# Properties

 Label 450.2.a.f.1.1 Level $450$ Weight $2$ Character 450.1 Self dual yes Analytic conductor $3.593$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{11} +6.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -2.00000 q^{22} +4.00000 q^{23} +6.00000 q^{26} +2.00000 q^{28} -8.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{37} -2.00000 q^{41} -4.00000 q^{43} -2.00000 q^{44} +4.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} +6.00000 q^{52} -6.00000 q^{53} +2.00000 q^{56} -10.0000 q^{59} +2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -8.00000 q^{67} -2.00000 q^{68} -12.0000 q^{71} -4.00000 q^{73} +2.00000 q^{74} -4.00000 q^{77} -2.00000 q^{82} +4.00000 q^{83} -4.00000 q^{86} -2.00000 q^{88} +10.0000 q^{89} +12.0000 q^{91} +4.00000 q^{92} +8.00000 q^{94} -8.00000 q^{97} -3.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 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
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 6.00000 0.832050
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −2.00000 −0.220863
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ −2.00000 −0.213201
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ 0 0
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.00000 0.188982
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ −10.0000 −0.920575
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 2.00000 0.181071
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 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
$$142$$ −12.0000 −1.00702
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ 20.0000 1.63846 0.819232 0.573462i $$-0.194400\pi$$
0.819232 + 0.573462i $$0.194400\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 12.0000 0.889499
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 8.00000 0.562878
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 0 0
$$208$$ 6.00000 0.416025
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 10.0000 0.677285
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ 26.0000 1.74109 0.870544 0.492090i $$-0.163767\pi$$
0.870544 + 0.492090i $$0.163767\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 28.0000 1.85843 0.929213 0.369546i $$-0.120487\pi$$
0.929213 + 0.369546i $$0.120487\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −10.0000 −0.650945
$$237$$ 0 0
$$238$$ −4.00000 −0.259281
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −8.00000 −0.508001
$$249$$ 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
$$253$$ −8.00000 −0.502956
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 18.0000 1.11204
$$263$$ 4.00000 0.246651 0.123325 0.992366i $$-0.460644\pi$$
0.123325 + 0.992366i $$0.460644\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −8.00000 −0.488678
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 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
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ −4.00000 −0.236113
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ 20.0000 1.15857
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −4.00000 −0.226093 −0.113047 0.993590i $$-0.536061\pi$$
−0.113047 + 0.993590i $$0.536061\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 8.00000 0.445823
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ −2.00000 −0.110432
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −28.0000 −1.52526 −0.762629 0.646837i $$-0.776092\pi$$
−0.762629 + 0.646837i $$0.776092\pi$$
$$338$$ 23.0000 1.25104
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 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
$$352$$ −2.00000 −0.106600
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ −10.0000 −0.528516
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 2.00000 0.105118
$$363$$ 0 0
$$364$$ 12.0000 0.628971
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 2.00000 0.104399 0.0521996 0.998637i $$-0.483377\pi$$
0.0521996 + 0.998637i $$0.483377\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −12.0000 −0.613973
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −4.00000 −0.203595
$$387$$ 0 0
$$388$$ −8.00000 −0.406138
$$389$$ −20.0000 −1.01404 −0.507020 0.861934i $$-0.669253\pi$$
−0.507020 + 0.861934i $$0.669253\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ −3.00000 −0.151523
$$393$$ 0 0
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 0 0
$$403$$ −48.0000 −2.39105
$$404$$ 8.00000 0.398015
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.00000 −0.198273
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −14.0000 −0.689730
$$413$$ −20.0000 −0.984136
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 10.0000 0.488532 0.244266 0.969708i $$-0.421453\pi$$
0.244266 + 0.969708i $$0.421453\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000 0.193574
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ 0 0
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −12.0000 −0.570782
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 26.0000 1.23114
$$447$$ 0 0
$$448$$ 2.00000 0.0944911
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 32.0000 1.49690 0.748448 0.663193i $$-0.230799\pi$$
0.748448 + 0.663193i $$0.230799\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 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
$$463$$ 6.00000 0.278844 0.139422 0.990233i $$-0.455476\pi$$
0.139422 + 0.990233i $$0.455476\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 14.0000 0.648537
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −10.0000 −0.460287
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ 0 0
$$478$$ −20.0000 −0.914779
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ 22.0000 1.00207
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −18.0000 −0.815658 −0.407829 0.913058i $$-0.633714\pi$$
−0.407829 + 0.913058i $$0.633714\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 18.0000 0.812329 0.406164 0.913800i $$-0.366866\pi$$
0.406164 + 0.913800i $$0.366866\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ −24.0000 −1.07655
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 18.0000 0.803379
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −8.00000 −0.355643
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −16.0000 −0.703679
$$518$$ 4.00000 0.175750
$$519$$ 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
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ 4.00000 0.174408
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −8.00000 −0.345547
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −38.0000 −1.63375 −0.816874 0.576816i $$-0.804295\pi$$
−0.816874 + 0.576816i $$0.804295\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 18.0000 0.759284
$$563$$ 44.0000 1.85438 0.927189 0.374593i $$-0.122217\pi$$
0.927189 + 0.374593i $$0.122217\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ 0 0
$$568$$ −12.0000 −0.503509
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −8.00000 −0.334790 −0.167395 0.985890i $$-0.553535\pi$$
−0.167395 + 0.985890i $$0.553535\pi$$
$$572$$ −12.0000 −0.501745
$$573$$ 0 0
$$574$$ −4.00000 −0.166957
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 32.0000 1.33218 0.666089 0.745873i $$-0.267967\pi$$
0.666089 + 0.745873i $$0.267967\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 8.00000 0.331896
$$582$$ 0 0
$$583$$ 12.0000 0.496989
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 20.0000 0.819232
$$597$$ 0 0
$$598$$ 24.0000 0.981433
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 22.0000 0.892952 0.446476 0.894795i $$-0.352679\pi$$
0.446476 + 0.894795i $$0.352679\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ 0 0
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −12.0000 −0.481156
$$623$$ 20.0000 0.801283
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −4.00000 −0.159872
$$627$$ 0 0
$$628$$ 22.0000 0.877896
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −18.0000 −0.713186
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ −24.0000 −0.946468 −0.473234 0.880937i $$-0.656913\pi$$
−0.473234 + 0.880937i $$0.656913\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 0 0
$$649$$ 20.0000 0.785069
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16.0000 0.626608
$$653$$ −26.0000 −1.01746 −0.508729 0.860927i $$-0.669885\pi$$
−0.508729 + 0.860927i $$0.669885\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 0 0
$$658$$ 16.0000 0.623745
$$659$$ 50.0000 1.94772 0.973862 0.227142i $$-0.0729380\pi$$
0.973862 + 0.227142i $$0.0729380\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ 36.0000 1.38770 0.693849 0.720121i $$-0.255914\pi$$
0.693849 + 0.720121i $$0.255914\pi$$
$$674$$ −28.0000 −1.07852
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ −2.00000 −0.0768662 −0.0384331 0.999261i $$-0.512237\pi$$
−0.0384331 + 0.999261i $$0.512237\pi$$
$$678$$ 0 0
$$679$$ −16.0000 −0.614024
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 16.0000 0.612672
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 14.0000 0.532200
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 10.0000 0.378506
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −32.0000 −1.20862 −0.604312 0.796748i $$-0.706552\pi$$
−0.604312 + 0.796748i $$0.706552\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ 16.0000 0.601742
$$708$$ 0 0
$$709$$ 30.0000 1.12667 0.563337 0.826227i $$-0.309517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ −32.0000 −1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ 0 0
$$721$$ −28.0000 −1.04277
$$722$$ −19.0000 −0.707107
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −18.0000 −0.667583 −0.333792 0.942647i $$-0.608328\pi$$
−0.333792 + 0.942647i $$0.608328\pi$$
$$728$$ 12.0000 0.444750
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 6.00000 0.219676
$$747$$ 0 0
$$748$$ 4.00000 0.146254
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ 0 0
$$763$$ 20.0000 0.724049
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ −60.0000 −2.16647
$$768$$ 0 0
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ 54.0000 1.94225 0.971123 0.238581i $$-0.0766824\pi$$
0.971123 + 0.238581i $$0.0766824\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −8.00000 −0.287183
$$777$$ 0 0
$$778$$ −20.0000 −0.717035
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ −8.00000 −0.286079
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 12.0000 0.426132
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −22.0000 −0.776847
$$803$$ 8.00000 0.282314
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −48.0000 −1.69073
$$807$$ 0 0
$$808$$ 8.00000 0.281439
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 52.0000 1.82597 0.912983 0.407997i $$-0.133772\pi$$
0.912983 + 0.407997i $$0.133772\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −4.00000 −0.140200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −10.0000 −0.349642
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8.00000 0.279202 0.139601 0.990208i $$-0.455418\pi$$
0.139601 + 0.990208i $$0.455418\pi$$
$$822$$ 0 0
$$823$$ 6.00000 0.209147 0.104573 0.994517i $$-0.466652\pi$$
0.104573 + 0.994517i $$0.466652\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ −20.0000 −0.695889
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 0 0
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 6.00000 0.208013
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 10.0000 0.345444
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 22.0000 0.758170
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −14.0000 −0.481046
$$848$$ −6.00000 −0.206041
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 4.00000 0.136877
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −32.0000 −1.08992
$$863$$ −36.0000 −1.22545 −0.612727 0.790295i $$-0.709928\pi$$
−0.612727 + 0.790295i $$0.709928\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −4.00000 −0.135926
$$867$$ 0 0
$$868$$ −16.0000 −0.543075
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ 10.0000 0.338643
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ −24.0000 −0.807664 −0.403832 0.914833i $$-0.632322\pi$$
−0.403832 + 0.914833i $$0.632322\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 26.0000 0.870544
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 4.00000 0.133185
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ 28.0000 0.929213
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ 32.0000 1.05847
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 36.0000 1.18882
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −12.0000 −0.395199
$$923$$ −72.0000 −2.36991
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 6.00000 0.197172
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 14.0000 0.458585
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −8.00000 −0.261349 −0.130674 0.991425i $$-0.541714\pi$$
−0.130674 + 0.991425i $$0.541714\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 28.0000 0.912774 0.456387 0.889781i $$-0.349143\pi$$
0.456387 + 0.889781i $$0.349143\pi$$
$$942$$ 0 0
$$943$$ −8.00000 −0.260516
$$944$$ −10.0000 −0.325472
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ −24.0000 −0.779073
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −4.00000 −0.129641
$$953$$ −46.0000 −1.49009 −0.745043 0.667016i $$-0.767571\pi$$
−0.745043 + 0.667016i $$0.767571\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −20.0000 −0.646846
$$957$$ 0 0
$$958$$ −20.0000 −0.646171
$$959$$ 36.0000 1.16250
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 12.0000 0.386896
$$963$$ 0 0
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −38.0000 −1.22200 −0.610999 0.791632i $$-0.709232\pi$$
−0.610999 + 0.791632i $$0.709232\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ 0 0
$$973$$ −40.0000 −1.28234
$$974$$ −18.0000 −0.576757
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ −20.0000 −0.639203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 18.0000 0.574403
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ 0 0
$$994$$ −24.0000 −0.761234
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −18.0000 −0.570066 −0.285033 0.958518i $$-0.592005\pi$$
−0.285033 + 0.958518i $$0.592005\pi$$
$$998$$ 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 450.2.a.f.1.1 1
3.2 odd 2 150.2.a.a.1.1 1
4.3 odd 2 3600.2.a.o.1.1 1
5.2 odd 4 90.2.c.a.19.2 2
5.3 odd 4 90.2.c.a.19.1 2
5.4 even 2 450.2.a.b.1.1 1
12.11 even 2 1200.2.a.m.1.1 1
15.2 even 4 30.2.c.a.19.1 2
15.8 even 4 30.2.c.a.19.2 yes 2
15.14 odd 2 150.2.a.c.1.1 1
20.3 even 4 720.2.f.f.289.1 2
20.7 even 4 720.2.f.f.289.2 2
20.19 odd 2 3600.2.a.bg.1.1 1
21.20 even 2 7350.2.a.bg.1.1 1
24.5 odd 2 4800.2.a.cg.1.1 1
24.11 even 2 4800.2.a.m.1.1 1
40.3 even 4 2880.2.f.c.1729.2 2
40.13 odd 4 2880.2.f.e.1729.2 2
40.27 even 4 2880.2.f.c.1729.1 2
40.37 odd 4 2880.2.f.e.1729.1 2
45.2 even 12 810.2.i.e.109.1 4
45.7 odd 12 810.2.i.b.109.2 4
45.13 odd 12 810.2.i.b.379.2 4
45.22 odd 12 810.2.i.b.379.1 4
45.23 even 12 810.2.i.e.379.1 4
45.32 even 12 810.2.i.e.379.2 4
45.38 even 12 810.2.i.e.109.2 4
45.43 odd 12 810.2.i.b.109.1 4
60.23 odd 4 240.2.f.a.49.2 2
60.47 odd 4 240.2.f.a.49.1 2
60.59 even 2 1200.2.a.g.1.1 1
105.2 even 12 1470.2.n.h.949.1 4
105.17 odd 12 1470.2.n.a.79.2 4
105.23 even 12 1470.2.n.h.949.2 4
105.32 even 12 1470.2.n.h.79.2 4
105.38 odd 12 1470.2.n.a.79.1 4
105.47 odd 12 1470.2.n.a.949.1 4
105.53 even 12 1470.2.n.h.79.1 4
105.62 odd 4 1470.2.g.g.589.1 2
105.68 odd 12 1470.2.n.a.949.2 4
105.83 odd 4 1470.2.g.g.589.2 2
105.104 even 2 7350.2.a.cc.1.1 1
120.29 odd 2 4800.2.a.l.1.1 1
120.53 even 4 960.2.f.h.769.2 2
120.59 even 2 4800.2.a.cj.1.1 1
120.77 even 4 960.2.f.h.769.1 2
120.83 odd 4 960.2.f.i.769.1 2
120.107 odd 4 960.2.f.i.769.2 2
240.53 even 4 3840.2.d.g.2689.1 2
240.77 even 4 3840.2.d.g.2689.2 2
240.83 odd 4 3840.2.d.j.2689.2 2
240.107 odd 4 3840.2.d.j.2689.1 2
240.173 even 4 3840.2.d.y.2689.2 2
240.197 even 4 3840.2.d.y.2689.1 2
240.203 odd 4 3840.2.d.x.2689.1 2
240.227 odd 4 3840.2.d.x.2689.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.c.a.19.1 2 15.2 even 4
30.2.c.a.19.2 yes 2 15.8 even 4
90.2.c.a.19.1 2 5.3 odd 4
90.2.c.a.19.2 2 5.2 odd 4
150.2.a.a.1.1 1 3.2 odd 2
150.2.a.c.1.1 1 15.14 odd 2
240.2.f.a.49.1 2 60.47 odd 4
240.2.f.a.49.2 2 60.23 odd 4
450.2.a.b.1.1 1 5.4 even 2
450.2.a.f.1.1 1 1.1 even 1 trivial
720.2.f.f.289.1 2 20.3 even 4
720.2.f.f.289.2 2 20.7 even 4
810.2.i.b.109.1 4 45.43 odd 12
810.2.i.b.109.2 4 45.7 odd 12
810.2.i.b.379.1 4 45.22 odd 12
810.2.i.b.379.2 4 45.13 odd 12
810.2.i.e.109.1 4 45.2 even 12
810.2.i.e.109.2 4 45.38 even 12
810.2.i.e.379.1 4 45.23 even 12
810.2.i.e.379.2 4 45.32 even 12
960.2.f.h.769.1 2 120.77 even 4
960.2.f.h.769.2 2 120.53 even 4
960.2.f.i.769.1 2 120.83 odd 4
960.2.f.i.769.2 2 120.107 odd 4
1200.2.a.g.1.1 1 60.59 even 2
1200.2.a.m.1.1 1 12.11 even 2
1470.2.g.g.589.1 2 105.62 odd 4
1470.2.g.g.589.2 2 105.83 odd 4
1470.2.n.a.79.1 4 105.38 odd 12
1470.2.n.a.79.2 4 105.17 odd 12
1470.2.n.a.949.1 4 105.47 odd 12
1470.2.n.a.949.2 4 105.68 odd 12
1470.2.n.h.79.1 4 105.53 even 12
1470.2.n.h.79.2 4 105.32 even 12
1470.2.n.h.949.1 4 105.2 even 12
1470.2.n.h.949.2 4 105.23 even 12
2880.2.f.c.1729.1 2 40.27 even 4
2880.2.f.c.1729.2 2 40.3 even 4
2880.2.f.e.1729.1 2 40.37 odd 4
2880.2.f.e.1729.2 2 40.13 odd 4
3600.2.a.o.1.1 1 4.3 odd 2
3600.2.a.bg.1.1 1 20.19 odd 2
3840.2.d.g.2689.1 2 240.53 even 4
3840.2.d.g.2689.2 2 240.77 even 4
3840.2.d.j.2689.1 2 240.107 odd 4
3840.2.d.j.2689.2 2 240.83 odd 4
3840.2.d.x.2689.1 2 240.203 odd 4
3840.2.d.x.2689.2 2 240.227 odd 4
3840.2.d.y.2689.1 2 240.197 even 4
3840.2.d.y.2689.2 2 240.173 even 4
4800.2.a.l.1.1 1 120.29 odd 2
4800.2.a.m.1.1 1 24.11 even 2
4800.2.a.cg.1.1 1 24.5 odd 2
4800.2.a.cj.1.1 1 120.59 even 2
7350.2.a.bg.1.1 1 21.20 even 2
7350.2.a.cc.1.1 1 105.104 even 2