# Properties

 Label 75.2.b.a.49.2 Level $75$ Weight $2$ Character 75.49 Analytic conductor $0.599$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 75.49 Dual form 75.2.b.a.49.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} -2.00000i q^{18} +5.00000 q^{19} +3.00000 q^{21} +4.00000i q^{22} -6.00000i q^{23} +2.00000 q^{26} -1.00000i q^{27} +6.00000i q^{28} -10.0000 q^{29} -3.00000 q^{31} -8.00000i q^{32} +2.00000i q^{33} -4.00000 q^{34} +2.00000 q^{36} +2.00000i q^{37} +10.0000i q^{38} +1.00000 q^{39} -8.00000 q^{41} +6.00000i q^{42} -1.00000i q^{43} -4.00000 q^{44} +12.0000 q^{46} +2.00000i q^{47} -4.00000i q^{48} -2.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} +4.00000i q^{53} +2.00000 q^{54} +5.00000i q^{57} -20.0000i q^{58} +10.0000 q^{59} +7.00000 q^{61} -6.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} -4.00000 q^{66} -3.00000i q^{67} -4.00000i q^{68} +6.00000 q^{69} -8.00000 q^{71} +14.0000i q^{73} -4.00000 q^{74} -10.0000 q^{76} -6.00000i q^{77} +2.00000i q^{78} +1.00000 q^{81} -16.0000i q^{82} -6.00000i q^{83} -6.00000 q^{84} +2.00000 q^{86} -10.0000i q^{87} -3.00000 q^{91} +12.0000i q^{92} -3.00000i q^{93} -4.00000 q^{94} +8.00000 q^{96} +17.0000i q^{97} -4.00000i q^{98} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} - 4q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 4q^{4} - 4q^{6} - 2q^{9} + 4q^{11} + 12q^{14} - 8q^{16} + 10q^{19} + 6q^{21} + 4q^{26} - 20q^{29} - 6q^{31} - 8q^{34} + 4q^{36} + 2q^{39} - 16q^{41} - 8q^{44} + 24q^{46} - 4q^{49} - 4q^{51} + 4q^{54} + 20q^{59} + 14q^{61} + 16q^{64} - 8q^{66} + 12q^{69} - 16q^{71} - 8q^{74} - 20q^{76} + 2q^{81} - 12q^{84} + 4q^{86} - 6q^{91} - 8q^{94} + 16q^{96} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000i 1.41421i 0.707107 + 0.707107i $$0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ − 2.00000i − 0.577350i
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 6.00000 1.60357
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ − 2.00000i − 0.471405i
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 4.00000i 0.852803i
$$23$$ − 6.00000i − 1.25109i −0.780189 0.625543i $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ 6.00000i 1.13389i
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ − 8.00000i − 1.41421i
$$33$$ 2.00000i 0.348155i
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 10.0000i 1.62221i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 6.00000i 0.925820i
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 12.0000 1.76930
$$47$$ 2.00000i 0.291730i 0.989305 + 0.145865i $$0.0465965\pi$$
−0.989305 + 0.145865i $$0.953403\pi$$
$$48$$ − 4.00000i − 0.577350i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 2.00000i 0.277350i
$$53$$ 4.00000i 0.549442i 0.961524 + 0.274721i $$0.0885855\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 2.00000 0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 5.00000i 0.662266i
$$58$$ − 20.0000i − 2.62613i
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ − 6.00000i − 0.762001i
$$63$$ 3.00000i 0.377964i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 3.00000i − 0.366508i −0.983066 0.183254i $$-0.941337\pi$$
0.983066 0.183254i $$-0.0586631\pi$$
$$68$$ − 4.00000i − 0.485071i
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 14.0000i 1.63858i 0.573382 + 0.819288i $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ −10.0000 −1.14708
$$77$$ − 6.00000i − 0.683763i
$$78$$ 2.00000i 0.226455i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 16.0000i − 1.76690i
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ −6.00000 −0.654654
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ − 10.0000i − 1.07211i
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 12.0000i 1.25109i
$$93$$ − 3.00000i − 0.311086i
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ 8.00000 0.816497
$$97$$ 17.0000i 1.72609i 0.505128 + 0.863044i $$0.331445\pi$$
−0.505128 + 0.863044i $$0.668555\pi$$
$$98$$ − 4.00000i − 0.404061i
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ − 4.00000i − 0.396059i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −8.00000 −0.777029
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 2.00000i 0.192450i
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 12.0000i 1.13389i
$$113$$ 4.00000i 0.376288i 0.982141 + 0.188144i $$0.0602472\pi$$
−0.982141 + 0.188144i $$0.939753\pi$$
$$114$$ −10.0000 −0.936586
$$115$$ 0 0
$$116$$ 20.0000 1.85695
$$117$$ 1.00000i 0.0924500i
$$118$$ 20.0000i 1.84115i
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 14.0000i 1.26750i
$$123$$ − 8.00000i − 0.721336i
$$124$$ 6.00000 0.538816
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 0 0
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ − 15.0000i − 1.30066i
$$134$$ 6.00000 0.518321
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 12.0000i 1.02151i
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ − 16.0000i − 1.34269i
$$143$$ − 2.00000i − 0.167248i
$$144$$ 4.00000 0.333333
$$145$$ 0 0
$$146$$ −28.0000 −2.31730
$$147$$ − 2.00000i − 0.164957i
$$148$$ − 4.00000i − 0.328798i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 7.00000 0.569652 0.284826 0.958579i $$-0.408064\pi$$
0.284826 + 0.958579i $$0.408064\pi$$
$$152$$ 0 0
$$153$$ − 2.00000i − 0.161690i
$$154$$ 12.0000 0.966988
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ − 13.0000i − 1.03751i −0.854922 0.518756i $$-0.826395\pi$$
0.854922 0.518756i $$-0.173605\pi$$
$$158$$ 0 0
$$159$$ −4.00000 −0.317221
$$160$$ 0 0
$$161$$ −18.0000 −1.41860
$$162$$ 2.00000i 0.157135i
$$163$$ − 11.0000i − 0.861586i −0.902451 0.430793i $$-0.858234\pi$$
0.902451 0.430793i $$-0.141766\pi$$
$$164$$ 16.0000 1.24939
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −5.00000 −0.382360
$$172$$ 2.00000i 0.152499i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 20.0000 1.51620
$$175$$ 0 0
$$176$$ −8.00000 −0.603023
$$177$$ 10.0000i 0.751646i
$$178$$ 0 0
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 17.0000 1.26360 0.631800 0.775131i $$-0.282316\pi$$
0.631800 + 0.775131i $$0.282316\pi$$
$$182$$ − 6.00000i − 0.444750i
$$183$$ 7.00000i 0.517455i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ 4.00000i 0.292509i
$$188$$ − 4.00000i − 0.291730i
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ 22.0000 1.59186 0.795932 0.605386i $$-0.206981\pi$$
0.795932 + 0.605386i $$0.206981\pi$$
$$192$$ 8.00000i 0.577350i
$$193$$ − 11.0000i − 0.791797i −0.918294 0.395899i $$-0.870433\pi$$
0.918294 0.395899i $$-0.129567\pi$$
$$194$$ −34.0000 −2.44106
$$195$$ 0 0
$$196$$ 4.00000 0.285714
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 0 0
$$201$$ 3.00000 0.211604
$$202$$ 24.0000i 1.68863i
$$203$$ 30.0000i 2.10559i
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 6.00000i 0.417029i
$$208$$ 4.00000i 0.277350i
$$209$$ 10.0000 0.691714
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ − 8.00000i − 0.549442i
$$213$$ − 8.00000i − 0.548151i
$$214$$ −24.0000 −1.64061
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 9.00000i 0.610960i
$$218$$ − 10.0000i − 0.677285i
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ − 4.00000i − 0.268462i
$$223$$ 19.0000i 1.27233i 0.771551 + 0.636167i $$0.219481\pi$$
−0.771551 + 0.636167i $$0.780519\pi$$
$$224$$ −24.0000 −1.60357
$$225$$ 0 0
$$226$$ −8.00000 −0.532152
$$227$$ − 8.00000i − 0.530979i −0.964114 0.265489i $$-0.914466\pi$$
0.964114 0.265489i $$-0.0855335\pi$$
$$228$$ − 10.0000i − 0.662266i
$$229$$ 15.0000 0.991228 0.495614 0.868543i $$-0.334943\pi$$
0.495614 + 0.868543i $$0.334943\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ 24.0000i 1.57229i 0.618041 + 0.786146i $$0.287927\pi$$
−0.618041 + 0.786146i $$0.712073\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −20.0000 −1.30189
$$237$$ 0 0
$$238$$ 12.0000i 0.777844i
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ −23.0000 −1.48156 −0.740780 0.671748i $$-0.765544\pi$$
−0.740780 + 0.671748i $$0.765544\pi$$
$$242$$ − 14.0000i − 0.899954i
$$243$$ 1.00000i 0.0641500i
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 16.0000 1.02012
$$247$$ − 5.00000i − 0.318142i
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ − 6.00000i − 0.377964i
$$253$$ − 12.0000i − 0.754434i
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 2.00000i 0.124515i
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 24.0000i 1.48272i
$$263$$ − 16.0000i − 0.986602i −0.869859 0.493301i $$-0.835790\pi$$
0.869859 0.493301i $$-0.164210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 30.0000 1.83942
$$267$$ 0 0
$$268$$ 6.00000i 0.366508i
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ − 8.00000i − 0.485071i
$$273$$ − 3.00000i − 0.181568i
$$274$$ 36.0000 2.17484
$$275$$ 0 0
$$276$$ −12.0000 −0.722315
$$277$$ − 3.00000i − 0.180253i −0.995930 0.0901263i $$-0.971273\pi$$
0.995930 0.0901263i $$-0.0287271\pi$$
$$278$$ − 40.0000i − 2.39904i
$$279$$ 3.00000 0.179605
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ 9.00000i 0.534994i 0.963559 + 0.267497i $$0.0861966\pi$$
−0.963559 + 0.267497i $$0.913803\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 24.0000i 1.41668i
$$288$$ 8.00000i 0.471405i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −17.0000 −0.996558
$$292$$ − 28.0000i − 1.63858i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 4.00000 0.233285
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 2.00000i − 0.116052i
$$298$$ − 20.0000i − 1.15857i
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 14.0000i 0.805609i
$$303$$ 12.0000i 0.689382i
$$304$$ −20.0000 −1.14708
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ 7.00000i 0.399511i 0.979846 + 0.199756i $$0.0640148\pi$$
−0.979846 + 0.199756i $$0.935985\pi$$
$$308$$ 12.0000i 0.683763i
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ − 11.0000i − 0.621757i −0.950450 0.310878i $$-0.899377\pi$$
0.950450 0.310878i $$-0.100623\pi$$
$$314$$ 26.0000 1.46726
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 8.00000i − 0.449325i −0.974437 0.224662i $$-0.927872\pi$$
0.974437 0.224662i $$-0.0721279\pi$$
$$318$$ − 8.00000i − 0.448618i
$$319$$ −20.0000 −1.11979
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ − 36.0000i − 2.00620i
$$323$$ 10.0000i 0.556415i
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ 22.0000 1.21847
$$327$$ − 5.00000i − 0.276501i
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ − 2.00000i − 0.109599i
$$334$$ −24.0000 −1.31322
$$335$$ 0 0
$$336$$ −12.0000 −0.654654
$$337$$ − 23.0000i − 1.25289i −0.779466 0.626445i $$-0.784509\pi$$
0.779466 0.626445i $$-0.215491\pi$$
$$338$$ 24.0000i 1.30543i
$$339$$ −4.00000 −0.217250
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ − 10.0000i − 0.540738i
$$343$$ − 15.0000i − 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ 2.00000i 0.107366i 0.998558 + 0.0536828i $$0.0170960\pi$$
−0.998558 + 0.0536828i $$0.982904\pi$$
$$348$$ 20.0000i 1.07211i
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ − 16.0000i − 0.852803i
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ −20.0000 −1.06299
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 6.00000i 0.317554i
$$358$$ 20.0000i 1.05703i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 34.0000i 1.78700i
$$363$$ − 7.00000i − 0.367405i
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ −14.0000 −0.731792
$$367$$ 27.0000i 1.40939i 0.709511 + 0.704694i $$0.248916\pi$$
−0.709511 + 0.704694i $$0.751084\pi$$
$$368$$ 24.0000i 1.25109i
$$369$$ 8.00000 0.416463
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 6.00000i 0.311086i
$$373$$ 29.0000i 1.50156i 0.660551 + 0.750782i $$0.270323\pi$$
−0.660551 + 0.750782i $$0.729677\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 10.0000i 0.515026i
$$378$$ − 6.00000i − 0.308607i
$$379$$ −25.0000 −1.28416 −0.642082 0.766636i $$-0.721929\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 44.0000i 2.25124i
$$383$$ − 36.0000i − 1.83951i −0.392488 0.919757i $$-0.628386\pi$$
0.392488 0.919757i $$-0.371614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ 1.00000i 0.0508329i
$$388$$ − 34.0000i − 1.72609i
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 12.0000i 0.605320i
$$394$$ 36.0000 1.81365
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 7.00000i 0.351320i 0.984451 + 0.175660i $$0.0562059\pi$$
−0.984451 + 0.175660i $$0.943794\pi$$
$$398$$ 10.0000i 0.501255i
$$399$$ 15.0000 0.750939
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 6.00000i 0.299253i
$$403$$ 3.00000i 0.149441i
$$404$$ −24.0000 −1.19404
$$405$$ 0 0
$$406$$ −60.0000 −2.97775
$$407$$ 4.00000i 0.198273i
$$408$$ 0 0
$$409$$ −5.00000 −0.247234 −0.123617 0.992330i $$-0.539449\pi$$
−0.123617 + 0.992330i $$0.539449\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ − 8.00000i − 0.394132i
$$413$$ − 30.0000i − 1.47620i
$$414$$ −12.0000 −0.589768
$$415$$ 0 0
$$416$$ −8.00000 −0.392232
$$417$$ − 20.0000i − 0.979404i
$$418$$ 20.0000i 0.978232i
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ − 26.0000i − 1.26566i
$$423$$ − 2.00000i − 0.0972433i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 16.0000 0.775203
$$427$$ − 21.0000i − 1.01626i
$$428$$ − 24.0000i − 1.16008i
$$429$$ 2.00000 0.0965609
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ 29.0000i 1.39365i 0.717241 + 0.696826i $$0.245405\pi$$
−0.717241 + 0.696826i $$0.754595\pi$$
$$434$$ −18.0000 −0.864028
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ − 30.0000i − 1.43509i
$$438$$ − 28.0000i − 1.33789i
$$439$$ 35.0000 1.67046 0.835229 0.549902i $$-0.185335\pi$$
0.835229 + 0.549902i $$0.185335\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 4.00000i 0.190261i
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ −38.0000 −1.79935
$$447$$ − 10.0000i − 0.472984i
$$448$$ − 24.0000i − 1.13389i
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ − 8.00000i − 0.376288i
$$453$$ 7.00000i 0.328889i
$$454$$ 16.0000 0.750917
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000i 1.02912i 0.857455 + 0.514558i $$0.172044\pi$$
−0.857455 + 0.514558i $$0.827956\pi$$
$$458$$ 30.0000i 1.40181i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 12.0000i 0.558291i
$$463$$ 24.0000i 1.11537i 0.830051 + 0.557687i $$0.188311\pi$$
−0.830051 + 0.557687i $$0.811689\pi$$
$$464$$ 40.0000 1.85695
$$465$$ 0 0
$$466$$ −48.0000 −2.22356
$$467$$ − 38.0000i − 1.75843i −0.476425 0.879215i $$-0.658068\pi$$
0.476425 0.879215i $$-0.341932\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ −9.00000 −0.415581
$$470$$ 0 0
$$471$$ 13.0000 0.599008
$$472$$ 0 0
$$473$$ − 2.00000i − 0.0919601i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −12.0000 −0.550019
$$477$$ − 4.00000i − 0.183147i
$$478$$ − 40.0000i − 1.82956i
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ − 46.0000i − 2.09524i
$$483$$ − 18.0000i − 0.819028i
$$484$$ 14.0000 0.636364
$$485$$ 0 0
$$486$$ −2.00000 −0.0907218
$$487$$ − 13.0000i − 0.589086i −0.955638 0.294543i $$-0.904833\pi$$
0.955638 0.294543i $$-0.0951675\pi$$
$$488$$ 0 0
$$489$$ 11.0000 0.497437
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 16.0000i 0.721336i
$$493$$ − 20.0000i − 0.900755i
$$494$$ 10.0000 0.449921
$$495$$ 0 0
$$496$$ 12.0000 0.538816
$$497$$ 24.0000i 1.07655i
$$498$$ 12.0000i 0.537733i
$$499$$ 5.00000 0.223831 0.111915 0.993718i $$-0.464301\pi$$
0.111915 + 0.993718i $$0.464301\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 24.0000i 1.07117i
$$503$$ − 16.0000i − 0.713405i −0.934218 0.356702i $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 24.0000 1.06693
$$507$$ 12.0000i 0.532939i
$$508$$ 16.0000i 0.709885i
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 42.0000 1.85797
$$512$$ 32.0000i 1.41421i
$$513$$ − 5.00000i − 0.220755i
$$514$$ −24.0000 −1.05859
$$515$$ 0 0
$$516$$ −2.00000 −0.0880451
$$517$$ 4.00000i 0.175920i
$$518$$ 12.0000i 0.527250i
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 20.0000i 0.875376i
$$523$$ − 31.0000i − 1.35554i −0.735276 0.677768i $$-0.762948\pi$$
0.735276 0.677768i $$-0.237052\pi$$
$$524$$ −24.0000 −1.04844
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ − 6.00000i − 0.261364i
$$528$$ − 8.00000i − 0.348155i
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −10.0000 −0.433963
$$532$$ 30.0000i 1.30066i
$$533$$ 8.00000i 0.346518i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 10.0000i 0.431532i
$$538$$ 20.0000i 0.862261i
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −3.00000 −0.128980 −0.0644900 0.997918i $$-0.520542\pi$$
−0.0644900 + 0.997918i $$0.520542\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ 17.0000i 0.729540i
$$544$$ 16.0000 0.685994
$$545$$ 0 0
$$546$$ 6.00000 0.256776
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ 36.0000i 1.53784i
$$549$$ −7.00000 −0.298753
$$550$$ 0 0
$$551$$ −50.0000 −2.13007
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 6.00000 0.254916
$$555$$ 0 0
$$556$$ 40.0000 1.69638
$$557$$ 42.0000i 1.77960i 0.456354 + 0.889799i $$0.349155\pi$$
−0.456354 + 0.889799i $$0.650845\pi$$
$$558$$ 6.00000i 0.254000i
$$559$$ −1.00000 −0.0422955
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ − 36.0000i − 1.51857i
$$563$$ − 6.00000i − 0.252870i −0.991975 0.126435i $$-0.959647\pi$$
0.991975 0.126435i $$-0.0403535\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 0 0
$$566$$ −18.0000 −0.756596
$$567$$ − 3.00000i − 0.125988i
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ −13.0000 −0.544033 −0.272017 0.962293i $$-0.587691\pi$$
−0.272017 + 0.962293i $$0.587691\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 22.0000i 0.919063i
$$574$$ −48.0000 −2.00348
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ − 13.0000i − 0.541197i −0.962692 0.270599i $$-0.912778\pi$$
0.962692 0.270599i $$-0.0872216\pi$$
$$578$$ 26.0000i 1.08146i
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ − 34.0000i − 1.40935i
$$583$$ 8.00000i 0.331326i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 12.0000 0.495715
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 4.00000i 0.164957i
$$589$$ −15.0000 −0.618064
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ − 8.00000i − 0.328798i
$$593$$ − 16.0000i − 0.657041i −0.944497 0.328521i $$-0.893450\pi$$
0.944497 0.328521i $$-0.106550\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 20.0000 0.819232
$$597$$ 5.00000i 0.204636i
$$598$$ − 12.0000i − 0.490716i
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ − 6.00000i − 0.244542i
$$603$$ 3.00000i 0.122169i
$$604$$ −14.0000 −0.569652
$$605$$ 0 0
$$606$$ −24.0000 −0.974933
$$607$$ − 8.00000i − 0.324710i −0.986732 0.162355i $$-0.948091\pi$$
0.986732 0.162355i $$-0.0519090\pi$$
$$608$$ − 40.0000i − 1.62221i
$$609$$ −30.0000 −1.21566
$$610$$ 0 0
$$611$$ 2.00000 0.0809113
$$612$$ 4.00000i 0.161690i
$$613$$ 14.0000i 0.565455i 0.959200 + 0.282727i $$0.0912392\pi$$
−0.959200 + 0.282727i $$0.908761\pi$$
$$614$$ −14.0000 −0.564994
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000i 0.483102i 0.970388 + 0.241551i $$0.0776561\pi$$
−0.970388 + 0.241551i $$0.922344\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ 25.0000 1.00483 0.502417 0.864625i $$-0.332444\pi$$
0.502417 + 0.864625i $$0.332444\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ − 36.0000i − 1.44347i
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ 22.0000 0.879297
$$627$$ 10.0000i 0.399362i
$$628$$ 26.0000i 1.03751i
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ −23.0000 −0.915616 −0.457808 0.889051i $$-0.651365\pi$$
−0.457808 + 0.889051i $$0.651365\pi$$
$$632$$ 0 0
$$633$$ − 13.0000i − 0.516704i
$$634$$ 16.0000 0.635441
$$635$$ 0 0
$$636$$ 8.00000 0.317221
$$637$$ 2.00000i 0.0792429i
$$638$$ − 40.0000i − 1.58362i
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ − 24.0000i − 0.947204i
$$643$$ − 36.0000i − 1.41970i −0.704352 0.709851i $$-0.748762\pi$$
0.704352 0.709851i $$-0.251238\pi$$
$$644$$ 36.0000 1.41860
$$645$$ 0 0
$$646$$ −20.0000 −0.786889
$$647$$ − 28.0000i − 1.10079i −0.834903 0.550397i $$-0.814476\pi$$
0.834903 0.550397i $$-0.185524\pi$$
$$648$$ 0 0
$$649$$ 20.0000 0.785069
$$650$$ 0 0
$$651$$ −9.00000 −0.352738
$$652$$ 22.0000i 0.861586i
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 32.0000 1.24939
$$657$$ − 14.0000i − 0.546192i
$$658$$ 12.0000i 0.467809i
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 24.0000i 0.932786i
$$663$$ 2.00000i 0.0776736i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ 60.0000i 2.32321i
$$668$$ − 24.0000i − 0.928588i
$$669$$ −19.0000 −0.734582
$$670$$ 0 0
$$671$$ 14.0000 0.540464
$$672$$ − 24.0000i − 0.925820i
$$673$$ − 6.00000i − 0.231283i −0.993291 0.115642i $$-0.963108\pi$$
0.993291 0.115642i $$-0.0368924\pi$$
$$674$$ 46.0000 1.77185
$$675$$ 0 0
$$676$$ −24.0000 −0.923077
$$677$$ 42.0000i 1.61419i 0.590421 + 0.807096i $$0.298962\pi$$
−0.590421 + 0.807096i $$0.701038\pi$$
$$678$$ − 8.00000i − 0.307238i
$$679$$ 51.0000 1.95720
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ − 12.0000i − 0.459504i
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 10.0000 0.382360
$$685$$ 0 0
$$686$$ 30.0000 1.14541
$$687$$ 15.0000i 0.572286i
$$688$$ 4.00000i 0.152499i
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 12.0000i 0.456172i
$$693$$ 6.00000i 0.227921i
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 16.0000i − 0.606043i
$$698$$ − 20.0000i − 0.757011i
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ 10.0000i 0.377157i
$$704$$ 16.0000 0.603023
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ − 36.0000i − 1.35392i
$$708$$ − 20.0000i − 0.751646i
$$709$$ −25.0000 −0.938895 −0.469447 0.882960i $$-0.655547\pi$$
−0.469447 + 0.882960i $$0.655547\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 18.0000i 0.674105i
$$714$$ −12.0000 −0.449089
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ − 20.0000i − 0.746914i
$$718$$ 0 0
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 12.0000i 0.446594i
$$723$$ − 23.0000i − 0.855379i
$$724$$ −34.0000 −1.26360
$$725$$ 0 0
$$726$$ 14.0000 0.519589
$$727$$ − 43.0000i − 1.59478i −0.603463 0.797391i $$-0.706213\pi$$
0.603463 0.797391i $$-0.293787\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 2.00000 0.0739727
$$732$$ − 14.0000i − 0.517455i
$$733$$ 34.0000i 1.25582i 0.778287 + 0.627909i $$0.216089\pi$$
−0.778287 + 0.627909i $$0.783911\pi$$
$$734$$ −54.0000 −1.99318
$$735$$ 0 0
$$736$$ −48.0000 −1.76930
$$737$$ − 6.00000i − 0.221013i
$$738$$ 16.0000i 0.588968i
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 5.00000 0.183680
$$742$$ 24.0000i 0.881068i
$$743$$ 4.00000i 0.146746i 0.997305 + 0.0733729i $$0.0233763\pi$$
−0.997305 + 0.0733729i $$0.976624\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −58.0000 −2.12353
$$747$$ 6.00000i 0.219529i
$$748$$ − 8.00000i − 0.292509i
$$749$$ 36.0000 1.31541
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ 12.0000i 0.437304i
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ 6.00000 0.218218
$$757$$ − 23.0000i − 0.835949i −0.908459 0.417975i $$-0.862740\pi$$
0.908459 0.417975i $$-0.137260\pi$$
$$758$$ − 50.0000i − 1.81608i
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ 15.0000i 0.543036i
$$764$$ −44.0000 −1.59186
$$765$$ 0 0
$$766$$ 72.0000 2.60147
$$767$$ − 10.0000i − 0.361079i
$$768$$ 16.0000i 0.577350i
$$769$$ −35.0000 −1.26213 −0.631066 0.775729i $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ 22.0000i 0.791797i
$$773$$ 24.0000i 0.863220i 0.902060 + 0.431610i $$0.142054\pi$$
−0.902060 + 0.431610i $$0.857946\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 6.00000i 0.215249i
$$778$$ 0 0
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 24.0000i 0.858238i
$$783$$ 10.0000i 0.357371i
$$784$$ 8.00000 0.285714
$$785$$ 0 0
$$786$$ −24.0000 −0.856052
$$787$$ 7.00000i 0.249523i 0.992187 + 0.124762i $$0.0398166\pi$$
−0.992187 + 0.124762i $$0.960183\pi$$
$$788$$ 36.0000i 1.28245i
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ − 7.00000i − 0.248577i
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ 52.0000i 1.84193i 0.389640 + 0.920967i $$0.372599\pi$$
−0.389640 + 0.920967i $$0.627401\pi$$
$$798$$ 30.0000i 1.06199i
$$799$$ −4.00000 −0.141510
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 24.0000i 0.847469i
$$803$$ 28.0000i 0.988099i
$$804$$ −6.00000 −0.211604
$$805$$ 0 0
$$806$$ −6.00000 −0.211341
$$807$$ 10.0000i 0.352017i
$$808$$ 0 0
$$809$$ 20.0000 0.703163 0.351581 0.936157i $$-0.385644\pi$$
0.351581 + 0.936157i $$0.385644\pi$$
$$810$$ 0 0
$$811$$ 27.0000 0.948098 0.474049 0.880498i $$-0.342792\pi$$
0.474049 + 0.880498i $$0.342792\pi$$
$$812$$ − 60.0000i − 2.10559i
$$813$$ − 8.00000i − 0.280572i
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 8.00000 0.280056
$$817$$ − 5.00000i − 0.174928i
$$818$$ − 10.0000i − 0.349642i
$$819$$ 3.00000 0.104828
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 36.0000i 1.25564i
$$823$$ − 41.0000i − 1.42917i −0.699549 0.714585i $$-0.746616\pi$$
0.699549 0.714585i $$-0.253384\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 60.0000 2.08767
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ − 12.0000i − 0.417029i
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ 3.00000 0.104069
$$832$$ − 8.00000i − 0.277350i
$$833$$ − 4.00000i − 0.138592i
$$834$$ 40.0000 1.38509
$$835$$ 0 0
$$836$$ −20.0000 −0.691714
$$837$$ 3.00000i 0.103695i
$$838$$ 40.0000i 1.38178i
$$839$$ −10.0000 −0.345238 −0.172619 0.984989i $$-0.555223\pi$$
−0.172619 + 0.984989i $$0.555223\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 44.0000i 1.51634i
$$843$$ − 18.0000i − 0.619953i
$$844$$ 26.0000 0.894957
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ 21.0000i 0.721569i
$$848$$ − 16.0000i − 0.549442i
$$849$$ −9.00000 −0.308879
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 16.0000i 0.548151i
$$853$$ − 51.0000i − 1.74621i −0.487535 0.873103i $$-0.662104\pi$$
0.487535 0.873103i $$-0.337896\pi$$
$$854$$ 42.0000 1.43721
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 28.0000i − 0.956462i −0.878234 0.478231i $$-0.841278\pi$$
0.878234 0.478231i $$-0.158722\pi$$
$$858$$ 4.00000i 0.136558i
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ − 36.0000i − 1.22616i
$$863$$ − 16.0000i − 0.544646i −0.962206 0.272323i $$-0.912208\pi$$
0.962206 0.272323i $$-0.0877920\pi$$
$$864$$ −8.00000 −0.272166
$$865$$ 0 0
$$866$$ −58.0000 −1.97092
$$867$$ 13.0000i 0.441503i
$$868$$ − 18.0000i − 0.610960i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ 0 0
$$873$$ − 17.0000i − 0.575363i
$$874$$ 60.0000 2.02953
$$875$$ 0 0
$$876$$ 28.0000 0.946032
$$877$$ 27.0000i 0.911725i 0.890050 + 0.455863i $$0.150669\pi$$
−0.890050 + 0.455863i $$0.849331\pi$$
$$878$$ 70.0000i 2.36239i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 32.0000 1.07811 0.539054 0.842271i $$-0.318782\pi$$
0.539054 + 0.842271i $$0.318782\pi$$
$$882$$ 4.00000i 0.134687i
$$883$$ − 41.0000i − 1.37976i −0.723924 0.689880i $$-0.757663\pi$$
0.723924 0.689880i $$-0.242337\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −48.0000 −1.61259
$$887$$ − 18.0000i − 0.604381i −0.953248 0.302190i $$-0.902282\pi$$
0.953248 0.302190i $$-0.0977178\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ − 38.0000i − 1.27233i
$$893$$ 10.0000i 0.334637i
$$894$$ 20.0000 0.668900
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 6.00000i − 0.200334i
$$898$$ − 40.0000i − 1.33482i
$$899$$ 30.0000 1.00056
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ − 32.0000i − 1.06548i
$$903$$ − 3.00000i − 0.0998337i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ −14.0000 −0.465119
$$907$$ 12.0000i 0.398453i 0.979953 + 0.199227i $$0.0638430\pi$$
−0.979953 + 0.199227i $$0.936157\pi$$
$$908$$ 16.0000i 0.530979i
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −58.0000 −1.92163 −0.960813 0.277198i $$-0.910594\pi$$
−0.960813 + 0.277198i $$0.910594\pi$$
$$912$$ − 20.0000i − 0.662266i
$$913$$ − 12.0000i − 0.397142i
$$914$$ −44.0000 −1.45539
$$915$$ 0 0
$$916$$ −30.0000 −0.991228
$$917$$ − 36.0000i − 1.18882i
$$918$$ 4.00000i 0.132020i
$$919$$ −55.0000 −1.81428 −0.907141 0.420826i $$-0.861740\pi$$
−0.907141 + 0.420826i $$0.861740\pi$$
$$920$$ 0 0
$$921$$ −7.00000 −0.230658
$$922$$ 24.0000i 0.790398i
$$923$$ 8.00000i 0.263323i
$$924$$ −12.0000 −0.394771
$$925$$ 0 0
$$926$$ −48.0000 −1.57738
$$927$$ − 4.00000i − 0.131377i
$$928$$ 80.0000i 2.62613i
$$929$$ 50.0000 1.64045 0.820223 0.572043i $$-0.193849\pi$$
0.820223 + 0.572043i $$0.193849\pi$$
$$930$$ 0 0
$$931$$ −10.0000 −0.327737
$$932$$ − 48.0000i − 1.57229i
$$933$$ − 18.0000i − 0.589294i
$$934$$ 76.0000 2.48680
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 33.0000i − 1.07806i −0.842286 0.539032i $$-0.818790\pi$$
0.842286 0.539032i $$-0.181210\pi$$
$$938$$ − 18.0000i − 0.587721i
$$939$$ 11.0000 0.358971
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ 26.0000i 0.847126i
$$943$$ 48.0000i 1.56310i
$$944$$ −40.0000 −1.30189
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ − 18.0000i − 0.584921i −0.956278 0.292461i $$-0.905526\pi$$
0.956278 0.292461i $$-0.0944741\pi$$
$$948$$ 0 0
$$949$$ 14.0000 0.454459
$$950$$ 0 0
$$951$$ 8.00000 0.259418
$$952$$ 0 0
$$953$$ − 56.0000i − 1.81402i −0.421111 0.907009i $$-0.638360\pi$$
0.421111 0.907009i $$-0.361640\pi$$
$$954$$ 8.00000 0.259010
$$955$$ 0 0
$$956$$ 40.0000 1.29369
$$957$$ − 20.0000i − 0.646508i
$$958$$ − 60.0000i − 1.93851i
$$959$$ −54.0000 −1.74375
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 4.00000i 0.128965i
$$963$$ − 12.0000i − 0.386695i
$$964$$ 46.0000 1.48156
$$965$$ 0 0
$$966$$ 36.0000 1.15828
$$967$$ 32.0000i 1.02905i 0.857475 + 0.514525i $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 0 0
$$969$$ −10.0000 −0.321246
$$970$$ 0 0
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ − 2.00000i − 0.0641500i
$$973$$ 60.0000i 1.92351i
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ −28.0000 −0.896258
$$977$$ 2.00000i 0.0639857i 0.999488 + 0.0319928i $$0.0101854\pi$$
−0.999488 + 0.0319928i $$0.989815\pi$$
$$978$$ 22.0000i 0.703482i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 5.00000 0.159638
$$982$$ − 16.0000i − 0.510581i
$$983$$ − 36.0000i − 1.14822i −0.818778 0.574111i $$-0.805348\pi$$
0.818778 0.574111i $$-0.194652\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 40.0000 1.27386
$$987$$ 6.00000i 0.190982i
$$988$$ 10.0000i 0.318142i
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ 17.0000 0.540023 0.270011 0.962857i $$-0.412973\pi$$
0.270011 + 0.962857i $$0.412973\pi$$
$$992$$ 24.0000i 0.762001i
$$993$$ 12.0000i 0.380808i
$$994$$ −48.0000 −1.52247
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 42.0000i 1.33015i 0.746775 + 0.665077i $$0.231601\pi$$
−0.746775 + 0.665077i $$0.768399\pi$$
$$998$$ 10.0000i 0.316544i
$$999$$ 2.00000 0.0632772
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 75.2.b.a.49.2 2
3.2 odd 2 225.2.b.a.199.1 2
4.3 odd 2 1200.2.f.d.49.1 2
5.2 odd 4 75.2.a.a.1.1 1
5.3 odd 4 75.2.a.c.1.1 yes 1
5.4 even 2 inner 75.2.b.a.49.1 2
8.3 odd 2 4800.2.f.y.3649.2 2
8.5 even 2 4800.2.f.l.3649.1 2
12.11 even 2 3600.2.f.p.2449.2 2
15.2 even 4 225.2.a.e.1.1 1
15.8 even 4 225.2.a.a.1.1 1
15.14 odd 2 225.2.b.a.199.2 2
20.3 even 4 1200.2.a.p.1.1 1
20.7 even 4 1200.2.a.c.1.1 1
20.19 odd 2 1200.2.f.d.49.2 2
35.13 even 4 3675.2.a.q.1.1 1
35.27 even 4 3675.2.a.b.1.1 1
40.3 even 4 4800.2.a.be.1.1 1
40.13 odd 4 4800.2.a.bq.1.1 1
40.19 odd 2 4800.2.f.y.3649.1 2
40.27 even 4 4800.2.a.br.1.1 1
40.29 even 2 4800.2.f.l.3649.2 2
40.37 odd 4 4800.2.a.bb.1.1 1
55.32 even 4 9075.2.a.s.1.1 1
55.43 even 4 9075.2.a.a.1.1 1
60.23 odd 4 3600.2.a.bk.1.1 1
60.47 odd 4 3600.2.a.j.1.1 1
60.59 even 2 3600.2.f.p.2449.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 1 5.2 odd 4
75.2.a.c.1.1 yes 1 5.3 odd 4
75.2.b.a.49.1 2 5.4 even 2 inner
75.2.b.a.49.2 2 1.1 even 1 trivial
225.2.a.a.1.1 1 15.8 even 4
225.2.a.e.1.1 1 15.2 even 4
225.2.b.a.199.1 2 3.2 odd 2
225.2.b.a.199.2 2 15.14 odd 2
1200.2.a.c.1.1 1 20.7 even 4
1200.2.a.p.1.1 1 20.3 even 4
1200.2.f.d.49.1 2 4.3 odd 2
1200.2.f.d.49.2 2 20.19 odd 2
3600.2.a.j.1.1 1 60.47 odd 4
3600.2.a.bk.1.1 1 60.23 odd 4
3600.2.f.p.2449.1 2 60.59 even 2
3600.2.f.p.2449.2 2 12.11 even 2
3675.2.a.b.1.1 1 35.27 even 4
3675.2.a.q.1.1 1 35.13 even 4
4800.2.a.bb.1.1 1 40.37 odd 4
4800.2.a.be.1.1 1 40.3 even 4
4800.2.a.bq.1.1 1 40.13 odd 4
4800.2.a.br.1.1 1 40.27 even 4
4800.2.f.l.3649.1 2 8.5 even 2
4800.2.f.l.3649.2 2 40.29 even 2
4800.2.f.y.3649.1 2 40.19 odd 2
4800.2.f.y.3649.2 2 8.3 odd 2
9075.2.a.a.1.1 1 55.43 even 4
9075.2.a.s.1.1 1 55.32 even 4