# Properties

 Label 9075.2.a.br.1.2 Level $9075$ Weight $2$ Character 9075.1 Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 9075.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} +2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} +2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +3.00000 q^{12} +4.47214 q^{13} +5.00000 q^{14} -1.00000 q^{16} +2.23607 q^{17} +2.23607 q^{18} +2.23607 q^{19} +2.23607 q^{21} +1.00000 q^{23} +2.23607 q^{24} +10.0000 q^{26} +1.00000 q^{27} +6.70820 q^{28} -4.47214 q^{29} +10.0000 q^{31} -6.70820 q^{32} +5.00000 q^{34} +3.00000 q^{36} +7.00000 q^{37} +5.00000 q^{38} +4.47214 q^{39} -6.70820 q^{41} +5.00000 q^{42} +2.23607 q^{46} +3.00000 q^{47} -1.00000 q^{48} -2.00000 q^{49} +2.23607 q^{51} +13.4164 q^{52} -14.0000 q^{53} +2.23607 q^{54} +5.00000 q^{56} +2.23607 q^{57} -10.0000 q^{58} -5.00000 q^{59} -4.47214 q^{61} +22.3607 q^{62} +2.23607 q^{63} -13.0000 q^{64} -2.00000 q^{67} +6.70820 q^{68} +1.00000 q^{69} -13.0000 q^{71} +2.23607 q^{72} +13.4164 q^{73} +15.6525 q^{74} +6.70820 q^{76} +10.0000 q^{78} +15.6525 q^{79} +1.00000 q^{81} -15.0000 q^{82} +4.47214 q^{83} +6.70820 q^{84} -4.47214 q^{87} +6.00000 q^{89} +10.0000 q^{91} +3.00000 q^{92} +10.0000 q^{93} +6.70820 q^{94} -6.70820 q^{96} +17.0000 q^{97} -4.47214 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 6 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 6 * q^4 + 2 * q^9 $$2 q + 2 q^{3} + 6 q^{4} + 2 q^{9} + 6 q^{12} + 10 q^{14} - 2 q^{16} + 2 q^{23} + 20 q^{26} + 2 q^{27} + 20 q^{31} + 10 q^{34} + 6 q^{36} + 14 q^{37} + 10 q^{38} + 10 q^{42} + 6 q^{47} - 2 q^{48} - 4 q^{49} - 28 q^{53} + 10 q^{56} - 20 q^{58} - 10 q^{59} - 26 q^{64} - 4 q^{67} + 2 q^{69} - 26 q^{71} + 20 q^{78} + 2 q^{81} - 30 q^{82} + 12 q^{89} + 20 q^{91} + 6 q^{92} + 20 q^{93} + 34 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 6 * q^4 + 2 * q^9 + 6 * q^12 + 10 * q^14 - 2 * q^16 + 2 * q^23 + 20 * q^26 + 2 * q^27 + 20 * q^31 + 10 * q^34 + 6 * q^36 + 14 * q^37 + 10 * q^38 + 10 * q^42 + 6 * q^47 - 2 * q^48 - 4 * q^49 - 28 * q^53 + 10 * q^56 - 20 * q^58 - 10 * q^59 - 26 * q^64 - 4 * q^67 + 2 * q^69 - 26 * q^71 + 20 * q^78 + 2 * q^81 - 30 * q^82 + 12 * q^89 + 20 * q^91 + 6 * q^92 + 20 * q^93 + 34 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.23607 1.58114 0.790569 0.612372i $$-0.209785\pi$$
0.790569 + 0.612372i $$0.209785\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 3.00000 1.50000
$$5$$ 0 0
$$6$$ 2.23607 0.912871
$$7$$ 2.23607 0.845154 0.422577 0.906327i $$-0.361126\pi$$
0.422577 + 0.906327i $$0.361126\pi$$
$$8$$ 2.23607 0.790569
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 3.00000 0.866025
$$13$$ 4.47214 1.24035 0.620174 0.784465i $$-0.287062\pi$$
0.620174 + 0.784465i $$0.287062\pi$$
$$14$$ 5.00000 1.33631
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 2.23607 0.542326 0.271163 0.962533i $$-0.412592\pi$$
0.271163 + 0.962533i $$0.412592\pi$$
$$18$$ 2.23607 0.527046
$$19$$ 2.23607 0.512989 0.256495 0.966546i $$-0.417432\pi$$
0.256495 + 0.966546i $$0.417432\pi$$
$$20$$ 0 0
$$21$$ 2.23607 0.487950
$$22$$ 0 0
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ 10.0000 1.96116
$$27$$ 1.00000 0.192450
$$28$$ 6.70820 1.26773
$$29$$ −4.47214 −0.830455 −0.415227 0.909718i $$-0.636298\pi$$
−0.415227 + 0.909718i $$0.636298\pi$$
$$30$$ 0 0
$$31$$ 10.0000 1.79605 0.898027 0.439941i $$-0.145001\pi$$
0.898027 + 0.439941i $$0.145001\pi$$
$$32$$ −6.70820 −1.18585
$$33$$ 0 0
$$34$$ 5.00000 0.857493
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ 5.00000 0.811107
$$39$$ 4.47214 0.716115
$$40$$ 0 0
$$41$$ −6.70820 −1.04765 −0.523823 0.851827i $$-0.675495\pi$$
−0.523823 + 0.851827i $$0.675495\pi$$
$$42$$ 5.00000 0.771517
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 2.23607 0.329690
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 2.23607 0.313112
$$52$$ 13.4164 1.86052
$$53$$ −14.0000 −1.92305 −0.961524 0.274721i $$-0.911414\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 2.23607 0.304290
$$55$$ 0 0
$$56$$ 5.00000 0.668153
$$57$$ 2.23607 0.296174
$$58$$ −10.0000 −1.31306
$$59$$ −5.00000 −0.650945 −0.325472 0.945552i $$-0.605523\pi$$
−0.325472 + 0.945552i $$0.605523\pi$$
$$60$$ 0 0
$$61$$ −4.47214 −0.572598 −0.286299 0.958140i $$-0.592425\pi$$
−0.286299 + 0.958140i $$0.592425\pi$$
$$62$$ 22.3607 2.83981
$$63$$ 2.23607 0.281718
$$64$$ −13.0000 −1.62500
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ 6.70820 0.813489
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −13.0000 −1.54282 −0.771408 0.636341i $$-0.780447\pi$$
−0.771408 + 0.636341i $$0.780447\pi$$
$$72$$ 2.23607 0.263523
$$73$$ 13.4164 1.57027 0.785136 0.619324i $$-0.212593\pi$$
0.785136 + 0.619324i $$0.212593\pi$$
$$74$$ 15.6525 1.81956
$$75$$ 0 0
$$76$$ 6.70820 0.769484
$$77$$ 0 0
$$78$$ 10.0000 1.13228
$$79$$ 15.6525 1.76104 0.880521 0.474008i $$-0.157193\pi$$
0.880521 + 0.474008i $$0.157193\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −15.0000 −1.65647
$$83$$ 4.47214 0.490881 0.245440 0.969412i $$-0.421067\pi$$
0.245440 + 0.969412i $$0.421067\pi$$
$$84$$ 6.70820 0.731925
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −4.47214 −0.479463
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 10.0000 1.04828
$$92$$ 3.00000 0.312772
$$93$$ 10.0000 1.03695
$$94$$ 6.70820 0.691898
$$95$$ 0 0
$$96$$ −6.70820 −0.684653
$$97$$ 17.0000 1.72609 0.863044 0.505128i $$-0.168555\pi$$
0.863044 + 0.505128i $$0.168555\pi$$
$$98$$ −4.47214 −0.451754
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −11.1803 −1.11249 −0.556243 0.831020i $$-0.687758\pi$$
−0.556243 + 0.831020i $$0.687758\pi$$
$$102$$ 5.00000 0.495074
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 10.0000 0.980581
$$105$$ 0 0
$$106$$ −31.3050 −3.04061
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 3.00000 0.288675
$$109$$ −17.8885 −1.71341 −0.856706 0.515805i $$-0.827493\pi$$
−0.856706 + 0.515805i $$0.827493\pi$$
$$110$$ 0 0
$$111$$ 7.00000 0.664411
$$112$$ −2.23607 −0.211289
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 5.00000 0.468293
$$115$$ 0 0
$$116$$ −13.4164 −1.24568
$$117$$ 4.47214 0.413449
$$118$$ −11.1803 −1.02923
$$119$$ 5.00000 0.458349
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −10.0000 −0.905357
$$123$$ −6.70820 −0.604858
$$124$$ 30.0000 2.69408
$$125$$ 0 0
$$126$$ 5.00000 0.445435
$$127$$ 15.6525 1.38893 0.694466 0.719525i $$-0.255641\pi$$
0.694466 + 0.719525i $$0.255641\pi$$
$$128$$ −15.6525 −1.38350
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 13.4164 1.17220 0.586098 0.810240i $$-0.300663\pi$$
0.586098 + 0.810240i $$0.300663\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ −4.47214 −0.386334
$$135$$ 0 0
$$136$$ 5.00000 0.428746
$$137$$ −8.00000 −0.683486 −0.341743 0.939793i $$-0.611017\pi$$
−0.341743 + 0.939793i $$0.611017\pi$$
$$138$$ 2.23607 0.190347
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ −29.0689 −2.43941
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 30.0000 2.48282
$$147$$ −2.00000 −0.164957
$$148$$ 21.0000 1.72619
$$149$$ 20.1246 1.64867 0.824336 0.566101i $$-0.191549\pi$$
0.824336 + 0.566101i $$0.191549\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 5.00000 0.405554
$$153$$ 2.23607 0.180775
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 13.4164 1.07417
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 35.0000 2.78445
$$159$$ −14.0000 −1.11027
$$160$$ 0 0
$$161$$ 2.23607 0.176227
$$162$$ 2.23607 0.175682
$$163$$ 14.0000 1.09656 0.548282 0.836293i $$-0.315282\pi$$
0.548282 + 0.836293i $$0.315282\pi$$
$$164$$ −20.1246 −1.57147
$$165$$ 0 0
$$166$$ 10.0000 0.776151
$$167$$ 4.47214 0.346064 0.173032 0.984916i $$-0.444644\pi$$
0.173032 + 0.984916i $$0.444644\pi$$
$$168$$ 5.00000 0.385758
$$169$$ 7.00000 0.538462
$$170$$ 0 0
$$171$$ 2.23607 0.170996
$$172$$ 0 0
$$173$$ 20.1246 1.53005 0.765023 0.644003i $$-0.222728\pi$$
0.765023 + 0.644003i $$0.222728\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −5.00000 −0.375823
$$178$$ 13.4164 1.00560
$$179$$ 11.0000 0.822179 0.411089 0.911595i $$-0.365148\pi$$
0.411089 + 0.911595i $$0.365148\pi$$
$$180$$ 0 0
$$181$$ −5.00000 −0.371647 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$182$$ 22.3607 1.65748
$$183$$ −4.47214 −0.330590
$$184$$ 2.23607 0.164845
$$185$$ 0 0
$$186$$ 22.3607 1.63956
$$187$$ 0 0
$$188$$ 9.00000 0.656392
$$189$$ 2.23607 0.162650
$$190$$ 0 0
$$191$$ 15.0000 1.08536 0.542681 0.839939i $$-0.317409\pi$$
0.542681 + 0.839939i $$0.317409\pi$$
$$192$$ −13.0000 −0.938194
$$193$$ −8.94427 −0.643823 −0.321911 0.946770i $$-0.604325\pi$$
−0.321911 + 0.946770i $$0.604325\pi$$
$$194$$ 38.0132 2.72919
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ −15.6525 −1.11519 −0.557596 0.830112i $$-0.688276\pi$$
−0.557596 + 0.830112i $$0.688276\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ −25.0000 −1.75899
$$203$$ −10.0000 −0.701862
$$204$$ 6.70820 0.469668
$$205$$ 0 0
$$206$$ −8.94427 −0.623177
$$207$$ 1.00000 0.0695048
$$208$$ −4.47214 −0.310087
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −26.8328 −1.84725 −0.923624 0.383301i $$-0.874787\pi$$
−0.923624 + 0.383301i $$0.874787\pi$$
$$212$$ −42.0000 −2.88457
$$213$$ −13.0000 −0.890745
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 2.23607 0.152145
$$217$$ 22.3607 1.51794
$$218$$ −40.0000 −2.70914
$$219$$ 13.4164 0.906597
$$220$$ 0 0
$$221$$ 10.0000 0.672673
$$222$$ 15.6525 1.05053
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ −15.0000 −1.00223
$$225$$ 0 0
$$226$$ −31.3050 −2.08237
$$227$$ 13.4164 0.890478 0.445239 0.895412i $$-0.353119\pi$$
0.445239 + 0.895412i $$0.353119\pi$$
$$228$$ 6.70820 0.444262
$$229$$ −5.00000 −0.330409 −0.165205 0.986259i $$-0.552828\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −10.0000 −0.656532
$$233$$ −20.1246 −1.31841 −0.659204 0.751965i $$-0.729106\pi$$
−0.659204 + 0.751965i $$0.729106\pi$$
$$234$$ 10.0000 0.653720
$$235$$ 0 0
$$236$$ −15.0000 −0.976417
$$237$$ 15.6525 1.01674
$$238$$ 11.1803 0.724714
$$239$$ −8.94427 −0.578557 −0.289278 0.957245i $$-0.593415\pi$$
−0.289278 + 0.957245i $$0.593415\pi$$
$$240$$ 0 0
$$241$$ −22.3607 −1.44038 −0.720189 0.693778i $$-0.755945\pi$$
−0.720189 + 0.693778i $$0.755945\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ −13.4164 −0.858898
$$245$$ 0 0
$$246$$ −15.0000 −0.956365
$$247$$ 10.0000 0.636285
$$248$$ 22.3607 1.41990
$$249$$ 4.47214 0.283410
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 6.70820 0.422577
$$253$$ 0 0
$$254$$ 35.0000 2.19610
$$255$$ 0 0
$$256$$ −9.00000 −0.562500
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 15.6525 0.972598
$$260$$ 0 0
$$261$$ −4.47214 −0.276818
$$262$$ 30.0000 1.85341
$$263$$ −17.8885 −1.10305 −0.551527 0.834157i $$-0.685955\pi$$
−0.551527 + 0.834157i $$0.685955\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 11.1803 0.685511
$$267$$ 6.00000 0.367194
$$268$$ −6.00000 −0.366508
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ −20.1246 −1.22248 −0.611242 0.791444i $$-0.709330\pi$$
−0.611242 + 0.791444i $$0.709330\pi$$
$$272$$ −2.23607 −0.135582
$$273$$ 10.0000 0.605228
$$274$$ −17.8885 −1.08069
$$275$$ 0 0
$$276$$ 3.00000 0.180579
$$277$$ −13.4164 −0.806114 −0.403057 0.915175i $$-0.632052\pi$$
−0.403057 + 0.915175i $$0.632052\pi$$
$$278$$ 0 0
$$279$$ 10.0000 0.598684
$$280$$ 0 0
$$281$$ −24.5967 −1.46732 −0.733659 0.679517i $$-0.762189\pi$$
−0.733659 + 0.679517i $$0.762189\pi$$
$$282$$ 6.70820 0.399468
$$283$$ −6.70820 −0.398761 −0.199381 0.979922i $$-0.563893\pi$$
−0.199381 + 0.979922i $$0.563893\pi$$
$$284$$ −39.0000 −2.31422
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −15.0000 −0.885422
$$288$$ −6.70820 −0.395285
$$289$$ −12.0000 −0.705882
$$290$$ 0 0
$$291$$ 17.0000 0.996558
$$292$$ 40.2492 2.35541
$$293$$ 15.6525 0.914427 0.457214 0.889357i $$-0.348847\pi$$
0.457214 + 0.889357i $$0.348847\pi$$
$$294$$ −4.47214 −0.260820
$$295$$ 0 0
$$296$$ 15.6525 0.909782
$$297$$ 0 0
$$298$$ 45.0000 2.60678
$$299$$ 4.47214 0.258630
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −11.1803 −0.642294
$$304$$ −2.23607 −0.128247
$$305$$ 0 0
$$306$$ 5.00000 0.285831
$$307$$ −17.8885 −1.02095 −0.510477 0.859892i $$-0.670531\pi$$
−0.510477 + 0.859892i $$0.670531\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −32.0000 −1.81455 −0.907277 0.420534i $$-0.861843\pi$$
−0.907277 + 0.420534i $$0.861843\pi$$
$$312$$ 10.0000 0.566139
$$313$$ 9.00000 0.508710 0.254355 0.967111i $$-0.418137\pi$$
0.254355 + 0.967111i $$0.418137\pi$$
$$314$$ 4.47214 0.252377
$$315$$ 0 0
$$316$$ 46.9574 2.64156
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ −31.3050 −1.75549
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 5.00000 0.278639
$$323$$ 5.00000 0.278207
$$324$$ 3.00000 0.166667
$$325$$ 0 0
$$326$$ 31.3050 1.73382
$$327$$ −17.8885 −0.989239
$$328$$ −15.0000 −0.828236
$$329$$ 6.70820 0.369835
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 13.4164 0.736321
$$333$$ 7.00000 0.383598
$$334$$ 10.0000 0.547176
$$335$$ 0 0
$$336$$ −2.23607 −0.121988
$$337$$ −13.4164 −0.730838 −0.365419 0.930843i $$-0.619074\pi$$
−0.365419 + 0.930843i $$0.619074\pi$$
$$338$$ 15.6525 0.851382
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 5.00000 0.270369
$$343$$ −20.1246 −1.08663
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 45.0000 2.41921
$$347$$ 13.4164 0.720231 0.360115 0.932908i $$-0.382737\pi$$
0.360115 + 0.932908i $$0.382737\pi$$
$$348$$ −13.4164 −0.719195
$$349$$ 13.4164 0.718164 0.359082 0.933306i $$-0.383090\pi$$
0.359082 + 0.933306i $$0.383090\pi$$
$$350$$ 0 0
$$351$$ 4.47214 0.238705
$$352$$ 0 0
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ −11.1803 −0.594228
$$355$$ 0 0
$$356$$ 18.0000 0.953998
$$357$$ 5.00000 0.264628
$$358$$ 24.5967 1.29998
$$359$$ −4.47214 −0.236030 −0.118015 0.993012i $$-0.537653\pi$$
−0.118015 + 0.993012i $$0.537653\pi$$
$$360$$ 0 0
$$361$$ −14.0000 −0.736842
$$362$$ −11.1803 −0.587626
$$363$$ 0 0
$$364$$ 30.0000 1.57243
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 18.0000 0.939592 0.469796 0.882775i $$-0.344327\pi$$
0.469796 + 0.882775i $$0.344327\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −6.70820 −0.349215
$$370$$ 0 0
$$371$$ −31.3050 −1.62527
$$372$$ 30.0000 1.55543
$$373$$ −8.94427 −0.463117 −0.231558 0.972821i $$-0.574382\pi$$
−0.231558 + 0.972821i $$0.574382\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 6.70820 0.345949
$$377$$ −20.0000 −1.03005
$$378$$ 5.00000 0.257172
$$379$$ 10.0000 0.513665 0.256833 0.966456i $$-0.417321\pi$$
0.256833 + 0.966456i $$0.417321\pi$$
$$380$$ 0 0
$$381$$ 15.6525 0.801901
$$382$$ 33.5410 1.71611
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ −15.6525 −0.798762
$$385$$ 0 0
$$386$$ −20.0000 −1.01797
$$387$$ 0 0
$$388$$ 51.0000 2.58913
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 2.23607 0.113083
$$392$$ −4.47214 −0.225877
$$393$$ 13.4164 0.676768
$$394$$ −35.0000 −1.76327
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 22.3607 1.12084
$$399$$ 5.00000 0.250313
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ −4.47214 −0.223050
$$403$$ 44.7214 2.22773
$$404$$ −33.5410 −1.66873
$$405$$ 0 0
$$406$$ −22.3607 −1.10974
$$407$$ 0 0
$$408$$ 5.00000 0.247537
$$409$$ −13.4164 −0.663399 −0.331699 0.943385i $$-0.607622\pi$$
−0.331699 + 0.943385i $$0.607622\pi$$
$$410$$ 0 0
$$411$$ −8.00000 −0.394611
$$412$$ −12.0000 −0.591198
$$413$$ −11.1803 −0.550149
$$414$$ 2.23607 0.109897
$$415$$ 0 0
$$416$$ −30.0000 −1.47087
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 9.00000 0.439679 0.219839 0.975536i $$-0.429447\pi$$
0.219839 + 0.975536i $$0.429447\pi$$
$$420$$ 0 0
$$421$$ −15.0000 −0.731055 −0.365528 0.930800i $$-0.619111\pi$$
−0.365528 + 0.930800i $$0.619111\pi$$
$$422$$ −60.0000 −2.92075
$$423$$ 3.00000 0.145865
$$424$$ −31.3050 −1.52030
$$425$$ 0 0
$$426$$ −29.0689 −1.40839
$$427$$ −10.0000 −0.483934
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −22.3607 −1.07708 −0.538538 0.842601i $$-0.681023\pi$$
−0.538538 + 0.842601i $$0.681023\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 50.0000 2.40008
$$435$$ 0 0
$$436$$ −53.6656 −2.57012
$$437$$ 2.23607 0.106966
$$438$$ 30.0000 1.43346
$$439$$ −20.1246 −0.960495 −0.480248 0.877133i $$-0.659453\pi$$
−0.480248 + 0.877133i $$0.659453\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ 22.3607 1.06359
$$443$$ 21.0000 0.997740 0.498870 0.866677i $$-0.333748\pi$$
0.498870 + 0.866677i $$0.333748\pi$$
$$444$$ 21.0000 0.996616
$$445$$ 0 0
$$446$$ 31.3050 1.48233
$$447$$ 20.1246 0.951861
$$448$$ −29.0689 −1.37338
$$449$$ 4.00000 0.188772 0.0943858 0.995536i $$-0.469911\pi$$
0.0943858 + 0.995536i $$0.469911\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −42.0000 −1.97551
$$453$$ 0 0
$$454$$ 30.0000 1.40797
$$455$$ 0 0
$$456$$ 5.00000 0.234146
$$457$$ −22.3607 −1.04599 −0.522994 0.852336i $$-0.675185\pi$$
−0.522994 + 0.852336i $$0.675185\pi$$
$$458$$ −11.1803 −0.522423
$$459$$ 2.23607 0.104371
$$460$$ 0 0
$$461$$ 22.3607 1.04144 0.520720 0.853727i $$-0.325663\pi$$
0.520720 + 0.853727i $$0.325663\pi$$
$$462$$ 0 0
$$463$$ 14.0000 0.650635 0.325318 0.945605i $$-0.394529\pi$$
0.325318 + 0.945605i $$0.394529\pi$$
$$464$$ 4.47214 0.207614
$$465$$ 0 0
$$466$$ −45.0000 −2.08458
$$467$$ 32.0000 1.48078 0.740392 0.672176i $$-0.234640\pi$$
0.740392 + 0.672176i $$0.234640\pi$$
$$468$$ 13.4164 0.620174
$$469$$ −4.47214 −0.206504
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ −11.1803 −0.514617
$$473$$ 0 0
$$474$$ 35.0000 1.60760
$$475$$ 0 0
$$476$$ 15.0000 0.687524
$$477$$ −14.0000 −0.641016
$$478$$ −20.0000 −0.914779
$$479$$ −22.3607 −1.02169 −0.510843 0.859674i $$-0.670667\pi$$
−0.510843 + 0.859674i $$0.670667\pi$$
$$480$$ 0 0
$$481$$ 31.3050 1.42738
$$482$$ −50.0000 −2.27744
$$483$$ 2.23607 0.101745
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 2.23607 0.101430
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 14.0000 0.633102
$$490$$ 0 0
$$491$$ 4.47214 0.201825 0.100912 0.994895i $$-0.467824\pi$$
0.100912 + 0.994895i $$0.467824\pi$$
$$492$$ −20.1246 −0.907288
$$493$$ −10.0000 −0.450377
$$494$$ 22.3607 1.00605
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ −29.0689 −1.30392
$$498$$ 10.0000 0.448111
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ 4.47214 0.199800
$$502$$ −26.8328 −1.19761
$$503$$ −35.7771 −1.59522 −0.797611 0.603173i $$-0.793903\pi$$
−0.797611 + 0.603173i $$0.793903\pi$$
$$504$$ 5.00000 0.222718
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 7.00000 0.310881
$$508$$ 46.9574 2.08340
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 30.0000 1.32712
$$512$$ 11.1803 0.494106
$$513$$ 2.23607 0.0987248
$$514$$ 40.2492 1.77532
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 35.0000 1.53781
$$519$$ 20.1246 0.883372
$$520$$ 0 0
$$521$$ 20.0000 0.876216 0.438108 0.898922i $$-0.355649\pi$$
0.438108 + 0.898922i $$0.355649\pi$$
$$522$$ −10.0000 −0.437688
$$523$$ 6.70820 0.293329 0.146665 0.989186i $$-0.453146\pi$$
0.146665 + 0.989186i $$0.453146\pi$$
$$524$$ 40.2492 1.75830
$$525$$ 0 0
$$526$$ −40.0000 −1.74408
$$527$$ 22.3607 0.974047
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ −5.00000 −0.216982
$$532$$ 15.0000 0.650332
$$533$$ −30.0000 −1.29944
$$534$$ 13.4164 0.580585
$$535$$ 0 0
$$536$$ −4.47214 −0.193167
$$537$$ 11.0000 0.474685
$$538$$ 67.0820 2.89211
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −17.8885 −0.769089 −0.384544 0.923107i $$-0.625641\pi$$
−0.384544 + 0.923107i $$0.625641\pi$$
$$542$$ −45.0000 −1.93292
$$543$$ −5.00000 −0.214571
$$544$$ −15.0000 −0.643120
$$545$$ 0 0
$$546$$ 22.3607 0.956949
$$547$$ −20.1246 −0.860466 −0.430233 0.902718i $$-0.641569\pi$$
−0.430233 + 0.902718i $$0.641569\pi$$
$$548$$ −24.0000 −1.02523
$$549$$ −4.47214 −0.190866
$$550$$ 0 0
$$551$$ −10.0000 −0.426014
$$552$$ 2.23607 0.0951734
$$553$$ 35.0000 1.48835
$$554$$ −30.0000 −1.27458
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 22.3607 0.947452 0.473726 0.880672i $$-0.342909\pi$$
0.473726 + 0.880672i $$0.342909\pi$$
$$558$$ 22.3607 0.946603
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −55.0000 −2.32003
$$563$$ −35.7771 −1.50782 −0.753912 0.656975i $$-0.771836\pi$$
−0.753912 + 0.656975i $$0.771836\pi$$
$$564$$ 9.00000 0.378968
$$565$$ 0 0
$$566$$ −15.0000 −0.630497
$$567$$ 2.23607 0.0939060
$$568$$ −29.0689 −1.21970
$$569$$ −11.1803 −0.468704 −0.234352 0.972152i $$-0.575297\pi$$
−0.234352 + 0.972152i $$0.575297\pi$$
$$570$$ 0 0
$$571$$ −35.7771 −1.49722 −0.748612 0.663008i $$-0.769280\pi$$
−0.748612 + 0.663008i $$0.769280\pi$$
$$572$$ 0 0
$$573$$ 15.0000 0.626634
$$574$$ −33.5410 −1.39998
$$575$$ 0 0
$$576$$ −13.0000 −0.541667
$$577$$ 3.00000 0.124892 0.0624458 0.998048i $$-0.480110\pi$$
0.0624458 + 0.998048i $$0.480110\pi$$
$$578$$ −26.8328 −1.11610
$$579$$ −8.94427 −0.371711
$$580$$ 0 0
$$581$$ 10.0000 0.414870
$$582$$ 38.0132 1.57570
$$583$$ 0 0
$$584$$ 30.0000 1.24141
$$585$$ 0 0
$$586$$ 35.0000 1.44584
$$587$$ −23.0000 −0.949312 −0.474656 0.880172i $$-0.657427\pi$$
−0.474656 + 0.880172i $$0.657427\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ 22.3607 0.921356
$$590$$ 0 0
$$591$$ −15.6525 −0.643857
$$592$$ −7.00000 −0.287698
$$593$$ 4.47214 0.183649 0.0918243 0.995775i $$-0.470730\pi$$
0.0918243 + 0.995775i $$0.470730\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 60.3738 2.47301
$$597$$ 10.0000 0.409273
$$598$$ 10.0000 0.408930
$$599$$ 9.00000 0.367730 0.183865 0.982952i $$-0.441139\pi$$
0.183865 + 0.982952i $$0.441139\pi$$
$$600$$ 0 0
$$601$$ 13.4164 0.547267 0.273633 0.961834i $$-0.411775\pi$$
0.273633 + 0.961834i $$0.411775\pi$$
$$602$$ 0 0
$$603$$ −2.00000 −0.0814463
$$604$$ 0 0
$$605$$ 0 0
$$606$$ −25.0000 −1.01556
$$607$$ −26.8328 −1.08911 −0.544555 0.838725i $$-0.683302\pi$$
−0.544555 + 0.838725i $$0.683302\pi$$
$$608$$ −15.0000 −0.608330
$$609$$ −10.0000 −0.405220
$$610$$ 0 0
$$611$$ 13.4164 0.542770
$$612$$ 6.70820 0.271163
$$613$$ −8.94427 −0.361256 −0.180628 0.983552i $$-0.557813\pi$$
−0.180628 + 0.983552i $$0.557813\pi$$
$$614$$ −40.0000 −1.61427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −22.0000 −0.885687 −0.442843 0.896599i $$-0.646030\pi$$
−0.442843 + 0.896599i $$0.646030\pi$$
$$618$$ −8.94427 −0.359791
$$619$$ −16.0000 −0.643094 −0.321547 0.946894i $$-0.604203\pi$$
−0.321547 + 0.946894i $$0.604203\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ −71.5542 −2.86906
$$623$$ 13.4164 0.537517
$$624$$ −4.47214 −0.179029
$$625$$ 0 0
$$626$$ 20.1246 0.804341
$$627$$ 0 0
$$628$$ 6.00000 0.239426
$$629$$ 15.6525 0.624105
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ 35.0000 1.39223
$$633$$ −26.8328 −1.06651
$$634$$ −49.1935 −1.95372
$$635$$ 0 0
$$636$$ −42.0000 −1.66541
$$637$$ −8.94427 −0.354385
$$638$$ 0 0
$$639$$ −13.0000 −0.514272
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 16.0000 0.630978 0.315489 0.948929i $$-0.397831\pi$$
0.315489 + 0.948929i $$0.397831\pi$$
$$644$$ 6.70820 0.264340
$$645$$ 0 0
$$646$$ 11.1803 0.439885
$$647$$ 43.0000 1.69050 0.845252 0.534368i $$-0.179450\pi$$
0.845252 + 0.534368i $$0.179450\pi$$
$$648$$ 2.23607 0.0878410
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 22.3607 0.876384
$$652$$ 42.0000 1.64485
$$653$$ 26.0000 1.01746 0.508729 0.860927i $$-0.330115\pi$$
0.508729 + 0.860927i $$0.330115\pi$$
$$654$$ −40.0000 −1.56412
$$655$$ 0 0
$$656$$ 6.70820 0.261911
$$657$$ 13.4164 0.523424
$$658$$ 15.0000 0.584761
$$659$$ 31.3050 1.21947 0.609734 0.792606i $$-0.291276\pi$$
0.609734 + 0.792606i $$0.291276\pi$$
$$660$$ 0 0
$$661$$ 3.00000 0.116686 0.0583432 0.998297i $$-0.481418\pi$$
0.0583432 + 0.998297i $$0.481418\pi$$
$$662$$ 44.7214 1.73814
$$663$$ 10.0000 0.388368
$$664$$ 10.0000 0.388075
$$665$$ 0 0
$$666$$ 15.6525 0.606521
$$667$$ −4.47214 −0.173162
$$668$$ 13.4164 0.519096
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −15.0000 −0.578638
$$673$$ −8.94427 −0.344776 −0.172388 0.985029i $$-0.555148\pi$$
−0.172388 + 0.985029i $$0.555148\pi$$
$$674$$ −30.0000 −1.15556
$$675$$ 0 0
$$676$$ 21.0000 0.807692
$$677$$ −49.1935 −1.89066 −0.945330 0.326116i $$-0.894260\pi$$
−0.945330 + 0.326116i $$0.894260\pi$$
$$678$$ −31.3050 −1.20226
$$679$$ 38.0132 1.45881
$$680$$ 0 0
$$681$$ 13.4164 0.514118
$$682$$ 0 0
$$683$$ −21.0000 −0.803543 −0.401771 0.915740i $$-0.631605\pi$$
−0.401771 + 0.915740i $$0.631605\pi$$
$$684$$ 6.70820 0.256495
$$685$$ 0 0
$$686$$ −45.0000 −1.71811
$$687$$ −5.00000 −0.190762
$$688$$ 0 0
$$689$$ −62.6099 −2.38525
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 60.3738 2.29507
$$693$$ 0 0
$$694$$ 30.0000 1.13878
$$695$$ 0 0
$$696$$ −10.0000 −0.379049
$$697$$ −15.0000 −0.568166
$$698$$ 30.0000 1.13552
$$699$$ −20.1246 −0.761183
$$700$$ 0 0
$$701$$ −42.4853 −1.60465 −0.802324 0.596889i $$-0.796403\pi$$
−0.802324 + 0.596889i $$0.796403\pi$$
$$702$$ 10.0000 0.377426
$$703$$ 15.6525 0.590344
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −76.0263 −2.86129
$$707$$ −25.0000 −0.940222
$$708$$ −15.0000 −0.563735
$$709$$ 19.0000 0.713560 0.356780 0.934188i $$-0.383875\pi$$
0.356780 + 0.934188i $$0.383875\pi$$
$$710$$ 0 0
$$711$$ 15.6525 0.587014
$$712$$ 13.4164 0.502801
$$713$$ 10.0000 0.374503
$$714$$ 11.1803 0.418414
$$715$$ 0 0
$$716$$ 33.0000 1.23327
$$717$$ −8.94427 −0.334030
$$718$$ −10.0000 −0.373197
$$719$$ −4.00000 −0.149175 −0.0745874 0.997214i $$-0.523764\pi$$
−0.0745874 + 0.997214i $$0.523764\pi$$
$$720$$ 0 0
$$721$$ −8.94427 −0.333102
$$722$$ −31.3050 −1.16505
$$723$$ −22.3607 −0.831603
$$724$$ −15.0000 −0.557471
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 22.3607 0.828742
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ −13.4164 −0.495885
$$733$$ 40.2492 1.48664 0.743319 0.668937i $$-0.233250\pi$$
0.743319 + 0.668937i $$0.233250\pi$$
$$734$$ 40.2492 1.48563
$$735$$ 0 0
$$736$$ −6.70820 −0.247268
$$737$$ 0 0
$$738$$ −15.0000 −0.552158
$$739$$ 24.5967 0.904806 0.452403 0.891814i $$-0.350567\pi$$
0.452403 + 0.891814i $$0.350567\pi$$
$$740$$ 0 0
$$741$$ 10.0000 0.367359
$$742$$ −70.0000 −2.56978
$$743$$ −40.2492 −1.47660 −0.738300 0.674472i $$-0.764371\pi$$
−0.738300 + 0.674472i $$0.764371\pi$$
$$744$$ 22.3607 0.819782
$$745$$ 0 0
$$746$$ −20.0000 −0.732252
$$747$$ 4.47214 0.163627
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −48.0000 −1.75154 −0.875772 0.482724i $$-0.839647\pi$$
−0.875772 + 0.482724i $$0.839647\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ −12.0000 −0.437304
$$754$$ −44.7214 −1.62866
$$755$$ 0 0
$$756$$ 6.70820 0.243975
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ 22.3607 0.812176
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 40.2492 1.45903 0.729517 0.683963i $$-0.239745\pi$$
0.729517 + 0.683963i $$0.239745\pi$$
$$762$$ 35.0000 1.26792
$$763$$ −40.0000 −1.44810
$$764$$ 45.0000 1.62804
$$765$$ 0 0
$$766$$ −35.7771 −1.29268
$$767$$ −22.3607 −0.807397
$$768$$ −9.00000 −0.324760
$$769$$ 35.7771 1.29015 0.645077 0.764117i $$-0.276825\pi$$
0.645077 + 0.764117i $$0.276825\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ −26.8328 −0.965734
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 38.0132 1.36459
$$777$$ 15.6525 0.561529
$$778$$ 0 0
$$779$$ −15.0000 −0.537431
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 5.00000 0.178800
$$783$$ −4.47214 −0.159821
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ 30.0000 1.07006
$$787$$ −46.9574 −1.67385 −0.836926 0.547316i $$-0.815649\pi$$
−0.836926 + 0.547316i $$0.815649\pi$$
$$788$$ −46.9574 −1.67279
$$789$$ −17.8885 −0.636849
$$790$$ 0 0
$$791$$ −31.3050 −1.11308
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ 49.1935 1.74581
$$795$$ 0 0
$$796$$ 30.0000 1.06332
$$797$$ −8.00000 −0.283375 −0.141687 0.989911i $$-0.545253\pi$$
−0.141687 + 0.989911i $$0.545253\pi$$
$$798$$ 11.1803 0.395780
$$799$$ 6.70820 0.237319
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 22.3607 0.789583
$$803$$ 0 0
$$804$$ −6.00000 −0.211604
$$805$$ 0 0
$$806$$ 100.000 3.52235
$$807$$ 30.0000 1.05605
$$808$$ −25.0000 −0.879497
$$809$$ 24.5967 0.864776 0.432388 0.901688i $$-0.357671\pi$$
0.432388 + 0.901688i $$0.357671\pi$$
$$810$$ 0 0
$$811$$ 46.9574 1.64890 0.824449 0.565936i $$-0.191485\pi$$
0.824449 + 0.565936i $$0.191485\pi$$
$$812$$ −30.0000 −1.05279
$$813$$ −20.1246 −0.705801
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −2.23607 −0.0782780
$$817$$ 0 0
$$818$$ −30.0000 −1.04893
$$819$$ 10.0000 0.349428
$$820$$ 0 0
$$821$$ 31.3050 1.09255 0.546275 0.837606i $$-0.316045\pi$$
0.546275 + 0.837606i $$0.316045\pi$$
$$822$$ −17.8885 −0.623935
$$823$$ −14.0000 −0.488009 −0.244005 0.969774i $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ −8.94427 −0.311588
$$825$$ 0 0
$$826$$ −25.0000 −0.869861
$$827$$ 17.8885 0.622046 0.311023 0.950402i $$-0.399328\pi$$
0.311023 + 0.950402i $$0.399328\pi$$
$$828$$ 3.00000 0.104257
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ −13.4164 −0.465410
$$832$$ −58.1378 −2.01556
$$833$$ −4.47214 −0.154950
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 10.0000 0.345651
$$838$$ 20.1246 0.695193
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ −33.5410 −1.15590
$$843$$ −24.5967 −0.847157
$$844$$ −80.4984 −2.77087
$$845$$ 0 0
$$846$$ 6.70820 0.230633
$$847$$ 0 0
$$848$$ 14.0000 0.480762
$$849$$ −6.70820 −0.230225
$$850$$ 0 0
$$851$$ 7.00000 0.239957
$$852$$ −39.0000 −1.33612
$$853$$ −22.3607 −0.765615 −0.382808 0.923828i $$-0.625043\pi$$
−0.382808 + 0.923828i $$0.625043\pi$$
$$854$$ −22.3607 −0.765167
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 11.1803 0.381913 0.190957 0.981598i $$-0.438841\pi$$
0.190957 + 0.981598i $$0.438841\pi$$
$$858$$ 0 0
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ −15.0000 −0.511199
$$862$$ −50.0000 −1.70301
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ −6.70820 −0.228218
$$865$$ 0 0
$$866$$ 31.3050 1.06379
$$867$$ −12.0000 −0.407541
$$868$$ 67.0820 2.27691
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.94427 −0.303065
$$872$$ −40.0000 −1.35457
$$873$$ 17.0000 0.575363
$$874$$ 5.00000 0.169128
$$875$$ 0 0
$$876$$ 40.2492 1.35990
$$877$$ −31.3050 −1.05709 −0.528547 0.848904i $$-0.677263\pi$$
−0.528547 + 0.848904i $$0.677263\pi$$
$$878$$ −45.0000 −1.51868
$$879$$ 15.6525 0.527945
$$880$$ 0 0
$$881$$ 12.0000 0.404290 0.202145 0.979356i $$-0.435209\pi$$
0.202145 + 0.979356i $$0.435209\pi$$
$$882$$ −4.47214 −0.150585
$$883$$ −36.0000 −1.21150 −0.605748 0.795656i $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ 30.0000 1.00901
$$885$$ 0 0
$$886$$ 46.9574 1.57757
$$887$$ 4.47214 0.150160 0.0750798 0.997178i $$-0.476079\pi$$
0.0750798 + 0.997178i $$0.476079\pi$$
$$888$$ 15.6525 0.525263
$$889$$ 35.0000 1.17386
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 42.0000 1.40626
$$893$$ 6.70820 0.224481
$$894$$ 45.0000 1.50503
$$895$$ 0 0
$$896$$ −35.0000 −1.16927
$$897$$ 4.47214 0.149320
$$898$$ 8.94427 0.298474
$$899$$ −44.7214 −1.49154
$$900$$ 0 0
$$901$$ −31.3050 −1.04292
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −31.3050 −1.04119
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 8.00000 0.265636 0.132818 0.991140i $$-0.457597\pi$$
0.132818 + 0.991140i $$0.457597\pi$$
$$908$$ 40.2492 1.33572
$$909$$ −11.1803 −0.370828
$$910$$ 0 0
$$911$$ 55.0000 1.82223 0.911116 0.412151i $$-0.135222\pi$$
0.911116 + 0.412151i $$0.135222\pi$$
$$912$$ −2.23607 −0.0740436
$$913$$ 0 0
$$914$$ −50.0000 −1.65385
$$915$$ 0 0
$$916$$ −15.0000 −0.495614
$$917$$ 30.0000 0.990687
$$918$$ 5.00000 0.165025
$$919$$ 24.5967 0.811372 0.405686 0.914013i $$-0.367033\pi$$
0.405686 + 0.914013i $$0.367033\pi$$
$$920$$ 0 0
$$921$$ −17.8885 −0.589448
$$922$$ 50.0000 1.64666
$$923$$ −58.1378 −1.91363
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 31.3050 1.02874
$$927$$ −4.00000 −0.131377
$$928$$ 30.0000 0.984798
$$929$$ −20.0000 −0.656179 −0.328089 0.944647i $$-0.606405\pi$$
−0.328089 + 0.944647i $$0.606405\pi$$
$$930$$ 0 0
$$931$$ −4.47214 −0.146568
$$932$$ −60.3738 −1.97761
$$933$$ −32.0000 −1.04763
$$934$$ 71.5542 2.34132
$$935$$ 0 0
$$936$$ 10.0000 0.326860
$$937$$ −26.8328 −0.876590 −0.438295 0.898831i $$-0.644417\pi$$
−0.438295 + 0.898831i $$0.644417\pi$$
$$938$$ −10.0000 −0.326512
$$939$$ 9.00000 0.293704
$$940$$ 0 0
$$941$$ 24.5967 0.801831 0.400916 0.916115i $$-0.368692\pi$$
0.400916 + 0.916115i $$0.368692\pi$$
$$942$$ 4.47214 0.145710
$$943$$ −6.70820 −0.218449
$$944$$ 5.00000 0.162736
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 17.0000 0.552426 0.276213 0.961096i $$-0.410921\pi$$
0.276213 + 0.961096i $$0.410921\pi$$
$$948$$ 46.9574 1.52511
$$949$$ 60.0000 1.94768
$$950$$ 0 0
$$951$$ −22.0000 −0.713399
$$952$$ 11.1803 0.362357
$$953$$ −42.4853 −1.37623 −0.688117 0.725600i $$-0.741562\pi$$
−0.688117 + 0.725600i $$0.741562\pi$$
$$954$$ −31.3050 −1.01354
$$955$$ 0 0
$$956$$ −26.8328 −0.867835
$$957$$ 0 0
$$958$$ −50.0000 −1.61543
$$959$$ −17.8885 −0.577651
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 70.0000 2.25689
$$963$$ 0 0
$$964$$ −67.0820 −2.16057
$$965$$ 0 0
$$966$$ 5.00000 0.160872
$$967$$ 8.94427 0.287628 0.143814 0.989605i $$-0.454063\pi$$
0.143814 + 0.989605i $$0.454063\pi$$
$$968$$ 0 0
$$969$$ 5.00000 0.160623
$$970$$ 0 0
$$971$$ 5.00000 0.160458 0.0802288 0.996776i $$-0.474435\pi$$
0.0802288 + 0.996776i $$0.474435\pi$$
$$972$$ 3.00000 0.0962250
$$973$$ 0 0
$$974$$ −71.5542 −2.29274
$$975$$ 0 0
$$976$$ 4.47214 0.143150
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 31.3050 1.00102
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −17.8885 −0.571137
$$982$$ 10.0000 0.319113
$$983$$ −49.0000 −1.56286 −0.781429 0.623995i $$-0.785509\pi$$
−0.781429 + 0.623995i $$0.785509\pi$$
$$984$$ −15.0000 −0.478183
$$985$$ 0 0
$$986$$ −22.3607 −0.712109
$$987$$ 6.70820 0.213524
$$988$$ 30.0000 0.954427
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 10.0000 0.317660 0.158830 0.987306i $$-0.449228\pi$$
0.158830 + 0.987306i $$0.449228\pi$$
$$992$$ −67.0820 −2.12986
$$993$$ 20.0000 0.634681
$$994$$ −65.0000 −2.06167
$$995$$ 0 0
$$996$$ 13.4164 0.425115
$$997$$ −62.6099 −1.98288 −0.991438 0.130580i $$-0.958316\pi$$
−0.991438 + 0.130580i $$0.958316\pi$$
$$998$$ 44.7214 1.41563
$$999$$ 7.00000 0.221470
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 9075.2.a.br.1.2 yes 2
5.4 even 2 9075.2.a.bk.1.1 2
11.10 odd 2 inner 9075.2.a.br.1.1 yes 2
55.54 odd 2 9075.2.a.bk.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
9075.2.a.bk.1.1 2 5.4 even 2
9075.2.a.bk.1.2 yes 2 55.54 odd 2
9075.2.a.br.1.1 yes 2 11.10 odd 2 inner
9075.2.a.br.1.2 yes 2 1.1 even 1 trivial