# Properties

 Label 225.2.b.a.199.2 Level $225$ Weight $2$ Character 225.199 Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 225.199 Dual form 225.2.b.a.199.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} +O(q^{10})$$ $$q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} -2.00000 q^{11} +1.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} +5.00000 q^{19} -4.00000i q^{22} -6.00000i q^{23} -2.00000 q^{26} -6.00000i q^{28} +10.0000 q^{29} -3.00000 q^{31} -8.00000i q^{32} -4.00000 q^{34} -2.00000i q^{37} +10.0000i q^{38} +8.00000 q^{41} +1.00000i q^{43} +4.00000 q^{44} +12.0000 q^{46} +2.00000i q^{47} -2.00000 q^{49} -2.00000i q^{52} +4.00000i q^{53} +20.0000i q^{58} -10.0000 q^{59} +7.00000 q^{61} -6.00000i q^{62} +8.00000 q^{64} +3.00000i q^{67} -4.00000i q^{68} +8.00000 q^{71} -14.0000i q^{73} +4.00000 q^{74} -10.0000 q^{76} -6.00000i q^{77} +16.0000i q^{82} -6.00000i q^{83} -2.00000 q^{86} -3.00000 q^{91} +12.0000i q^{92} -4.00000 q^{94} -17.0000i q^{97} -4.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + O(q^{10})$$ $$2q - 4q^{4} - 4q^{11} - 12q^{14} - 8q^{16} + 10q^{19} - 4q^{26} + 20q^{29} - 6q^{31} - 8q^{34} + 16q^{41} + 8q^{44} + 24q^{46} - 4q^{49} - 20q^{59} + 14q^{61} + 16q^{64} + 16q^{71} + 8q^{74} - 20q^{76} - 4q^{86} - 6q^{91} - 8q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$0.750000\pi$$
$$3$$ 0 0
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 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 $$0.955716\pi$$
$$14$$ −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 $$0.922021\pi$$
$$18$$ 0 0
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 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
$$27$$ 0 0
$$28$$ − 6.00000i − 1.13389i
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 10.0000i 1.62221i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ 1.00000i 0.152499i 0.997089 + 0.0762493i $$0.0242945\pi$$
−0.997089 + 0.0762493i $$0.975706\pi$$
$$44$$ 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 $$0.953403\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 4.00000i 0.549442i 0.961524 + 0.274721i $$0.0885855\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 20.0000i 2.62613i
$$59$$ −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 $$0.352090\pi$$
$$62$$ − 6.00000i − 0.762001i
$$63$$ 0 0
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.00000i 0.366508i 0.983066 + 0.183254i $$0.0586631\pi$$
−0.983066 + 0.183254i $$0.941337\pi$$
$$68$$ − 4.00000i − 0.485071i
$$69$$ 0 0
$$70$$ 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 $$-0.305631\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ −10.0000 −1.14708
$$77$$ − 6.00000i − 0.683763i
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 16.0000i 1.76690i
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 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
$$93$$ 0 0
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 17.0000i − 1.72609i −0.505128 0.863044i $$-0.668555\pi$$
0.505128 0.863044i $$-0.331445\pi$$
$$98$$ − 4.00000i − 0.404061i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 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 $$0.803036\pi$$
$$108$$ 0 0
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 12.0000i − 1.13389i
$$113$$ 4.00000i 0.376288i 0.982141 + 0.188144i $$0.0602472\pi$$
−0.982141 + 0.188144i $$0.939753\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −20.0000 −1.85695
$$117$$ 0 0
$$118$$ − 20.0000i − 1.84115i
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 14.0000i 1.26750i
$$123$$ 0 0
$$124$$ 6.00000 0.538816
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 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
$$134$$ −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 $$-0.279207\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.0000i 1.34269i
$$143$$ − 2.00000i − 0.167248i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 28.0000 2.31730
$$147$$ 0 0
$$148$$ 4.00000i 0.328798i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 7.00000 0.569652 0.284826 0.958579i $$-0.408064\pi$$
0.284826 + 0.958579i $$0.408064\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 12.0000 0.966988
$$155$$ 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
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 18.0000 1.41860
$$162$$ 0 0
$$163$$ 11.0000i 0.861586i 0.902451 + 0.430793i $$0.141766\pi$$
−0.902451 + 0.430793i $$0.858234\pi$$
$$164$$ −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
$$171$$ 0 0
$$172$$ − 2.00000i − 0.152499i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 8.00000 0.603023
$$177$$ 0 0
$$178$$ 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 $$0.282316\pi$$
$$182$$ − 6.00000i − 0.444750i
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 4.00000i − 0.292509i
$$188$$ − 4.00000i − 0.291730i
$$189$$ 0 0
$$190$$ 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 $$0.870433\pi$$
$$194$$ 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 $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 24.0000i − 1.68863i
$$203$$ 30.0000i 2.10559i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ − 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
$$213$$ 0 0
$$214$$ −24.0000 −1.64061
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 9.00000i − 0.610960i
$$218$$ − 10.0000i − 0.677285i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ − 19.0000i − 1.27233i −0.771551 0.636167i $$-0.780519\pi$$
0.771551 0.636167i $$-0.219481\pi$$
$$224$$ 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 $$-0.0855335\pi$$
$$228$$ 0 0
$$229$$ 15.0000 0.991228 0.495614 0.868543i $$-0.334943\pi$$
0.495614 + 0.868543i $$0.334943\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 24.0000i 1.57229i 0.618041 + 0.786146i $$0.287927\pi$$
−0.618041 + 0.786146i $$0.712073\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 20.0000 1.30189
$$237$$ 0 0
$$238$$ − 12.0000i − 0.777844i
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
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 $$0.765544\pi$$
$$242$$ − 14.0000i − 0.899954i
$$243$$ 0 0
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000i 0.318142i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 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
$$254$$ −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 $$0.877893\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 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 $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 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
$$273$$ 0 0
$$274$$ 36.0000 2.17484
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.00000i 0.180253i 0.995930 + 0.0901263i $$0.0287271\pi$$
−0.995930 + 0.0901263i $$0.971273\pi$$
$$278$$ − 40.0000i − 2.39904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ − 9.00000i − 0.534994i −0.963559 0.267497i $$-0.913803\pi$$
0.963559 0.267497i $$-0.0861966\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 24.0000i 1.41668i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 28.0000i 1.63858i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 20.0000i 1.15857i
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 14.0000i 0.805609i
$$303$$ 0 0
$$304$$ −20.0000 −1.14708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 7.00000i − 0.399511i −0.979846 0.199756i $$-0.935985\pi$$
0.979846 0.199756i $$-0.0640148\pi$$
$$308$$ 12.0000i 0.683763i
$$309$$ 0 0
$$310$$ 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 $$0.899377\pi$$
$$314$$ −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 $$-0.0721279\pi$$
$$318$$ 0 0
$$319$$ −20.0000 −1.11979
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 36.0000i 2.00620i
$$323$$ 10.0000i 0.556415i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −22.0000 −1.21847
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ −24.0000 −1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.0000i 1.25289i 0.779466 + 0.626445i $$0.215491\pi$$
−0.779466 + 0.626445i $$0.784509\pi$$
$$338$$ 24.0000i 1.30543i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ 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 $$0.982904\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 16.0000i 0.852803i
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ − 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
$$363$$ 0 0
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 27.0000i − 1.40939i −0.709511 0.704694i $$-0.751084\pi$$
0.709511 0.704694i $$-0.248916\pi$$
$$368$$ 24.0000i 1.25109i
$$369$$ 0 0
$$370$$ 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 $$-0.270323\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 10.0000i 0.515026i
$$378$$ 0 0
$$379$$ −25.0000 −1.28416 −0.642082 0.766636i $$-0.721929\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 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
$$387$$ 0 0
$$388$$ 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
$$393$$ 0 0
$$394$$ 36.0000 1.81365
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 7.00000i − 0.351320i −0.984451 0.175660i $$-0.943794\pi$$
0.984451 0.175660i $$-0.0562059\pi$$
$$398$$ 10.0000i 0.501255i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ − 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
$$411$$ 0 0
$$412$$ 8.00000i 0.394132i
$$413$$ − 30.0000i − 1.47620i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 8.00000 0.392232
$$417$$ 0 0
$$418$$ − 20.0000i − 0.978232i
$$419$$ −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 $$0.319894\pi$$
$$422$$ − 26.0000i − 1.26566i
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 21.0000i 1.01626i
$$428$$ − 24.0000i − 1.16008i
$$429$$ 0 0
$$430$$ 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 $$-0.245405\pi$$
$$434$$ 18.0000 0.864028
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ − 30.0000i − 1.43509i
$$438$$ 0 0
$$439$$ 35.0000 1.67046 0.835229 0.549902i $$-0.185335\pi$$
0.835229 + 0.549902i $$0.185335\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 4.00000i − 0.190261i
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 38.0000 1.79935
$$447$$ 0 0
$$448$$ 24.0000i 1.13389i
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ − 8.00000i − 0.376288i
$$453$$ 0 0
$$454$$ 16.0000 0.750917
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ 30.0000i 1.40181i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ − 24.0000i − 1.11537i −0.830051 0.557687i $$-0.811689\pi$$
0.830051 0.557687i $$-0.188311\pi$$
$$464$$ −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 $$-0.341932\pi$$
$$468$$ 0 0
$$469$$ −9.00000 −0.415581
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 2.00000i − 0.0919601i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ 0 0
$$478$$ 40.0000i 1.82956i
$$479$$ 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
$$482$$ − 46.0000i − 2.09524i
$$483$$ 0 0
$$484$$ 14.0000 0.636364
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 13.0000i 0.589086i 0.955638 + 0.294543i $$0.0951675\pi$$
−0.955638 + 0.294543i $$0.904833\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 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
$$494$$ −10.0000 −0.449921
$$495$$ 0 0
$$496$$ 12.0000 0.538816
$$497$$ 24.0000i 1.07655i
$$498$$ 0 0
$$499$$ 5.00000 0.223831 0.111915 0.993718i $$-0.464301\pi$$
0.111915 + 0.993718i $$0.464301\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 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
$$507$$ 0 0
$$508$$ − 16.0000i − 0.709885i
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ 42.0000 1.85797
$$512$$ 32.0000i 1.41421i
$$513$$ 0 0
$$514$$ −24.0000 −1.05859
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 4.00000i − 0.175920i
$$518$$ 12.0000i 0.527250i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ 31.0000i 1.35554i 0.735276 + 0.677768i $$0.237052\pi$$
−0.735276 + 0.677768i $$0.762948\pi$$
$$524$$ 24.0000 1.04844
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ − 6.00000i − 0.261364i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 30.0000i − 1.30066i
$$533$$ 8.00000i 0.346518i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ − 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
$$543$$ 0 0
$$544$$ 16.0000 0.685994
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ 36.0000i 1.53784i
$$549$$ 0 0
$$550$$ 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 $$0.650845\pi$$
$$558$$ 0 0
$$559$$ −1.00000 −0.0422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 36.0000i 1.51857i
$$563$$ − 6.00000i − 0.252870i −0.991975 0.126435i $$-0.959647\pi$$
0.991975 0.126435i $$-0.0403535\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 18.0000 0.756596
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ −48.0000 −2.00348
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 13.0000i 0.541197i 0.962692 + 0.270599i $$0.0872216\pi$$
−0.962692 + 0.270599i $$0.912778\pi$$
$$578$$ 26.0000i 1.08146i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ − 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 $$0.920343\pi$$
$$588$$ 0 0
$$589$$ −15.0000 −0.618064
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 8.00000i 0.328798i
$$593$$ − 16.0000i − 0.657041i −0.944497 0.328521i $$-0.893450\pi$$
0.944497 0.328521i $$-0.106550\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −20.0000 −0.819232
$$597$$ 0 0
$$598$$ 12.0000i 0.490716i
$$599$$ −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 $$0.585418\pi$$
$$602$$ − 6.00000i − 0.244542i
$$603$$ 0 0
$$604$$ −14.0000 −0.569652
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8.00000i 0.324710i 0.986732 + 0.162355i $$0.0519090\pi$$
−0.986732 + 0.162355i $$0.948091\pi$$
$$608$$ − 40.0000i − 1.62221i
$$609$$ 0 0
$$610$$ 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 $$-0.0912392\pi$$
$$614$$ 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 $$0.922344\pi$$
$$618$$ 0 0
$$619$$ 25.0000 1.00483 0.502417 0.864625i $$-0.332444\pi$$
0.502417 + 0.864625i $$0.332444\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 36.0000i 1.44347i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ − 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
$$633$$ 0 0
$$634$$ 16.0000 0.635441
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ − 40.0000i − 1.58362i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ 0 0
$$643$$ 36.0000i 1.41970i 0.704352 + 0.709851i $$0.251238\pi$$
−0.704352 + 0.709851i $$0.748762\pi$$
$$644$$ −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
$$651$$ 0 0
$$652$$ − 22.0000i − 0.861586i
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −32.0000 −1.24939
$$657$$ 0 0
$$658$$ − 12.0000i − 0.467809i
$$659$$ −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 $$0.764708\pi$$
$$662$$ 24.0000i 0.932786i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 60.0000i − 2.32321i
$$668$$ − 24.0000i − 0.928588i
$$669$$ 0 0
$$670$$ 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 $$0.963108\pi$$
$$674$$ −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 $$0.701038\pi$$
$$678$$ 0 0
$$679$$ 51.0000 1.95720
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 12.0000i 0.459504i
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −30.0000 −1.14541
$$687$$ 0 0
$$688$$ − 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
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16.0000i 0.606043i
$$698$$ − 20.0000i − 0.757011i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ − 10.0000i − 0.377157i
$$704$$ −16.0000 −0.603023
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ − 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
$$716$$ 20.0000 0.747435
$$717$$ 0 0
$$718$$ 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
$$722$$ 12.0000i 0.446594i
$$723$$ 0 0
$$724$$ −34.0000 −1.26360
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 43.0000i 1.59478i 0.603463 + 0.797391i $$0.293787\pi$$
−0.603463 + 0.797391i $$0.706213\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 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 $$-0.216089\pi$$
$$734$$ 54.0000 1.99318
$$735$$ 0 0
$$736$$ −48.0000 −1.76930
$$737$$ − 6.00000i − 0.221013i
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 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
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 23.0000i 0.835949i 0.908459 + 0.417975i $$0.137260\pi$$
−0.908459 + 0.417975i $$0.862740\pi$$
$$758$$ − 50.0000i − 1.81608i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −12.0000 −0.435000 −0.217500 0.976060i $$-0.569790\pi$$
−0.217500 + 0.976060i $$0.569790\pi$$
$$762$$ 0 0
$$763$$ − 15.0000i − 0.543036i
$$764$$ 44.0000 1.59186
$$765$$ 0 0
$$766$$ 72.0000 2.60147
$$767$$ − 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
$$771$$ 0 0
$$772$$ − 22.0000i − 0.791797i
$$773$$ 24.0000i 0.863220i 0.902060 + 0.431610i $$0.142054\pi$$
−0.902060 + 0.431610i $$0.857946\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 24.0000i 0.858238i
$$783$$ 0 0
$$784$$ 8.00000 0.285714
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 7.00000i − 0.249523i −0.992187 0.124762i $$-0.960183\pi$$
0.992187 0.124762i $$-0.0398166\pi$$
$$788$$ 36.0000i 1.28245i
$$789$$ 0 0
$$790$$ 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 $$0.627401\pi$$
$$798$$ 0 0
$$799$$ −4.00000 −0.141510
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 24.0000i − 0.847469i
$$803$$ 28.0000i 0.988099i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 6.00000 0.211341
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −20.0000 −0.703163 −0.351581 0.936157i $$-0.614356\pi$$
−0.351581 + 0.936157i $$0.614356\pi$$
$$810$$ 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
$$813$$ 0 0
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 5.00000i 0.174928i
$$818$$ − 10.0000i − 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ 41.0000i 1.42917i 0.699549 + 0.714585i $$0.253384\pi$$
−0.699549 + 0.714585i $$0.746616\pi$$
$$824$$ 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 $$-0.161846\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 8.00000i 0.277350i
$$833$$ − 4.00000i − 0.138592i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 20.0000 0.691714
$$837$$ 0 0
$$838$$ − 40.0000i − 1.38178i
$$839$$ 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
$$842$$ 44.0000i 1.51634i
$$843$$ 0 0
$$844$$ 26.0000 0.894957
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 21.0000i − 0.721569i
$$848$$ − 16.0000i − 0.549442i
$$849$$ 0 0
$$850$$ 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 $$0.662104\pi$$
$$854$$ −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 $$-0.158722\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 36.0000i 1.22616i
$$863$$ − 16.0000i − 0.544646i −0.962206 0.272323i $$-0.912208\pi$$
0.962206 0.272323i $$-0.0877920\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 58.0000 1.97092
$$867$$ 0 0
$$868$$ 18.0000i 0.610960i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 60.0000 2.02953
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 27.0000i − 0.911725i −0.890050 0.455863i $$-0.849331\pi$$
0.890050 0.455863i $$-0.150669\pi$$
$$878$$ 70.0000i 2.36239i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −32.0000 −1.07811 −0.539054 0.842271i $$-0.681218\pi$$
−0.539054 + 0.842271i $$0.681218\pi$$
$$882$$ 0 0
$$883$$ 41.0000i 1.37976i 0.723924 + 0.689880i $$0.242337\pi$$
−0.723924 + 0.689880i $$0.757663\pi$$
$$884$$ 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
$$891$$ 0 0
$$892$$ 38.0000i 1.27233i
$$893$$ 10.0000i 0.334637i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 40.0000i 1.33482i
$$899$$ −30.0000 −1.00056
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ − 32.0000i − 1.06548i
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 12.0000i − 0.398453i −0.979953 0.199227i $$-0.936157\pi$$
0.979953 0.199227i $$-0.0638430\pi$$
$$908$$ 16.0000i 0.530979i
$$909$$ 0 0
$$910$$ 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
$$914$$ 44.0000 1.45539
$$915$$ 0 0
$$916$$ −30.0000 −0.991228
$$917$$ − 36.0000i − 1.18882i
$$918$$ 0 0
$$919$$ −55.0000 −1.81428 −0.907141 0.420826i $$-0.861740\pi$$
−0.907141 + 0.420826i $$0.861740\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 24.0000i − 0.790398i
$$923$$ 8.00000i 0.263323i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 48.0000 1.57738
$$927$$ 0 0
$$928$$ − 80.0000i − 2.62613i
$$929$$ −50.0000 −1.64045 −0.820223 0.572043i $$-0.806151\pi$$
−0.820223 + 0.572043i $$0.806151\pi$$
$$930$$ 0 0
$$931$$ −10.0000 −0.327737
$$932$$ − 48.0000i − 1.57229i
$$933$$ 0 0
$$934$$ 76.0000 2.48680
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 33.0000i 1.07806i 0.842286 + 0.539032i $$0.181210\pi$$
−0.842286 + 0.539032i $$0.818790\pi$$
$$938$$ − 18.0000i − 0.587721i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 0 0
$$943$$ − 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
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 56.0000i − 1.81402i −0.421111 0.907009i $$-0.638360\pi$$
0.421111 0.907009i $$-0.361640\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −40.0000 −1.29369
$$957$$ 0 0
$$958$$ 60.0000i 1.93851i
$$959$$ 54.0000 1.74375
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 4.00000i 0.128965i
$$963$$ 0 0
$$964$$ 46.0000 1.48156
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 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
$$974$$ −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 $$0.989815\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 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
$$987$$ 0 0
$$988$$ − 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
$$993$$ 0 0
$$994$$ −48.0000 −1.52247
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 42.0000i − 1.33015i −0.746775 0.665077i $$-0.768399\pi$$
0.746775 0.665077i $$-0.231601\pi$$
$$998$$ 10.0000i 0.316544i
$$999$$ 0 0
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 225.2.b.a.199.2 2
3.2 odd 2 75.2.b.a.49.1 2
4.3 odd 2 3600.2.f.p.2449.1 2
5.2 odd 4 225.2.a.a.1.1 1
5.3 odd 4 225.2.a.e.1.1 1
5.4 even 2 inner 225.2.b.a.199.1 2
12.11 even 2 1200.2.f.d.49.2 2
15.2 even 4 75.2.a.c.1.1 yes 1
15.8 even 4 75.2.a.a.1.1 1
15.14 odd 2 75.2.b.a.49.2 2
20.3 even 4 3600.2.a.j.1.1 1
20.7 even 4 3600.2.a.bk.1.1 1
20.19 odd 2 3600.2.f.p.2449.2 2
24.5 odd 2 4800.2.f.l.3649.2 2
24.11 even 2 4800.2.f.y.3649.1 2
60.23 odd 4 1200.2.a.c.1.1 1
60.47 odd 4 1200.2.a.p.1.1 1
60.59 even 2 1200.2.f.d.49.1 2
105.62 odd 4 3675.2.a.q.1.1 1
105.83 odd 4 3675.2.a.b.1.1 1
120.29 odd 2 4800.2.f.l.3649.1 2
120.53 even 4 4800.2.a.bb.1.1 1
120.59 even 2 4800.2.f.y.3649.2 2
120.77 even 4 4800.2.a.bq.1.1 1
120.83 odd 4 4800.2.a.br.1.1 1
120.107 odd 4 4800.2.a.be.1.1 1
165.32 odd 4 9075.2.a.a.1.1 1
165.98 odd 4 9075.2.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 1 15.8 even 4
75.2.a.c.1.1 yes 1 15.2 even 4
75.2.b.a.49.1 2 3.2 odd 2
75.2.b.a.49.2 2 15.14 odd 2
225.2.a.a.1.1 1 5.2 odd 4
225.2.a.e.1.1 1 5.3 odd 4
225.2.b.a.199.1 2 5.4 even 2 inner
225.2.b.a.199.2 2 1.1 even 1 trivial
1200.2.a.c.1.1 1 60.23 odd 4
1200.2.a.p.1.1 1 60.47 odd 4
1200.2.f.d.49.1 2 60.59 even 2
1200.2.f.d.49.2 2 12.11 even 2
3600.2.a.j.1.1 1 20.3 even 4
3600.2.a.bk.1.1 1 20.7 even 4
3600.2.f.p.2449.1 2 4.3 odd 2
3600.2.f.p.2449.2 2 20.19 odd 2
3675.2.a.b.1.1 1 105.83 odd 4
3675.2.a.q.1.1 1 105.62 odd 4
4800.2.a.bb.1.1 1 120.53 even 4
4800.2.a.be.1.1 1 120.107 odd 4
4800.2.a.bq.1.1 1 120.77 even 4
4800.2.a.br.1.1 1 120.83 odd 4
4800.2.f.l.3649.1 2 120.29 odd 2
4800.2.f.l.3649.2 2 24.5 odd 2
4800.2.f.y.3649.1 2 24.11 even 2
4800.2.f.y.3649.2 2 120.59 even 2
9075.2.a.a.1.1 1 165.32 odd 4
9075.2.a.s.1.1 1 165.98 odd 4