# Properties

 Label 1350.2.c.b.649.1 Level $1350$ Weight $2$ Character 1350.649 Analytic conductor $10.780$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1350.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1350.649 Dual form 1350.2.c.b.649.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{11} +4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} +3.00000i q^{22} +6.00000i q^{23} +4.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +2.00000i q^{37} +2.00000i q^{38} -6.00000 q^{41} +10.0000i q^{43} +3.00000 q^{44} +6.00000 q^{46} +6.00000i q^{47} +6.00000 q^{49} -4.00000i q^{52} -9.00000i q^{53} +1.00000 q^{56} +6.00000i q^{58} -12.0000 q^{59} +8.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +14.0000i q^{67} +7.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} +3.00000i q^{77} -8.00000 q^{79} +6.00000i q^{82} +3.00000i q^{83} +10.0000 q^{86} -3.00000i q^{88} +18.0000 q^{89} +4.00000 q^{91} -6.00000i q^{92} +6.00000 q^{94} -1.00000i q^{97} -6.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + O(q^{10})$$ $$2 q - 2 q^{4} - 6 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{19} + 8 q^{26} - 12 q^{29} + 10 q^{31} - 12 q^{41} + 6 q^{44} + 12 q^{46} + 12 q^{49} + 2 q^{56} - 24 q^{59} + 16 q^{61} - 2 q^{64} + 4 q^{74} + 4 q^{76} - 16 q^{79} + 20 q^{86} + 36 q^{89} + 8 q^{91} + 12 q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i −0.981981 0.188982i $$-0.939481\pi$$
0.981981 0.188982i $$-0.0605189\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.00000i 0.639602i
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ 0 0
$$28$$ 1.00000i 0.188982i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 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 $$0.947432\pi$$
$$38$$ 2.00000i 0.324443i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 10.0000i 1.52499i 0.646997 + 0.762493i $$0.276025\pi$$
−0.646997 + 0.762493i $$0.723975\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 6.00000i 0.875190i 0.899172 + 0.437595i $$0.144170\pi$$
−0.899172 + 0.437595i $$0.855830\pi$$
$$48$$ 0 0
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 4.00000i − 0.554700i
$$53$$ − 9.00000i − 1.23625i −0.786082 0.618123i $$-0.787894\pi$$
0.786082 0.618123i $$-0.212106\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 6.00000i 0.787839i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ − 5.00000i − 0.635001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 14.0000i 1.71037i 0.518321 + 0.855186i $$0.326557\pi$$
−0.518321 + 0.855186i $$0.673443\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 7.00000i 0.819288i 0.912245 + 0.409644i $$0.134347\pi$$
−0.912245 + 0.409644i $$0.865653\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 3.00000i 0.341882i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 6.00000i 0.662589i
$$83$$ 3.00000i 0.329293i 0.986353 + 0.164646i $$0.0526483\pi$$
−0.986353 + 0.164646i $$0.947352\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 10.0000 1.07833
$$87$$ 0 0
$$88$$ − 3.00000i − 0.319801i
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ − 6.00000i − 0.625543i
$$93$$ 0 0
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 1.00000i − 0.101535i −0.998711 0.0507673i $$-0.983833\pi$$
0.998711 0.0507673i $$-0.0161667\pi$$
$$98$$ − 6.00000i − 0.606092i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ −9.00000 −0.874157
$$107$$ 9.00000i 0.870063i 0.900415 + 0.435031i $$0.143263\pi$$
−0.900415 + 0.435031i $$0.856737\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 1.00000i − 0.0944911i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ 12.0000i 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ − 8.00000i − 0.724286i
$$123$$ 0 0
$$124$$ −5.00000 −0.449013
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 7.00000i − 0.621150i −0.950549 0.310575i $$-0.899478\pi$$
0.950549 0.310575i $$-0.100522\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ 0 0
$$133$$ 2.00000i 0.173422i
$$134$$ 14.0000 1.20942
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 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
$$143$$ − 12.0000i − 1.00349i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 7.00000 0.579324
$$147$$ 0 0
$$148$$ − 2.00000i − 0.164399i
$$149$$ −3.00000 −0.245770 −0.122885 0.992421i $$-0.539215\pi$$
−0.122885 + 0.992421i $$0.539215\pi$$
$$150$$ 0 0
$$151$$ 17.0000 1.38344 0.691720 0.722166i $$-0.256853\pi$$
0.691720 + 0.722166i $$0.256853\pi$$
$$152$$ − 2.00000i − 0.162221i
$$153$$ 0 0
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 4.00000i − 0.319235i −0.987179 0.159617i $$-0.948974\pi$$
0.987179 0.159617i $$-0.0510260\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 3.00000 0.232845
$$167$$ − 6.00000i − 0.464294i −0.972681 0.232147i $$-0.925425\pi$$
0.972681 0.232147i $$-0.0745750\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 10.0000i − 0.762493i
$$173$$ − 15.0000i − 1.14043i −0.821496 0.570214i $$-0.806860\pi$$
0.821496 0.570214i $$-0.193140\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ − 18.0000i − 1.34916i
$$179$$ 9.00000 0.672692 0.336346 0.941739i $$-0.390809\pi$$
0.336346 + 0.941739i $$0.390809\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ − 4.00000i − 0.296500i
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ − 6.00000i − 0.437595i
$$189$$ 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
$$193$$ − 5.00000i − 0.359908i −0.983675 0.179954i $$-0.942405\pi$$
0.983675 0.179954i $$-0.0575949\pi$$
$$194$$ −1.00000 −0.0717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 9.00000i 0.641223i 0.947211 + 0.320612i $$0.103888\pi$$
−0.947211 + 0.320612i $$0.896112\pi$$
$$198$$ 0 0
$$199$$ 7.00000 0.496217 0.248108 0.968732i $$-0.420191\pi$$
0.248108 + 0.968732i $$0.420191\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 3.00000i 0.211079i
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ 4.00000i 0.277350i
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −22.0000 −1.51454 −0.757271 0.653101i $$-0.773468\pi$$
−0.757271 + 0.653101i $$0.773468\pi$$
$$212$$ 9.00000i 0.618123i
$$213$$ 0 0
$$214$$ 9.00000 0.615227
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 5.00000i − 0.339422i
$$218$$ 2.00000i 0.135457i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 18.0000i − 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 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
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 2.00000i 0.128565i
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 8.00000i − 0.509028i
$$248$$ 5.00000i 0.317500i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ − 18.0000i − 1.13165i
$$254$$ −7.00000 −0.439219
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 15.0000i 0.926703i
$$263$$ 30.0000i 1.84988i 0.380114 + 0.924940i $$0.375885\pi$$
−0.380114 + 0.924940i $$0.624115\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ − 14.0000i − 0.855186i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ −25.0000 −1.51864 −0.759321 0.650716i $$-0.774469\pi$$
−0.759321 + 0.650716i $$0.774469\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8.00000i 0.480673i 0.970690 + 0.240337i $$0.0772579\pi$$
−0.970690 + 0.240337i $$0.922742\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 24.0000 1.43172 0.715860 0.698244i $$-0.246035\pi$$
0.715860 + 0.698244i $$0.246035\pi$$
$$282$$ 0 0
$$283$$ − 14.0000i − 0.832214i −0.909316 0.416107i $$-0.863394\pi$$
0.909316 0.416107i $$-0.136606\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ 6.00000i 0.354169i
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 7.00000i − 0.409644i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 3.00000i 0.173785i
$$299$$ −24.0000 −1.38796
$$300$$ 0 0
$$301$$ 10.0000 0.576390
$$302$$ − 17.0000i − 0.978240i
$$303$$ 0 0
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 16.0000i − 0.913168i −0.889680 0.456584i $$-0.849073\pi$$
0.889680 0.456584i $$-0.150927\pi$$
$$308$$ − 3.00000i − 0.170941i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 19.0000i 1.07394i 0.843600 + 0.536972i $$0.180432\pi$$
−0.843600 + 0.536972i $$0.819568\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ − 3.00000i − 0.168497i −0.996445 0.0842484i $$-0.973151\pi$$
0.996445 0.0842484i $$-0.0268489\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 6.00000i − 0.334367i
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ 0 0
$$328$$ − 6.00000i − 0.331295i
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ − 3.00000i − 0.164646i
$$333$$ 0 0
$$334$$ −6.00000 −0.328305
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 22.0000i − 1.19842i −0.800593 0.599208i $$-0.795482\pi$$
0.800593 0.599208i $$-0.204518\pi$$
$$338$$ 3.00000i 0.163178i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −15.0000 −0.812296
$$342$$ 0 0
$$343$$ − 13.0000i − 0.701934i
$$344$$ −10.0000 −0.539164
$$345$$ 0 0
$$346$$ −15.0000 −0.806405
$$347$$ 3.00000i 0.161048i 0.996753 + 0.0805242i $$0.0256594\pi$$
−0.996753 + 0.0805242i $$0.974341\pi$$
$$348$$ 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
$$352$$ 3.00000i 0.159901i
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ − 9.00000i − 0.475665i
$$359$$ −18.0000 −0.950004 −0.475002 0.879985i $$-0.657553\pi$$
−0.475002 + 0.879985i $$0.657553\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 16.0000i 0.840941i
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 17.0000i 0.887393i 0.896177 + 0.443696i $$0.146333\pi$$
−0.896177 + 0.443696i $$0.853667\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −9.00000 −0.467257
$$372$$ 0 0
$$373$$ − 32.0000i − 1.65690i −0.560065 0.828449i $$-0.689224\pi$$
0.560065 0.828449i $$-0.310776\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ − 24.0000i − 1.23606i
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 12.0000i 0.613973i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −5.00000 −0.254493
$$387$$ 0 0
$$388$$ 1.00000i 0.0507673i
$$389$$ 21.0000 1.06474 0.532371 0.846511i $$-0.321301\pi$$
0.532371 + 0.846511i $$0.321301\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 6.00000i 0.303046i
$$393$$ 0 0
$$394$$ 9.00000 0.453413
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 20.0000i 1.00377i 0.864934 + 0.501886i $$0.167360\pi$$
−0.864934 + 0.501886i $$0.832640\pi$$
$$398$$ − 7.00000i − 0.350878i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 0 0
$$403$$ 20.0000i 0.996271i
$$404$$ 3.00000 0.149256
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ − 6.00000i − 0.297409i
$$408$$ 0 0
$$409$$ −23.0000 −1.13728 −0.568638 0.822588i $$-0.692530\pi$$
−0.568638 + 0.822588i $$0.692530\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 4.00000i − 0.197066i
$$413$$ 12.0000i 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 0 0
$$418$$ − 6.00000i − 0.293470i
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ 22.0000i 1.07094i
$$423$$ 0 0
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 8.00000i − 0.387147i
$$428$$ − 9.00000i − 0.435031i
$$429$$ 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 $$-0.245405\pi$$
$$434$$ −5.00000 −0.240008
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ − 12.0000i − 0.574038i
$$438$$ 0 0
$$439$$ 19.0000 0.906821 0.453410 0.891302i $$-0.350207\pi$$
0.453410 + 0.891302i $$0.350207\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 0 0
$$448$$ 1.00000i 0.0472456i
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 18.0000 0.847587
$$452$$ − 6.00000i − 0.282216i
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 1.00000i − 0.0467780i −0.999726 0.0233890i $$-0.992554\pi$$
0.999726 0.0233890i $$-0.00744563\pi$$
$$458$$ 14.0000i 0.654177i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −21.0000 −0.978068 −0.489034 0.872265i $$-0.662651\pi$$
−0.489034 + 0.872265i $$0.662651\pi$$
$$462$$ 0 0
$$463$$ 13.0000i 0.604161i 0.953282 + 0.302081i $$0.0976812\pi$$
−0.953282 + 0.302081i $$0.902319\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ − 27.0000i − 1.24941i −0.780860 0.624705i $$-0.785219\pi$$
0.780860 0.624705i $$-0.214781\pi$$
$$468$$ 0 0
$$469$$ 14.0000 0.646460
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 12.0000i − 0.552345i
$$473$$ − 30.0000i − 1.37940i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 30.0000i 1.37217i
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 10.0000i 0.455488i
$$483$$ 0 0
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 8.00000i 0.362143i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 39.0000 1.76005 0.880023 0.474932i $$-0.157527\pi$$
0.880023 + 0.474932i $$0.157527\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 5.00000 0.224507
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 18.0000i 0.802580i 0.915951 + 0.401290i $$0.131438\pi$$
−0.915951 + 0.401290i $$0.868562\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −18.0000 −0.800198
$$507$$ 0 0
$$508$$ 7.00000i 0.310575i
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.309662
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ 12.0000 0.529297
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 18.0000i − 0.791639i
$$518$$ − 2.00000i − 0.0878750i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ 0 0
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ 15.0000 0.655278
$$525$$ 0 0
$$526$$ 30.0000 1.30806
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 2.00000i − 0.0867110i
$$533$$ − 24.0000i − 1.03956i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −14.0000 −0.604708
$$537$$ 0 0
$$538$$ − 18.0000i − 0.776035i
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ 25.0000i 1.07384i
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 8.00000i 0.340195i
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 27.0000i 1.14403i 0.820244 + 0.572013i $$0.193837\pi$$
−0.820244 + 0.572013i $$0.806163\pi$$
$$558$$ 0 0
$$559$$ −40.0000 −1.69182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 24.0000i − 1.01238i
$$563$$ − 3.00000i − 0.126435i −0.998000 0.0632175i $$-0.979864\pi$$
0.998000 0.0632175i $$-0.0201362\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −14.0000 −0.588464
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 12.0000 0.503066 0.251533 0.967849i $$-0.419065\pi$$
0.251533 + 0.967849i $$0.419065\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 12.0000i 0.501745i
$$573$$ 0 0
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 38.0000i 1.58196i 0.611842 + 0.790980i $$0.290429\pi$$
−0.611842 + 0.790980i $$0.709571\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 3.00000 0.124461
$$582$$ 0 0
$$583$$ 27.0000i 1.11823i
$$584$$ −7.00000 −0.289662
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 3.00000i − 0.123823i −0.998082 0.0619116i $$-0.980280\pi$$
0.998082 0.0619116i $$-0.0197197\pi$$
$$588$$ 0 0
$$589$$ −10.0000 −0.412043
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000i 0.0821995i
$$593$$ − 18.0000i − 0.739171i −0.929197 0.369586i $$-0.879500\pi$$
0.929197 0.369586i $$-0.120500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 3.00000 0.122885
$$597$$ 0 0
$$598$$ 24.0000i 0.981433i
$$599$$ 42.0000 1.71607 0.858037 0.513588i $$-0.171684\pi$$
0.858037 + 0.513588i $$0.171684\pi$$
$$600$$ 0 0
$$601$$ 35.0000 1.42768 0.713840 0.700309i $$-0.246954\pi$$
0.713840 + 0.700309i $$0.246954\pi$$
$$602$$ − 10.0000i − 0.407570i
$$603$$ 0 0
$$604$$ −17.0000 −0.691720
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ 2.00000i 0.0811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 34.0000i 1.37325i 0.727013 + 0.686624i $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 6.00000i 0.240578i
$$623$$ − 18.0000i − 0.721155i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 19.0000 0.759393
$$627$$ 0 0
$$628$$ 4.00000i 0.159617i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −25.0000 −0.995234 −0.497617 0.867397i $$-0.665792\pi$$
−0.497617 + 0.867397i $$0.665792\pi$$
$$632$$ − 8.00000i − 0.318223i
$$633$$ 0 0
$$634$$ −3.00000 −0.119145
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 24.0000i 0.950915i
$$638$$ − 18.0000i − 0.712627i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ 0 0
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.0000i 0.783260i
$$653$$ − 39.0000i − 1.52619i −0.646288 0.763094i $$-0.723679\pi$$
0.646288 0.763094i $$-0.276321\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 0 0
$$658$$ − 6.00000i − 0.233904i
$$659$$ 21.0000 0.818044 0.409022 0.912525i $$-0.365870\pi$$
0.409022 + 0.912525i $$0.365870\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 10.0000i 0.388661i
$$663$$ 0 0
$$664$$ −3.00000 −0.116423
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 36.0000i − 1.39393i
$$668$$ 6.00000i 0.232147i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ 0 0
$$673$$ 19.0000i 0.732396i 0.930537 + 0.366198i $$0.119341\pi$$
−0.930537 + 0.366198i $$0.880659\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 42.0000i 1.61419i 0.590421 + 0.807096i $$0.298962\pi$$
−0.590421 + 0.807096i $$0.701038\pi$$
$$678$$ 0 0
$$679$$ −1.00000 −0.0383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 15.0000i 0.574380i
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ 0 0
$$688$$ 10.0000i 0.381246i
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ 15.0000i 0.570214i
$$693$$ 0 0
$$694$$ 3.00000 0.113878
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ − 10.0000i − 0.378506i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 9.00000 0.339925 0.169963 0.985451i $$-0.445635\pi$$
0.169963 + 0.985451i $$0.445635\pi$$
$$702$$ 0 0
$$703$$ − 4.00000i − 0.150863i
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 3.00000i 0.112827i
$$708$$ 0 0
$$709$$ −44.0000 −1.65245 −0.826227 0.563337i $$-0.809517\pi$$
−0.826227 + 0.563337i $$0.809517\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 18.0000i 0.674579i
$$713$$ 30.0000i 1.12351i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −9.00000 −0.336346
$$717$$ 0 0
$$718$$ 18.0000i 0.671754i
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ 15.0000i 0.558242i
$$723$$ 0 0
$$724$$ 16.0000 0.594635
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 1.00000i − 0.0370879i −0.999828 0.0185440i $$-0.994097\pi$$
0.999828 0.0185440i $$-0.00590307\pi$$
$$728$$ 4.00000i 0.148250i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 22.0000i 0.812589i 0.913742 + 0.406294i $$0.133179\pi$$
−0.913742 + 0.406294i $$0.866821\pi$$
$$734$$ 17.0000 0.627481
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ − 42.0000i − 1.54709i
$$738$$ 0 0
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 9.00000i 0.330400i
$$743$$ − 12.0000i − 0.440237i −0.975473 0.220119i $$-0.929356\pi$$
0.975473 0.220119i $$-0.0706445\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −32.0000 −1.17160
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 9.00000 0.328853
$$750$$ 0 0
$$751$$ 41.0000 1.49611 0.748056 0.663636i $$-0.230988\pi$$
0.748056 + 0.663636i $$0.230988\pi$$
$$752$$ 6.00000i 0.218797i
$$753$$ 0 0
$$754$$ −24.0000 −0.874028
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 48.0000 1.74000 0.869999 0.493053i $$-0.164119\pi$$
0.869999 + 0.493053i $$0.164119\pi$$
$$762$$ 0 0
$$763$$ 2.00000i 0.0724049i
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ − 48.0000i − 1.73318i
$$768$$ 0 0
$$769$$ 31.0000 1.11789 0.558944 0.829205i $$-0.311207\pi$$
0.558944 + 0.829205i $$0.311207\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 5.00000i 0.179954i
$$773$$ 18.0000i 0.647415i 0.946157 + 0.323708i $$0.104929\pi$$
−0.946157 + 0.323708i $$0.895071\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 1.00000 0.0358979
$$777$$ 0 0
$$778$$ − 21.0000i − 0.752886i
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 6.00000 0.214286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ − 9.00000i − 0.320612i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ 32.0000i 1.13635i
$$794$$ 20.0000 0.709773
$$795$$ 0 0
$$796$$ −7.00000 −0.248108
$$797$$ 39.0000i 1.38145i 0.723117 + 0.690725i $$0.242709\pi$$
−0.723117 + 0.690725i $$0.757291\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 12.0000i − 0.423735i
$$803$$ − 21.0000i − 0.741074i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 20.0000 0.704470
$$807$$ 0 0
$$808$$ − 3.00000i − 0.105540i
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ − 6.00000i − 0.210559i
$$813$$ 0 0
$$814$$ −6.00000 −0.210300
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 20.0000i − 0.699711i
$$818$$ 23.0000i 0.804176i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ 31.0000i 1.08059i 0.841475 + 0.540296i $$0.181688\pi$$
−0.841475 + 0.540296i $$0.818312\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 4.00000i − 0.138675i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −6.00000 −0.207514
$$837$$ 0 0
$$838$$ 12.0000i 0.414533i
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 8.00000i − 0.275698i
$$843$$ 0 0
$$844$$ 22.0000 0.757271
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.00000i 0.0687208i
$$848$$ − 9.00000i − 0.309061i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ 10.0000i 0.342393i 0.985237 + 0.171197i $$0.0547634\pi$$
−0.985237 + 0.171197i $$0.945237\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ − 48.0000i − 1.63965i −0.572615 0.819824i $$-0.694071\pi$$
0.572615 0.819824i $$-0.305929\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 18.0000i − 0.613082i
$$863$$ 36.0000i 1.22545i 0.790295 + 0.612727i $$0.209928\pi$$
−0.790295 + 0.612727i $$0.790072\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −29.0000 −0.985460
$$867$$ 0 0
$$868$$ 5.00000i 0.169711i
$$869$$ 24.0000 0.814144
$$870$$ 0 0
$$871$$ −56.0000 −1.89749
$$872$$ − 2.00000i − 0.0677285i
$$873$$ 0 0
$$874$$ −12.0000 −0.405906
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ − 19.0000i − 0.641219i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −36.0000 −1.21287 −0.606435 0.795133i $$-0.707401\pi$$
−0.606435 + 0.795133i $$0.707401\pi$$
$$882$$ 0 0
$$883$$ − 2.00000i − 0.0673054i −0.999434 0.0336527i $$-0.989286\pi$$
0.999434 0.0336527i $$-0.0107140\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 12.0000i 0.402921i 0.979497 + 0.201460i $$0.0645687\pi$$
−0.979497 + 0.201460i $$0.935431\pi$$
$$888$$ 0 0
$$889$$ −7.00000 −0.234772
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8.00000i 0.267860i
$$893$$ − 12.0000i − 0.401565i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ − 18.0000i − 0.600668i
$$899$$ −30.0000 −1.00056
$$900$$ 0 0
$$901$$ 0 0
$$902$$ − 18.0000i − 0.599334i
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 26.0000i 0.863316i 0.902037 + 0.431658i $$0.142071\pi$$
−0.902037 + 0.431658i $$0.857929\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 0 0
$$913$$ − 9.00000i − 0.297857i
$$914$$ −1.00000 −0.0330771
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 15.0000i 0.495344i
$$918$$ 0 0
$$919$$ −29.0000 −0.956622 −0.478311 0.878191i $$-0.658751\pi$$
−0.478311 + 0.878191i $$0.658751\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 21.0000i 0.691598i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 13.0000 0.427207
$$927$$ 0 0
$$928$$ 6.00000i 0.196960i
$$929$$ 12.0000 0.393707 0.196854 0.980433i $$-0.436928\pi$$
0.196854 + 0.980433i $$0.436928\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 18.0000i 0.589610i
$$933$$ 0 0
$$934$$ −27.0000 −0.883467
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 11.0000i 0.359354i 0.983726 + 0.179677i $$0.0575053\pi$$
−0.983726 + 0.179677i $$0.942495\pi$$
$$938$$ − 14.0000i − 0.457116i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −33.0000 −1.07577 −0.537885 0.843018i $$-0.680776\pi$$
−0.537885 + 0.843018i $$0.680776\pi$$
$$942$$ 0 0
$$943$$ − 36.0000i − 1.17232i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −30.0000 −0.975384
$$947$$ 51.0000i 1.65728i 0.559784 + 0.828639i $$0.310884\pi$$
−0.559784 + 0.828639i $$0.689116\pi$$
$$948$$ 0 0
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 36.0000i 1.16615i 0.812417 + 0.583077i $$0.198151\pi$$
−0.812417 + 0.583077i $$0.801849\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 30.0000 0.970269
$$957$$ 0 0
$$958$$ 6.00000i 0.193851i
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 8.00000i 0.257930i
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 23.0000i 0.739630i 0.929105 + 0.369815i $$0.120579\pi$$
−0.929105 + 0.369815i $$0.879421\pi$$
$$968$$ − 2.00000i − 0.0642824i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −27.0000 −0.866471 −0.433236 0.901281i $$-0.642628\pi$$
−0.433236 + 0.901281i $$0.642628\pi$$
$$972$$ 0 0
$$973$$ − 4.00000i − 0.128234i
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 0 0
$$979$$ −54.0000 −1.72585
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 39.0000i − 1.24454i
$$983$$ − 6.00000i − 0.191370i −0.995412 0.0956851i $$-0.969496\pi$$
0.995412 0.0956851i $$-0.0305042\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 8.00000i 0.254514i
$$989$$ −60.0000 −1.90789
$$990$$ 0 0
$$991$$ 47.0000 1.49300 0.746502 0.665383i $$-0.231732\pi$$
0.746502 + 0.665383i $$0.231732\pi$$
$$992$$ − 5.00000i − 0.158750i
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 28.0000i − 0.886769i −0.896332 0.443384i $$-0.853778\pi$$
0.896332 0.443384i $$-0.146222\pi$$
$$998$$ 14.0000i 0.443162i
$$999$$ 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 1350.2.c.b.649.1 2
3.2 odd 2 1350.2.c.k.649.2 2
5.2 odd 4 1350.2.a.r.1.1 1
5.3 odd 4 54.2.a.a.1.1 1
5.4 even 2 inner 1350.2.c.b.649.2 2
15.2 even 4 1350.2.a.h.1.1 1
15.8 even 4 54.2.a.b.1.1 yes 1
15.14 odd 2 1350.2.c.k.649.1 2
20.3 even 4 432.2.a.g.1.1 1
35.13 even 4 2646.2.a.a.1.1 1
40.3 even 4 1728.2.a.d.1.1 1
40.13 odd 4 1728.2.a.c.1.1 1
45.13 odd 12 162.2.c.c.55.1 2
45.23 even 12 162.2.c.b.55.1 2
45.38 even 12 162.2.c.b.109.1 2
45.43 odd 12 162.2.c.c.109.1 2
55.43 even 4 6534.2.a.bc.1.1 1
60.23 odd 4 432.2.a.b.1.1 1
65.38 odd 4 9126.2.a.u.1.1 1
105.83 odd 4 2646.2.a.bd.1.1 1
120.53 even 4 1728.2.a.y.1.1 1
120.83 odd 4 1728.2.a.z.1.1 1
165.98 odd 4 6534.2.a.b.1.1 1
180.23 odd 12 1296.2.i.o.865.1 2
180.43 even 12 1296.2.i.c.433.1 2
180.83 odd 12 1296.2.i.o.433.1 2
180.103 even 12 1296.2.i.c.865.1 2
195.38 even 4 9126.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
54.2.a.a.1.1 1 5.3 odd 4
54.2.a.b.1.1 yes 1 15.8 even 4
162.2.c.b.55.1 2 45.23 even 12
162.2.c.b.109.1 2 45.38 even 12
162.2.c.c.55.1 2 45.13 odd 12
162.2.c.c.109.1 2 45.43 odd 12
432.2.a.b.1.1 1 60.23 odd 4
432.2.a.g.1.1 1 20.3 even 4
1296.2.i.c.433.1 2 180.43 even 12
1296.2.i.c.865.1 2 180.103 even 12
1296.2.i.o.433.1 2 180.83 odd 12
1296.2.i.o.865.1 2 180.23 odd 12
1350.2.a.h.1.1 1 15.2 even 4
1350.2.a.r.1.1 1 5.2 odd 4
1350.2.c.b.649.1 2 1.1 even 1 trivial
1350.2.c.b.649.2 2 5.4 even 2 inner
1350.2.c.k.649.1 2 15.14 odd 2
1350.2.c.k.649.2 2 3.2 odd 2
1728.2.a.c.1.1 1 40.13 odd 4
1728.2.a.d.1.1 1 40.3 even 4
1728.2.a.y.1.1 1 120.53 even 4
1728.2.a.z.1.1 1 120.83 odd 4
2646.2.a.a.1.1 1 35.13 even 4
2646.2.a.bd.1.1 1 105.83 odd 4
6534.2.a.b.1.1 1 165.98 odd 4
6534.2.a.bc.1.1 1 55.43 even 4
9126.2.a.r.1.1 1 195.38 even 4
9126.2.a.u.1.1 1 65.38 odd 4