Properties

 Label 1200.2.f.d.49.2 Level $1200$ Weight $2$ Character 1200.49 Analytic conductor $9.582$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ 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: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 49.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.49 Dual form 1200.2.f.d.49.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{13} -2.00000i q^{17} -5.00000 q^{19} +3.00000 q^{21} -6.00000i q^{23} -1.00000i q^{27} -10.0000 q^{29} +3.00000 q^{31} -2.00000i q^{33} -2.00000i q^{37} -1.00000 q^{39} -8.00000 q^{41} -1.00000i q^{43} +2.00000i q^{47} -2.00000 q^{49} +2.00000 q^{51} -4.00000i q^{53} -5.00000i q^{57} -10.0000 q^{59} +7.00000 q^{61} +3.00000i q^{63} -3.00000i q^{67} +6.00000 q^{69} +8.00000 q^{71} -14.0000i q^{73} +6.00000i q^{77} +1.00000 q^{81} -6.00000i q^{83} -10.0000i q^{87} +3.00000 q^{91} +3.00000i q^{93} -17.0000i q^{97} +2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 4q^{11} - 10q^{19} + 6q^{21} - 20q^{29} + 6q^{31} - 2q^{39} - 16q^{41} - 4q^{49} + 4q^{51} - 20q^{59} + 14q^{61} + 12q^{69} + 16q^{71} + 2q^{81} + 6q^{91} + 4q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000i 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 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 $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 0 0
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ − 6.00000i − 1.25109i −0.780189 0.625543i $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ −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 $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ − 2.00000i − 0.348155i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 0 0
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.00000i 0.291730i 0.989305 + 0.145865i $$0.0465965\pi$$
−0.989305 + 0.145865i $$0.953403\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ − 4.00000i − 0.549442i −0.961524 0.274721i $$-0.911414\pi$$
0.961524 0.274721i $$-0.0885855\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 5.00000i − 0.662266i
$$58$$ 0 0
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 0 0
$$63$$ 3.00000i 0.377964i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 3.00000i − 0.366508i −0.983066 0.183254i $$-0.941337\pi$$
0.983066 0.183254i $$-0.0586631\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.00000i 0.683763i
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 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
$$92$$ 0 0
$$93$$ 3.00000i 0.311086i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 17.0000i − 1.72609i −0.505128 0.863044i $$-0.668555\pi$$
0.505128 0.863044i $$-0.331445\pi$$
$$98$$ 0 0
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ −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
$$112$$ 0 0
$$113$$ − 4.00000i − 0.376288i −0.982141 0.188144i $$-0.939753\pi$$
0.982141 0.188144i $$-0.0602472\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 1.00000i − 0.0924500i
$$118$$ 0 0
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ − 8.00000i − 0.721336i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 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 $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 15.0000i 1.30066i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ 0 0
$$143$$ − 2.00000i − 0.167248i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 2.00000i − 0.164957i
$$148$$ 0 0
$$149$$ −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 $$-0.591936\pi$$
−0.284826 + 0.958579i $$0.591936\pi$$
$$152$$ 0 0
$$153$$ 2.00000i 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.0000i 1.03751i 0.854922 + 0.518756i $$0.173605\pi$$
−0.854922 + 0.518756i $$0.826395\pi$$
$$158$$ 0 0
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ −18.0000 −1.41860
$$162$$ 0 0
$$163$$ − 11.0000i − 0.861586i −0.902451 0.430793i $$-0.858234\pi$$
0.902451 0.430793i $$-0.141766\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 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
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 10.0000i − 0.751646i
$$178$$ 0 0
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ 17.0000 1.26360 0.631800 0.775131i $$-0.282316\pi$$
0.631800 + 0.775131i $$0.282316\pi$$
$$182$$ 0 0
$$183$$ 7.00000i 0.517455i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ 0 0
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ −22.0000 −1.59186 −0.795932 0.605386i $$-0.793019\pi$$
−0.795932 + 0.605386i $$0.793019\pi$$
$$192$$ 0 0
$$193$$ 11.0000i 0.791797i 0.918294 + 0.395899i $$0.129567\pi$$
−0.918294 + 0.395899i $$0.870433\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ 0 0
$$201$$ 3.00000 0.211604
$$202$$ 0 0
$$203$$ 30.0000i 2.10559i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.00000i 0.417029i
$$208$$ 0 0
$$209$$ 10.0000 0.691714
$$210$$ 0 0
$$211$$ 13.0000 0.894957 0.447478 0.894295i $$-0.352322\pi$$
0.447478 + 0.894295i $$0.352322\pi$$
$$212$$ 0 0
$$213$$ 8.00000i 0.548151i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 9.00000i − 0.610960i
$$218$$ 0 0
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 0 0
$$223$$ 19.0000i 1.27233i 0.771551 + 0.636167i $$0.219481\pi$$
−0.771551 + 0.636167i $$0.780519\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 8.00000i − 0.530979i −0.964114 0.265489i $$-0.914466\pi$$
0.964114 0.265489i $$-0.0855335\pi$$
$$228$$ 0 0
$$229$$ 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 $$-0.712073\pi$$
0.618041 0.786146i $$-0.287927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ −23.0000 −1.48156 −0.740780 0.671748i $$-0.765544\pi$$
−0.740780 + 0.671748i $$0.765544\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 5.00000i − 0.318142i
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 12.0000i 0.754434i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 12.0000i − 0.748539i −0.927320 0.374270i $$-0.877893\pi$$
0.927320 0.374270i $$-0.122107\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 0 0
$$263$$ − 16.0000i − 0.986602i −0.869859 0.493301i $$-0.835790\pi$$
0.869859 0.493301i $$-0.164210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 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 $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 3.00000i 0.181568i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.00000i 0.180253i 0.995930 + 0.0901263i $$0.0287271\pi$$
−0.995930 + 0.0901263i $$0.971273\pi$$
$$278$$ 0 0
$$279$$ −3.00000 −0.179605
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ 9.00000i 0.534994i 0.963559 + 0.267497i $$0.0861966\pi$$
−0.963559 + 0.267497i $$0.913803\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000i 1.41668i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 17.0000 0.996558
$$292$$ 0 0
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.00000i 0.116052i
$$298$$ 0 0
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 0 0
$$303$$ 12.0000i 0.689382i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 7.00000i 0.399511i 0.979846 + 0.199756i $$0.0640148\pi$$
−0.979846 + 0.199756i $$0.935985\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 11.0000i 0.621757i 0.950450 + 0.310878i $$0.100623\pi$$
−0.950450 + 0.310878i $$0.899377\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.00000i 0.449325i 0.974437 + 0.224662i $$0.0721279\pi$$
−0.974437 + 0.224662i $$0.927872\pi$$
$$318$$ 0 0
$$319$$ 20.0000 1.11979
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 10.0000i 0.556415i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 5.00000i − 0.276501i
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 0 0
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.0000i 1.25289i 0.779466 + 0.626445i $$0.215491\pi$$
−0.779466 + 0.626445i $$0.784509\pi$$
$$338$$ 0 0
$$339$$ 4.00000 0.217250
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ − 15.0000i − 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.00000i 0.107366i 0.998558 + 0.0536828i $$0.0170960\pi$$
−0.998558 + 0.0536828i $$0.982904\pi$$
$$348$$ 0 0
$$349$$ −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
$$352$$ 0 0
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 6.00000i − 0.317554i
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ − 7.00000i − 0.367405i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 27.0000i 1.40939i 0.709511 + 0.704694i $$0.248916\pi$$
−0.709511 + 0.704694i $$0.751084\pi$$
$$368$$ 0 0
$$369$$ 8.00000 0.416463
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ − 29.0000i − 1.50156i −0.660551 0.750782i $$-0.729677\pi$$
0.660551 0.750782i $$-0.270323\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 10.0000i − 0.515026i
$$378$$ 0 0
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ − 36.0000i − 1.83951i −0.392488 0.919757i $$-0.628386\pi$$
0.392488 0.919757i $$-0.371614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.00000i 0.0508329i
$$388$$ 0 0
$$389$$ 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
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 7.00000i − 0.351320i −0.984451 0.175660i $$-0.943794\pi$$
0.984451 0.175660i $$-0.0562059\pi$$
$$398$$ 0 0
$$399$$ −15.0000 −0.750939
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 0 0
$$403$$ 3.00000i 0.149441i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 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
$$412$$ 0 0
$$413$$ 30.0000i 1.47620i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 20.0000i 0.979404i
$$418$$ 0 0
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ − 2.00000i − 0.0972433i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 21.0000i − 1.01626i
$$428$$ 0 0
$$429$$ 2.00000 0.0965609
$$430$$ 0 0
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ − 29.0000i − 1.39365i −0.717241 0.696826i $$-0.754595\pi$$
0.717241 0.696826i $$-0.245405\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 30.0000i 1.43509i
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 10.0000i − 0.472984i
$$448$$ 0 0
$$449$$ −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
$$452$$ 0 0
$$453$$ − 7.00000i − 0.328889i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ 0 0
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ 24.0000i 1.11537i 0.830051 + 0.557687i $$0.188311\pi$$
−0.830051 + 0.557687i $$0.811689\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 38.0000i − 1.75843i −0.476425 0.879215i $$-0.658068\pi$$
0.476425 0.879215i $$-0.341932\pi$$
$$468$$ 0 0
$$469$$ −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
$$476$$ 0 0
$$477$$ 4.00000i 0.183147i
$$478$$ 0 0
$$479$$ 30.0000 1.37073 0.685367 0.728197i $$-0.259642\pi$$
0.685367 + 0.728197i $$0.259642\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 0 0
$$483$$ − 18.0000i − 0.819028i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 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 $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 0 0
$$493$$ 20.0000i 0.900755i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 24.0000i − 1.07655i
$$498$$ 0 0
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ − 16.0000i − 0.713405i −0.934218 0.356702i $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 12.0000i 0.532939i
$$508$$ 0 0
$$509$$ 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
$$512$$ 0 0
$$513$$ 5.00000i 0.220755i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 4.00000i − 0.175920i
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ − 31.0000i − 1.35554i −0.735276 0.677768i $$-0.762948\pi$$
0.735276 0.677768i $$-0.237052\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 6.00000i − 0.261364i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 10.0000 0.433963
$$532$$ 0 0
$$533$$ − 8.00000i − 0.346518i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 10.0000i − 0.431532i
$$538$$ 0 0
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ −3.00000 −0.128980 −0.0644900 0.997918i $$-0.520542\pi$$
−0.0644900 + 0.997918i $$0.520542\pi$$
$$542$$ 0 0
$$543$$ 17.0000i 0.729540i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ 0 0
$$549$$ −7.00000 −0.298753
$$550$$ 0 0
$$551$$ 50.0000 2.13007
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 42.0000i − 1.77960i −0.456354 0.889799i $$-0.650845\pi$$
0.456354 0.889799i $$-0.349155\pi$$
$$558$$ 0 0
$$559$$ 1.00000 0.0422955
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 0 0
$$563$$ − 6.00000i − 0.252870i −0.991975 0.126435i $$-0.959647\pi$$
0.991975 0.126435i $$-0.0403535\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 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 $$-0.412309\pi$$
0.272017 + 0.962293i $$0.412309\pi$$
$$572$$ 0 0
$$573$$ − 22.0000i − 0.919063i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 13.0000i 0.541197i 0.962692 + 0.270599i $$0.0872216\pi$$
−0.962692 + 0.270599i $$0.912778\pi$$
$$578$$ 0 0
$$579$$ −11.0000 −0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ 0 0
$$583$$ 8.00000i 0.331326i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ −15.0000 −0.618064
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ 0 0
$$593$$ 16.0000i 0.657041i 0.944497 + 0.328521i $$0.106550\pi$$
−0.944497 + 0.328521i $$0.893450\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 5.00000i − 0.204636i
$$598$$ 0 0
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ 0 0
$$603$$ 3.00000i 0.122169i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 8.00000i − 0.324710i −0.986732 0.162355i $$-0.948091\pi$$
0.986732 0.162355i $$-0.0519090\pi$$
$$608$$ 0 0
$$609$$ −30.0000 −1.21566
$$610$$ 0 0
$$611$$ −2.00000 −0.0809113
$$612$$ 0 0
$$613$$ − 14.0000i − 0.565455i −0.959200 0.282727i $$-0.908761\pi$$
0.959200 0.282727i $$-0.0912392\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 12.0000i − 0.483102i −0.970388 0.241551i $$-0.922344\pi$$
0.970388 0.241551i $$-0.0776561\pi$$
$$618$$ 0 0
$$619$$ −25.0000 −1.00483 −0.502417 0.864625i $$-0.667556\pi$$
−0.502417 + 0.864625i $$0.667556\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 10.0000i 0.399362i
$$628$$ 0 0
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 23.0000 0.915616 0.457808 0.889051i $$-0.348635\pi$$
0.457808 + 0.889051i $$0.348635\pi$$
$$632$$ 0 0
$$633$$ 13.0000i 0.516704i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ 0 0
$$643$$ − 36.0000i − 1.41970i −0.704352 0.709851i $$-0.748762\pi$$
0.704352 0.709851i $$-0.251238\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 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
$$652$$ 0 0
$$653$$ − 14.0000i − 0.547862i −0.961749 0.273931i $$-0.911676\pi$$
0.961749 0.273931i $$-0.0883240\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 14.0000i 0.546192i
$$658$$ 0 0
$$659$$ −40.0000 −1.55818 −0.779089 0.626913i $$-0.784318\pi$$
−0.779089 + 0.626913i $$0.784318\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 0 0
$$663$$ 2.00000i 0.0776736i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 60.0000i 2.32321i
$$668$$ 0 0
$$669$$ −19.0000 −0.734582
$$670$$ 0 0
$$671$$ −14.0000 −0.540464
$$672$$ 0 0
$$673$$ 6.00000i 0.231283i 0.993291 + 0.115642i $$0.0368924\pi$$
−0.993291 + 0.115642i $$0.963108\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 42.0000i − 1.61419i −0.590421 0.807096i $$-0.701038\pi$$
0.590421 0.807096i $$-0.298962\pi$$
$$678$$ 0 0
$$679$$ −51.0000 −1.95720
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 0 0
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 15.0000i 0.572286i
$$688$$ 0 0
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ − 6.00000i − 0.227921i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16.0000i 0.606043i
$$698$$ 0 0
$$699$$ 24.0000 0.907763
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ 10.0000i 0.377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 36.0000i − 1.35392i
$$708$$ 0 0
$$709$$ −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
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 20.0000i 0.746914i
$$718$$ 0 0
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 0 0
$$723$$ − 23.0000i − 0.855379i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 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
$$732$$ 0 0
$$733$$ − 34.0000i − 1.25582i −0.778287 0.627909i $$-0.783911\pi$$
0.778287 0.627909i $$-0.216089\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.00000i 0.221013i
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 5.00000 0.183680
$$742$$ 0 0
$$743$$ 4.00000i 0.146746i 0.997305 + 0.0733729i $$0.0233763\pi$$
−0.997305 + 0.0733729i $$0.976624\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.00000i 0.219529i
$$748$$ 0 0
$$749$$ 36.0000 1.31541
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ − 12.0000i − 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 23.0000i 0.835949i 0.908459 + 0.417975i $$0.137260\pi$$
−0.908459 + 0.417975i $$0.862740\pi$$
$$758$$ 0 0
$$759$$ −12.0000 −0.435572
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 0 0
$$763$$ 15.0000i 0.543036i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 10.0000i − 0.361079i
$$768$$ 0 0
$$769$$ −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
$$772$$ 0 0
$$773$$ − 24.0000i − 0.863220i −0.902060 0.431610i $$-0.857946\pi$$
0.902060 0.431610i $$-0.142054\pi$$
$$774$$ 0 0
$$775$$ 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
$$782$$ 0 0
$$783$$ 10.0000i 0.357371i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 7.00000i 0.249523i 0.992187 + 0.124762i $$0.0398166\pi$$
−0.992187 + 0.124762i $$0.960183\pi$$
$$788$$ 0 0
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 7.00000i 0.248577i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 52.0000i − 1.84193i −0.389640 0.920967i $$-0.627401\pi$$
0.389640 0.920967i $$-0.372599\pi$$
$$798$$ 0 0
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 28.0000i 0.988099i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 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 $$-0.657208\pi$$
−0.474049 + 0.880498i $$0.657208\pi$$
$$812$$ 0 0
$$813$$ 8.00000i 0.280572i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 5.00000i 0.174928i
$$818$$ 0 0
$$819$$ −3.00000 −0.104828
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ − 41.0000i − 1.42917i −0.699549 0.714585i $$-0.746616\pi$$
0.699549 0.714585i $$-0.253384\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ 0 0
$$829$$ 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
$$832$$ 0 0
$$833$$ 4.00000i 0.138592i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 3.00000i − 0.103695i
$$838$$ 0 0
$$839$$ 10.0000 0.345238 0.172619 0.984989i $$-0.444777\pi$$
0.172619 + 0.984989i $$0.444777\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 0 0
$$843$$ − 18.0000i − 0.619953i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 21.0000i 0.721569i
$$848$$ 0 0
$$849$$ −9.00000 −0.308879
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ 51.0000i 1.74621i 0.487535 + 0.873103i $$0.337896\pi$$
−0.487535 + 0.873103i $$0.662104\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 28.0000i 0.956462i 0.878234 + 0.478231i $$0.158722\pi$$
−0.878234 + 0.478231i $$0.841278\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ 0 0
$$863$$ − 16.0000i − 0.544646i −0.962206 0.272323i $$-0.912208\pi$$
0.962206 0.272323i $$-0.0877920\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ 0 0
$$873$$ 17.0000i 0.575363i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 27.0000i − 0.911725i −0.890050 0.455863i $$-0.849331\pi$$
0.890050 0.455863i $$-0.150669\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 32.0000 1.07811 0.539054 0.842271i $$-0.318782\pi$$
0.539054 + 0.842271i $$0.318782\pi$$
$$882$$ 0 0
$$883$$ − 41.0000i − 1.37976i −0.723924 0.689880i $$-0.757663\pi$$
0.723924 0.689880i $$-0.242337\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 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
$$892$$ 0 0
$$893$$ − 10.0000i − 0.334637i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000i 0.200334i
$$898$$ 0 0
$$899$$ −30.0000 −1.00056
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ 0 0
$$903$$ − 3.00000i − 0.0998337i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12.0000i 0.398453i 0.979953 + 0.199227i $$0.0638430\pi$$
−0.979953 + 0.199227i $$0.936157\pi$$
$$908$$ 0 0
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ 58.0000 1.92163 0.960813 0.277198i $$-0.0894057\pi$$
0.960813 + 0.277198i $$0.0894057\pi$$
$$912$$ 0 0
$$913$$ 12.0000i 0.397142i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 36.0000i 1.18882i
$$918$$ 0 0
$$919$$ 55.0000 1.81428 0.907141 0.420826i $$-0.138260\pi$$
0.907141 + 0.420826i $$0.138260\pi$$
$$920$$ 0 0
$$921$$ −7.00000 −0.230658
$$922$$ 0 0
$$923$$ 8.00000i 0.263323i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 4.00000i − 0.131377i
$$928$$ 0 0
$$929$$ 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
$$932$$ 0 0
$$933$$ 18.0000i 0.589294i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 33.0000i 1.07806i 0.842286 + 0.539032i $$0.181210\pi$$
−0.842286 + 0.539032i $$0.818790\pi$$
$$938$$ 0 0
$$939$$ −11.0000 −0.358971
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ 0 0
$$943$$ 48.0000i 1.56310i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 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 $$0.361640\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 20.0000i 0.646508i
$$958$$ 0 0
$$959$$ 54.0000 1.74375
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 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 $$-0.735392\pi$$
−0.673922 + 0.738802i $$0.735392\pi$$
$$972$$ 0 0
$$973$$ − 60.0000i − 1.92351i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 2.00000i − 0.0639857i −0.999488 0.0319928i $$-0.989815\pi$$
0.999488 0.0319928i $$-0.0101854\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 5.00000 0.159638
$$982$$ 0 0
$$983$$ − 36.0000i − 1.14822i −0.818778 0.574111i $$-0.805348\pi$$
0.818778 0.574111i $$-0.194652\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 6.00000i 0.190982i
$$988$$ 0 0
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ −17.0000 −0.540023 −0.270011 0.962857i $$-0.587027\pi$$
−0.270011 + 0.962857i $$0.587027\pi$$
$$992$$ 0 0
$$993$$ − 12.0000i − 0.380808i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 42.0000i − 1.33015i −0.746775 0.665077i $$-0.768399\pi$$
0.746775 0.665077i $$-0.231601\pi$$
$$998$$ 0 0
$$999$$ −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 1200.2.f.d.49.2 2
3.2 odd 2 3600.2.f.p.2449.1 2
4.3 odd 2 75.2.b.a.49.1 2
5.2 odd 4 1200.2.a.p.1.1 1
5.3 odd 4 1200.2.a.c.1.1 1
5.4 even 2 inner 1200.2.f.d.49.1 2
8.3 odd 2 4800.2.f.l.3649.2 2
8.5 even 2 4800.2.f.y.3649.1 2
12.11 even 2 225.2.b.a.199.2 2
15.2 even 4 3600.2.a.bk.1.1 1
15.8 even 4 3600.2.a.j.1.1 1
15.14 odd 2 3600.2.f.p.2449.2 2
20.3 even 4 75.2.a.a.1.1 1
20.7 even 4 75.2.a.c.1.1 yes 1
20.19 odd 2 75.2.b.a.49.2 2
40.3 even 4 4800.2.a.bb.1.1 1
40.13 odd 4 4800.2.a.br.1.1 1
40.19 odd 2 4800.2.f.l.3649.1 2
40.27 even 4 4800.2.a.bq.1.1 1
40.29 even 2 4800.2.f.y.3649.2 2
40.37 odd 4 4800.2.a.be.1.1 1
60.23 odd 4 225.2.a.e.1.1 1
60.47 odd 4 225.2.a.a.1.1 1
60.59 even 2 225.2.b.a.199.1 2
140.27 odd 4 3675.2.a.q.1.1 1
140.83 odd 4 3675.2.a.b.1.1 1
220.43 odd 4 9075.2.a.s.1.1 1
220.87 odd 4 9075.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 1 20.3 even 4
75.2.a.c.1.1 yes 1 20.7 even 4
75.2.b.a.49.1 2 4.3 odd 2
75.2.b.a.49.2 2 20.19 odd 2
225.2.a.a.1.1 1 60.47 odd 4
225.2.a.e.1.1 1 60.23 odd 4
225.2.b.a.199.1 2 60.59 even 2
225.2.b.a.199.2 2 12.11 even 2
1200.2.a.c.1.1 1 5.3 odd 4
1200.2.a.p.1.1 1 5.2 odd 4
1200.2.f.d.49.1 2 5.4 even 2 inner
1200.2.f.d.49.2 2 1.1 even 1 trivial
3600.2.a.j.1.1 1 15.8 even 4
3600.2.a.bk.1.1 1 15.2 even 4
3600.2.f.p.2449.1 2 3.2 odd 2
3600.2.f.p.2449.2 2 15.14 odd 2
3675.2.a.b.1.1 1 140.83 odd 4
3675.2.a.q.1.1 1 140.27 odd 4
4800.2.a.bb.1.1 1 40.3 even 4
4800.2.a.be.1.1 1 40.37 odd 4
4800.2.a.bq.1.1 1 40.27 even 4
4800.2.a.br.1.1 1 40.13 odd 4
4800.2.f.l.3649.1 2 40.19 odd 2
4800.2.f.l.3649.2 2 8.3 odd 2
4800.2.f.y.3649.1 2 8.5 even 2
4800.2.f.y.3649.2 2 40.29 even 2
9075.2.a.a.1.1 1 220.87 odd 4
9075.2.a.s.1.1 1 220.43 odd 4