# Properties

 Label 2475.2.c.h.199.1 Level $2475$ Weight $2$ Character 2475.199 Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.199 Dual form 2475.2.c.h.199.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{4} -1.00000i q^{7} +O(q^{10})$$ $$q+2.00000 q^{4} -1.00000i q^{7} +1.00000 q^{11} +1.00000i q^{13} +4.00000 q^{16} +6.00000i q^{17} +7.00000 q^{19} +6.00000i q^{23} -2.00000i q^{28} -6.00000 q^{29} -7.00000 q^{31} +2.00000i q^{37} +6.00000 q^{41} +1.00000i q^{43} +2.00000 q^{44} +6.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +5.00000 q^{61} +8.00000 q^{64} +5.00000i q^{67} +12.0000i q^{68} +12.0000 q^{71} -14.0000i q^{73} +14.0000 q^{76} -1.00000i q^{77} +4.00000 q^{79} -6.00000i q^{83} +6.00000 q^{89} +1.00000 q^{91} +12.0000i q^{92} +17.0000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4}+O(q^{10})$$ 2 * q + 4 * q^4 $$2 q + 4 q^{4} + 2 q^{11} + 8 q^{16} + 14 q^{19} - 12 q^{29} - 14 q^{31} + 12 q^{41} + 4 q^{44} + 12 q^{49} + 10 q^{61} + 16 q^{64} + 24 q^{71} + 28 q^{76} + 8 q^{79} + 12 q^{89} + 2 q^{91}+O(q^{100})$$ 2 * q + 4 * q^4 + 2 * q^11 + 8 * q^16 + 14 * q^19 - 12 * q^29 - 14 * q^31 + 12 * q^41 + 4 * q^44 + 12 * q^49 + 10 * q^61 + 16 * q^64 + 24 * q^71 + 28 * q^76 + 8 * q^79 + 12 * q^89 + 2 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i −0.981981 0.188982i $$-0.939481\pi$$
0.981981 0.188982i $$-0.0605189\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 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
$$16$$ 4.00000 1.00000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ 0 0
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ − 2.00000i − 0.377964i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −7.00000 −1.25724 −0.628619 0.777714i $$-0.716379\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 1.00000i 0.152499i 0.997089 + 0.0762493i $$0.0242945\pi$$
−0.997089 + 0.0762493i $$0.975706\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.00000i 0.610847i 0.952217 + 0.305424i $$0.0987981\pi$$
−0.952217 + 0.305424i $$0.901202\pi$$
$$68$$ 12.0000i 1.45521i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 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
$$76$$ 14.0000 1.60591
$$77$$ − 1.00000i − 0.113961i
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 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
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 12.0000i 1.25109i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 17.0000i 1.72609i 0.505128 + 0.863044i $$0.331445\pi$$
−0.505128 + 0.863044i $$0.668555\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 6.00000i 0.580042i 0.957020 + 0.290021i $$0.0936623\pi$$
−0.957020 + 0.290021i $$0.906338\pi$$
$$108$$ 0 0
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 4.00000i − 0.377964i
$$113$$ − 18.0000i − 1.69330i −0.532152 0.846649i $$-0.678617\pi$$
0.532152 0.846649i $$-0.321383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −12.0000 −1.11417
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −14.0000 −1.25724
$$125$$ 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
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ 0 0
$$133$$ − 7.00000i − 0.606977i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1.00000i 0.0836242i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 4.00000i 0.328798i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −7.00000 −0.569652 −0.284826 0.958579i $$-0.591936\pi$$
−0.284826 + 0.958579i $$0.591936\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 23.0000i 1.83560i 0.397043 + 0.917800i $$0.370036\pi$$
−0.397043 + 0.917800i $$0.629964\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ 13.0000i 1.01824i 0.860696 + 0.509119i $$0.170029\pi$$
−0.860696 + 0.509119i $$0.829971\pi$$
$$164$$ 12.0000 0.937043
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2.00000i 0.152499i
$$173$$ − 24.0000i − 1.82469i −0.409426 0.912343i $$-0.634271\pi$$
0.409426 0.912343i $$-0.365729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.00000i 0.438763i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ 7.00000i 0.503871i 0.967744 + 0.251936i $$0.0810671\pi$$
−0.967744 + 0.251936i $$0.918933\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 12.0000 0.857143
$$197$$ − 24.0000i − 1.70993i −0.518686 0.854965i $$-0.673579\pi$$
0.518686 0.854965i $$-0.326421\pi$$
$$198$$ 0 0
$$199$$ −11.0000 −0.779769 −0.389885 0.920864i $$-0.627485\pi$$
−0.389885 + 0.920864i $$0.627485\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 4.00000i 0.277350i
$$209$$ 7.00000 0.484200
$$210$$ 0 0
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ − 12.0000i − 0.824163i
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 7.00000i 0.475191i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ − 5.00000i − 0.334825i −0.985887 0.167412i $$-0.946459\pi$$
0.985887 0.167412i $$-0.0535411\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 24.0000i 1.59294i 0.604681 + 0.796468i $$0.293301\pi$$
−0.604681 + 0.796468i $$0.706699\pi$$
$$228$$ 0 0
$$229$$ −17.0000 −1.12339 −0.561696 0.827344i $$-0.689851\pi$$
−0.561696 + 0.827344i $$0.689851\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 12.0000i 0.786146i 0.919507 + 0.393073i $$0.128588\pi$$
−0.919507 + 0.393073i $$0.871412\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −30.0000 −1.94054 −0.970269 0.242028i $$-0.922188\pi$$
−0.970269 + 0.242028i $$0.922188\pi$$
$$240$$ 0 0
$$241$$ −13.0000 −0.837404 −0.418702 0.908124i $$-0.637515\pi$$
−0.418702 + 0.908124i $$0.637515\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 7.00000i 0.445399i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 0 0
$$253$$ 6.00000i 0.377217i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ − 24.0000i − 1.49708i −0.663090 0.748539i $$-0.730755\pi$$
0.663090 0.748539i $$-0.269245\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 18.0000i − 1.10993i −0.831875 0.554964i $$-0.812732\pi$$
0.831875 0.554964i $$-0.187268\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 10.0000i 0.610847i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 24.0000i 1.45521i
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 13.0000i − 0.781094i −0.920583 0.390547i $$-0.872286\pi$$
0.920583 0.390547i $$-0.127714\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 0 0
$$283$$ − 29.0000i − 1.72387i −0.507018 0.861936i $$-0.669252\pi$$
0.507018 0.861936i $$-0.330748\pi$$
$$284$$ 24.0000 1.42414
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6.00000i − 0.354169i
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 28.0000i − 1.63858i
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 28.0000 1.60591
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5.00000i 0.285365i 0.989769 + 0.142683i $$0.0455728\pi$$
−0.989769 + 0.142683i $$0.954427\pi$$
$$308$$ − 2.00000i − 0.113961i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ 13.0000i 0.734803i 0.930062 + 0.367402i $$0.119753\pi$$
−0.930062 + 0.367402i $$0.880247\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 30.0000i 1.68497i 0.538721 + 0.842484i $$0.318908\pi$$
−0.538721 + 0.842484i $$0.681092\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 42.0000i 2.33694i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 13.0000i − 0.708155i −0.935216 0.354078i $$-0.884795\pi$$
0.935216 0.354078i $$-0.115205\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −7.00000 −0.379071
$$342$$ 0 0
$$343$$ − 13.0000i − 0.701934i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 24.0000i − 1.28839i −0.764862 0.644194i $$-0.777193\pi$$
0.764862 0.644194i $$-0.222807\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 12.0000 0.635999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 37.0000i − 1.93138i −0.259690 0.965692i $$-0.583620\pi$$
0.259690 0.965692i $$-0.416380\pi$$
$$368$$ 24.0000i 1.25109i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 31.0000i 1.60512i 0.596572 + 0.802560i $$0.296529\pi$$
−0.596572 + 0.802560i $$0.703471\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 6.00000i − 0.309016i
$$378$$ 0 0
$$379$$ 19.0000 0.975964 0.487982 0.872854i $$-0.337733\pi$$
0.487982 + 0.872854i $$0.337733\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 6.00000i − 0.306586i −0.988181 0.153293i $$-0.951012\pi$$
0.988181 0.153293i $$-0.0489878\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 34.0000i 1.72609i
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 25.0000i − 1.25471i −0.778732 0.627357i $$-0.784137\pi$$
0.778732 0.627357i $$-0.215863\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ − 7.00000i − 0.348695i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2.00000i 0.0991363i
$$408$$ 0 0
$$409$$ 13.0000 0.642809 0.321404 0.946942i $$-0.395845\pi$$
0.321404 + 0.946942i $$0.395845\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 16.0000i − 0.788263i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 5.00000i − 0.241967i
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 0 0
$$433$$ − 11.0000i − 0.528626i −0.964437 0.264313i $$-0.914855\pi$$
0.964437 0.264313i $$-0.0851452\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −22.0000 −1.05361
$$437$$ 42.0000i 2.00913i
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 30.0000i − 1.42534i −0.701498 0.712672i $$-0.747485\pi$$
0.701498 0.712672i $$-0.252515\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ − 8.00000i − 0.377964i
$$449$$ 24.0000 1.13263 0.566315 0.824189i $$-0.308369\pi$$
0.566315 + 0.824189i $$0.308369\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ − 36.0000i − 1.69330i
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 26.0000i 1.21623i 0.793849 + 0.608114i $$0.208074\pi$$
−0.793849 + 0.608114i $$0.791926\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ −24.0000 −1.11417
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 18.0000i 0.832941i 0.909149 + 0.416470i $$0.136733\pi$$
−0.909149 + 0.416470i $$0.863267\pi$$
$$468$$ 0 0
$$469$$ 5.00000 0.230879
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1.00000i 0.0459800i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 2.00000 0.0909091
$$485$$ 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
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −18.0000 −0.812329 −0.406164 0.913800i $$-0.633134\pi$$
−0.406164 + 0.913800i $$0.633134\pi$$
$$492$$ 0 0
$$493$$ − 36.0000i − 1.62136i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −28.0000 −1.25724
$$497$$ − 12.0000i − 0.538274i
$$498$$ 0 0
$$499$$ 13.0000 0.581960 0.290980 0.956729i $$-0.406019\pi$$
0.290980 + 0.956729i $$0.406019\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 6.00000i 0.267527i 0.991013 + 0.133763i $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 16.0000i 0.709885i
$$509$$ 36.0000 1.59567 0.797836 0.602875i $$-0.205978\pi$$
0.797836 + 0.602875i $$0.205978\pi$$
$$510$$ 0 0
$$511$$ −14.0000 −0.619324
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ − 17.0000i − 0.743358i −0.928361 0.371679i $$-0.878782\pi$$
0.928361 0.371679i $$-0.121218\pi$$
$$524$$ −36.0000 −1.57267
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 42.0000i − 1.82955i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 14.0000i − 0.606977i
$$533$$ 6.00000i 0.259889i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −37.0000 −1.59075 −0.795377 0.606115i $$-0.792727\pi$$
−0.795377 + 0.606115i $$0.792727\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 28.0000i − 1.19719i −0.801050 0.598597i $$-0.795725\pi$$
0.801050 0.598597i $$-0.204275\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −42.0000 −1.78926
$$552$$ 0 0
$$553$$ − 4.00000i − 0.170097i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 12.0000i 0.508456i 0.967144 + 0.254228i $$0.0818214\pi$$
−0.967144 + 0.254228i $$0.918179\pi$$
$$558$$ 0 0
$$559$$ −1.00000 −0.0422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 30.0000i 1.26435i 0.774826 + 0.632175i $$0.217837\pi$$
−0.774826 + 0.632175i $$0.782163\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 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
$$571$$ −31.0000 −1.29731 −0.648655 0.761083i $$-0.724668\pi$$
−0.648655 + 0.761083i $$0.724668\pi$$
$$572$$ 2.00000i 0.0836242i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 1.00000i − 0.0416305i −0.999783 0.0208153i $$-0.993374\pi$$
0.999783 0.0208153i $$-0.00662619\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −6.00000 −0.248922
$$582$$ 0 0
$$583$$ − 6.00000i − 0.248495i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 18.0000i 0.742940i 0.928445 + 0.371470i $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ 0 0
$$589$$ −49.0000 −2.01901
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 8.00000i 0.328798i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −12.0000 −0.491539
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 17.0000 0.693444 0.346722 0.937968i $$-0.387295\pi$$
0.346722 + 0.937968i $$0.387295\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −14.0000 −0.569652
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 40.0000i − 1.62355i −0.583970 0.811775i $$-0.698502\pi$$
0.583970 0.811775i $$-0.301498\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 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
$$619$$ −47.0000 −1.88909 −0.944545 0.328383i $$-0.893496\pi$$
−0.944545 + 0.328383i $$0.893496\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 6.00000i − 0.240385i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 46.0000i 1.83560i
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ −19.0000 −0.756378 −0.378189 0.925728i $$-0.623453\pi$$
−0.378189 + 0.925728i $$0.623453\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.00000i 0.237729i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ − 32.0000i − 1.26196i −0.775800 0.630978i $$-0.782654\pi$$
0.775800 0.630978i $$-0.217346\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000i 0.943537i 0.881722 + 0.471769i $$0.156384\pi$$
−0.881722 + 0.471769i $$0.843616\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 26.0000i 1.01824i
$$653$$ − 24.0000i − 0.939193i −0.882881 0.469596i $$-0.844399\pi$$
0.882881 0.469596i $$-0.155601\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 24.0000 0.937043
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 36.0000i − 1.39393i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 5.00000 0.193023
$$672$$ 0 0
$$673$$ 22.0000i 0.848038i 0.905653 + 0.424019i $$0.139381\pi$$
−0.905653 + 0.424019i $$0.860619\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 24.0000 0.923077
$$677$$ − 30.0000i − 1.15299i −0.817099 0.576497i $$-0.804419\pi$$
0.817099 0.576497i $$-0.195581\pi$$
$$678$$ 0 0
$$679$$ 17.0000 0.652400
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 30.0000i 1.14792i 0.818884 + 0.573959i $$0.194593\pi$$
−0.818884 + 0.573959i $$0.805407\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000i 0.152499i
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ − 48.0000i − 1.82469i
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000i 1.36360i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 36.0000 1.35970 0.679851 0.733351i $$-0.262045\pi$$
0.679851 + 0.733351i $$0.262045\pi$$
$$702$$ 0 0
$$703$$ 14.0000i 0.528020i
$$704$$ 8.00000 0.301511
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −17.0000 −0.638448 −0.319224 0.947679i $$-0.603422\pi$$
−0.319224 + 0.947679i $$0.603422\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 42.0000i − 1.57291i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 36.0000 1.34538
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 5.00000i 0.185440i 0.995692 + 0.0927199i $$0.0295561\pi$$
−0.995692 + 0.0927199i $$0.970444\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −6.00000 −0.221918
$$732$$ 0 0
$$733$$ − 14.0000i − 0.517102i −0.965998 0.258551i $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 5.00000i 0.184177i
$$738$$ 0 0
$$739$$ −32.0000 −1.17714 −0.588570 0.808447i $$-0.700309\pi$$
−0.588570 + 0.808447i $$0.700309\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 6.00000i − 0.220119i −0.993925 0.110059i $$-0.964896\pi$$
0.993925 0.110059i $$-0.0351041\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 12.0000i 0.438763i
$$749$$ 6.00000 0.219235
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 29.0000i 1.05402i 0.849858 + 0.527011i $$0.176688\pi$$
−0.849858 + 0.527011i $$0.823312\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 11.0000i 0.398227i
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −29.0000 −1.04577 −0.522883 0.852404i $$-0.675144\pi$$
−0.522883 + 0.852404i $$0.675144\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 14.0000i 0.503871i
$$773$$ − 18.0000i − 0.647415i −0.946157 0.323708i $$-0.895071\pi$$
0.946157 0.323708i $$-0.104929\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 42.0000 1.50481
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 24.0000 0.857143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 55.0000i − 1.96054i −0.197667 0.980269i $$-0.563337\pi$$
0.197667 0.980269i $$-0.436663\pi$$
$$788$$ − 48.0000i − 1.70993i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 0 0
$$793$$ 5.00000i 0.177555i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −22.0000 −0.779769
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 14.0000i − 0.494049i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 12.0000 0.421898 0.210949 0.977497i $$-0.432345\pi$$
0.210949 + 0.977497i $$0.432345\pi$$
$$810$$ 0 0
$$811$$ −7.00000 −0.245803 −0.122902 0.992419i $$-0.539220\pi$$
−0.122902 + 0.992419i $$0.539220\pi$$
$$812$$ 12.0000i 0.421117i
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.00000i 0.244899i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ 31.0000i 1.08059i 0.841475 + 0.540296i $$0.181688\pi$$
−0.841475 + 0.540296i $$0.818312\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 42.0000i − 1.46048i −0.683189 0.730242i $$-0.739408\pi$$
0.683189 0.730242i $$-0.260592\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 8.00000i 0.277350i
$$833$$ 36.0000i 1.24733i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 14.0000 0.484200
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 10.0000 0.344214
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 1.00000i − 0.0343604i
$$848$$ − 24.0000i − 0.824163i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ 19.0000i 0.650548i 0.945620 + 0.325274i $$0.105456\pi$$
−0.945620 + 0.325274i $$0.894544\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 30.0000i − 1.02478i −0.858753 0.512390i $$-0.828760\pi$$
0.858753 0.512390i $$-0.171240\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 12.0000i − 0.408485i −0.978920 0.204242i $$-0.934527\pi$$
0.978920 0.204242i $$-0.0654731\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 14.0000i 0.475191i
$$869$$ 4.00000 0.135691
$$870$$ 0 0
$$871$$ −5.00000 −0.169419
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 53.0000i 1.78968i 0.446384 + 0.894841i $$0.352711\pi$$
−0.446384 + 0.894841i $$0.647289\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ 7.00000i 0.235569i 0.993039 + 0.117784i $$0.0375792\pi$$
−0.993039 + 0.117784i $$0.962421\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 6.00000i 0.201460i 0.994914 + 0.100730i $$0.0321179\pi$$
−0.994914 + 0.100730i $$0.967882\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 10.0000i − 0.334825i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 42.0000 1.40078
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 20.0000i 0.664089i 0.943264 + 0.332045i $$0.107738\pi$$
−0.943264 + 0.332045i $$0.892262\pi$$
$$908$$ 48.0000i 1.59294i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 30.0000 0.993944 0.496972 0.867766i $$-0.334445\pi$$
0.496972 + 0.867766i $$0.334445\pi$$
$$912$$ 0 0
$$913$$ − 6.00000i − 0.198571i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −34.0000 −1.12339
$$917$$ 18.0000i 0.594412i
$$918$$ 0 0
$$919$$ 43.0000 1.41844 0.709220 0.704988i $$-0.249047\pi$$
0.709220 + 0.704988i $$0.249047\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 12.0000i 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −24.0000 −0.787414 −0.393707 0.919236i $$-0.628808\pi$$
−0.393707 + 0.919236i $$0.628808\pi$$
$$930$$ 0 0
$$931$$ 42.0000 1.37649
$$932$$ 24.0000i 0.786146i
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 7.00000i − 0.228680i −0.993442 0.114340i $$-0.963525\pi$$
0.993442 0.114340i $$-0.0364753\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ 0 0
$$943$$ 36.0000i 1.17232i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 6.00000i 0.194974i 0.995237 + 0.0974869i $$0.0310804\pi$$
−0.995237 + 0.0974869i $$0.968920\pi$$
$$948$$ 0 0
$$949$$ 14.0000 0.454459
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 42.0000i 1.36051i 0.732974 + 0.680257i $$0.238132\pi$$
−0.732974 + 0.680257i $$0.761868\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −60.0000 −1.94054
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 52.0000i − 1.67221i −0.548572 0.836104i $$-0.684828\pi$$
0.548572 0.836104i $$-0.315172\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 6.00000 0.192549 0.0962746 0.995355i $$-0.469307\pi$$
0.0962746 + 0.995355i $$0.469307\pi$$
$$972$$ 0 0
$$973$$ − 4.00000i − 0.128234i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 20.0000 0.640184
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 0 0
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 14.0000i 0.445399i
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ 5.00000 0.158830 0.0794151 0.996842i $$-0.474695\pi$$
0.0794151 + 0.996842i $$0.474695\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 10.0000i − 0.316703i −0.987383 0.158352i $$-0.949382\pi$$
0.987383 0.158352i $$-0.0506179\pi$$
$$998$$ 0 0
$$999$$ 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 2475.2.c.h.199.1 2
3.2 odd 2 825.2.c.b.199.1 2
5.2 odd 4 2475.2.a.f.1.1 1
5.3 odd 4 2475.2.a.e.1.1 1
5.4 even 2 inner 2475.2.c.h.199.2 2
15.2 even 4 825.2.a.b.1.1 1
15.8 even 4 825.2.a.c.1.1 yes 1
15.14 odd 2 825.2.c.b.199.2 2
165.32 odd 4 9075.2.a.i.1.1 1
165.98 odd 4 9075.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.b.1.1 1 15.2 even 4
825.2.a.c.1.1 yes 1 15.8 even 4
825.2.c.b.199.1 2 3.2 odd 2
825.2.c.b.199.2 2 15.14 odd 2
2475.2.a.e.1.1 1 5.3 odd 4
2475.2.a.f.1.1 1 5.2 odd 4
2475.2.c.h.199.1 2 1.1 even 1 trivial
2475.2.c.h.199.2 2 5.4 even 2 inner
9075.2.a.i.1.1 1 165.32 odd 4
9075.2.a.l.1.1 1 165.98 odd 4