0691280
$$982$$ − 16.7477i − 0.534441i
$$983$$ − 0.626136i − 0.0199707i −0.999950 0.00998533i $$-0.996822\pi$$
0.999950 0.00998533i $$-0.00317848\pi$$
$$984$$ −3.95644 −0.126127
$$985$$ 0 0
$$986$$ −6.00000 −0.191079
$$987$$ 67.9129i 2.16169i
$$988$$ 1.46099i 0.0464801i
$$989$$ 7.16515 0.227839
$$990$$ 0 0
$$991$$ −37.7913 −1.20048 −0.600240 0.799820i $$-0.704928\pi$$
−0.600240 + 0.799820i $$0.704928\pi$$
$$992$$ 10.3739i 0.329370i
$$993$$ 12.0871i 0.383573i
$$994$$ −15.0000 −0.475771
$$995$$ 0 0
$$996$$ −10.7477 −0.340555
$$997$$ − 11.4955i − 0.364065i −0.983293 0.182032i $$-0.941732\pi$$
0.983293 0.182032i $$-0.0582676\pi$$
$$998$$ − 23.1652i − 0.733280i
$$999$$ −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.b.g.599.4 4
5.2 odd 4 230.2.a.a.1.2 2
5.3 odd 4 1150.2.a.o.1.1 2
5.4 even 2 inner 1150.2.b.g.599.1 4
15.2 even 4 2070.2.a.x.1.2 2
20.3 even 4 9200.2.a.bs.1.2 2
20.7 even 4 1840.2.a.n.1.1 2
40.27 even 4 7360.2.a.bk.1.2 2
40.37 odd 4 7360.2.a.bq.1.1 2
115.22 even 4 5290.2.a.e.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.2 2 5.2 odd 4
1150.2.a.o.1.1 2 5.3 odd 4
1150.2.b.g.599.1 4 5.4 even 2 inner
1150.2.b.g.599.4 4 1.1 even 1 trivial
1840.2.a.n.1.1 2 20.7 even 4
2070.2.a.x.1.2 2 15.2 even 4
5290.2.a.e.1.2 2 115.22 even 4
7360.2.a.bk.1.2 2 40.27 even 4
7360.2.a.bq.1.1 2 40.37 odd 4
9200.2.a.bs.1.2 2 20.3 even 4