# Properties

 Label 3840.2.k.f.1921.1 Level $3840$ Weight $2$ Character 3840.1921 Analytic conductor $30.663$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3840 = 2^{8} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3840.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.6625543762$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1921.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3840.1921 Dual form 3840.2.k.f.1921.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} +1.00000 q^{15} +6.00000 q^{17} +4.00000i q^{19} +4.00000i q^{21} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} -8.00000 q^{31} -4.00000i q^{35} -2.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} -1.00000i q^{45} +9.00000 q^{49} -6.00000i q^{51} +6.00000i q^{53} +4.00000 q^{57} -10.0000i q^{61} +4.00000 q^{63} -2.00000 q^{65} +4.00000i q^{67} -2.00000 q^{73} +1.00000i q^{75} -8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +6.00000i q^{85} -6.00000 q^{87} -18.0000 q^{89} -8.00000i q^{91} +8.00000i q^{93} -4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 8q^{7} - 2q^{9} + 2q^{15} + 12q^{17} - 2q^{25} - 16q^{31} + 4q^{39} + 12q^{41} + 18q^{49} + 8q^{57} + 8q^{63} - 4q^{65} - 4q^{73} - 16q^{79} + 2q^{81} - 12q^{87} - 36q^{89} - 8q^{95} + 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$1537$$ $$2561$$ $$2821$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 1.00000i 0.447214i
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 4.00000i 0.917663i 0.888523 + 0.458831i $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ 0 0
$$21$$ 4.00000i 0.872872i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ − 6.00000i − 1.11417i −0.830455 0.557086i $$-0.811919\pi$$
0.830455 0.557086i $$-0.188081\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 4.00000i − 0.676123i
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ − 1.00000i − 0.149071i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ − 6.00000i − 0.840168i
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ − 10.0000i − 1.28037i −0.768221 0.640184i $$-0.778858\pi$$
0.768221 0.640184i $$-0.221142\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 1.00000i 0.115470i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 6.00000i 0.650791i
$$86$$ 0 0
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ − 8.00000i − 0.838628i
$$92$$ 0 0
$$93$$ 8.00000i 0.829561i
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 18.0000i − 1.79107i −0.444994 0.895533i $$-0.646794\pi$$
0.444994 0.895533i $$-0.353206\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ 0 0
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ − 10.0000i − 0.957826i −0.877862 0.478913i $$-0.841031\pi$$
0.877862 0.478913i $$-0.158969\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ −24.0000 −2.20008
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ − 6.00000i − 0.541002i
$$124$$ 0 0
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ −20.0000 −1.77471 −0.887357 0.461084i $$-0.847461\pi$$
−0.887357 + 0.461084i $$0.847461\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ − 16.0000i − 1.38738i
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ − 4.00000i − 0.339276i −0.985506 0.169638i $$-0.945740\pi$$
0.985506 0.169638i $$-0.0542598\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 0 0
$$147$$ − 9.00000i − 0.742307i
$$148$$ 0 0
$$149$$ 6.00000i 0.491539i 0.969328 + 0.245770i $$0.0790407\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ − 8.00000i − 0.642575i
$$156$$ 0 0
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ − 4.00000i − 0.305888i
$$172$$ 0 0
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 24.0000i − 1.79384i −0.442189 0.896922i $$-0.645798\pi$$
0.442189 0.896922i $$-0.354202\pi$$
$$180$$ 0 0
$$181$$ − 14.0000i − 1.04061i −0.853980 0.520306i $$-0.825818\pi$$
0.853980 0.520306i $$-0.174182\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ − 4.00000i − 0.290957i
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 0 0
$$193$$ −22.0000 −1.58359 −0.791797 0.610784i $$-0.790854\pi$$
−0.791797 + 0.610784i $$0.790854\pi$$
$$194$$ 0 0
$$195$$ 2.00000i 0.143223i
$$196$$ 0 0
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 24.0000i 1.68447i
$$204$$ 0 0
$$205$$ 6.00000i 0.419058i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ − 20.0000i − 1.37686i −0.725304 0.688428i $$-0.758301\pi$$
0.725304 0.688428i $$-0.241699\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 32.0000 2.17230
$$218$$ 0 0
$$219$$ 2.00000i 0.135147i
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ −20.0000 −1.33930 −0.669650 0.742677i $$-0.733556\pi$$
−0.669650 + 0.742677i $$0.733556\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ 10.0000i 0.660819i 0.943838 + 0.330409i $$0.107187\pi$$
−0.943838 + 0.330409i $$0.892813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000i 0.519656i
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 9.00000i 0.574989i
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ − 24.0000i − 1.51487i −0.652913 0.757433i $$-0.726453\pi$$
0.652913 0.757433i $$-0.273547\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 6.00000 0.375735
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 8.00000i 0.497096i
$$260$$ 0 0
$$261$$ 6.00000i 0.371391i
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 18.0000i 1.10158i
$$268$$ 0 0
$$269$$ − 6.00000i − 0.365826i −0.983129 0.182913i $$-0.941447\pi$$
0.983129 0.182913i $$-0.0585527\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ −8.00000 −0.484182
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ 0 0
$$285$$ 4.00000i 0.236940i
$$286$$ 0 0
$$287$$ −24.0000 −1.41668
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ − 2.00000i − 0.117242i
$$292$$ 0 0
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 16.0000i 0.922225i
$$302$$ 0 0
$$303$$ −18.0000 −1.03407
$$304$$ 0 0
$$305$$ 10.0000 0.572598
$$306$$ 0 0
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 0 0
$$309$$ 4.00000i 0.227552i
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −2.00000 −0.113047 −0.0565233 0.998401i $$-0.518002\pi$$
−0.0565233 + 0.998401i $$0.518002\pi$$
$$314$$ 0 0
$$315$$ 4.00000i 0.225374i
$$316$$ 0 0
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ 0 0
$$325$$ − 2.00000i − 0.110940i
$$326$$ 0 0
$$327$$ −10.0000 −0.553001
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 28.0000i − 1.53902i −0.638635 0.769510i $$-0.720501\pi$$
0.638635 0.769510i $$-0.279499\pi$$
$$332$$ 0 0
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 0 0
$$339$$ 18.0000i 0.977626i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 0 0
$$349$$ − 10.0000i − 0.535288i −0.963518 0.267644i $$-0.913755\pi$$
0.963518 0.267644i $$-0.0862451\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 24.0000i 1.27021i
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ − 11.0000i − 0.577350i
$$364$$ 0 0
$$365$$ − 2.00000i − 0.104685i
$$366$$ 0 0
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ − 24.0000i − 1.24602i
$$372$$ 0 0
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ − 4.00000i − 0.205466i −0.994709 0.102733i $$-0.967241\pi$$
0.994709 0.102733i $$-0.0327588\pi$$
$$380$$ 0 0
$$381$$ 20.0000i 1.02463i
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.00000i 0.203331i
$$388$$ 0 0
$$389$$ 6.00000i 0.304212i 0.988364 + 0.152106i $$0.0486055\pi$$
−0.988364 + 0.152106i $$0.951394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 8.00000i − 0.402524i
$$396$$ 0 0
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ 0 0
$$399$$ −16.0000 −0.801002
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ − 16.0000i − 0.797017i
$$404$$ 0 0
$$405$$ 1.00000i 0.0496904i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 6.00000i 0.295958i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 10.0000i 0.487370i 0.969854 + 0.243685i $$0.0783563\pi$$
−0.969854 + 0.243685i $$0.921644\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 40.0000i 1.93574i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ − 6.00000i − 0.287678i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −9.00000 −0.428571
$$442$$ 0 0
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 0 0
$$445$$ − 18.0000i − 0.853282i
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 8.00000i − 0.375873i
$$454$$ 0 0
$$455$$ 8.00000 0.375046
$$456$$ 0 0
$$457$$ −26.0000 −1.21623 −0.608114 0.793849i $$-0.708074\pi$$
−0.608114 + 0.793849i $$0.708074\pi$$
$$458$$ 0 0
$$459$$ 6.00000i 0.280056i
$$460$$ 0 0
$$461$$ − 30.0000i − 1.39724i −0.715493 0.698620i $$-0.753798\pi$$
0.715493 0.698620i $$-0.246202\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ −8.00000 −0.370991
$$466$$ 0 0
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ 0 0
$$469$$ − 16.0000i − 0.738811i
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ − 4.00000i − 0.183533i
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2.00000i 0.0908153i
$$486$$ 0 0
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ 0 0
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 24.0000i 1.08310i 0.840667 + 0.541552i $$0.182163\pi$$
−0.840667 + 0.541552i $$0.817837\pi$$
$$492$$ 0 0
$$493$$ − 36.0000i − 1.62136i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000i 0.179065i 0.995984 + 0.0895323i $$0.0285372\pi$$
−0.995984 + 0.0895323i $$0.971463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 0 0
$$507$$ − 9.00000i − 0.399704i
$$508$$ 0 0
$$509$$ − 6.00000i − 0.265945i −0.991120 0.132973i $$-0.957548\pi$$
0.991120 0.132973i $$-0.0424523\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ − 4.00000i − 0.176261i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 0 0
$$525$$ − 4.00000i − 0.174574i
$$526$$ 0 0
$$527$$ −48.0000 −2.09091
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 0 0
$$537$$ −24.0000 −1.03568
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 10.0000i − 0.429934i −0.976621 0.214967i $$-0.931036\pi$$
0.976621 0.214967i $$-0.0689643\pi$$
$$542$$ 0 0
$$543$$ −14.0000 −0.600798
$$544$$ 0 0
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ 28.0000i 1.19719i 0.801050 + 0.598597i $$0.204275\pi$$
−0.801050 + 0.598597i $$0.795725\pi$$
$$548$$ 0 0
$$549$$ 10.0000i 0.426790i
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ 32.0000 1.36078
$$554$$ 0 0
$$555$$ − 2.00000i − 0.0848953i
$$556$$ 0 0
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 0 0
$$565$$ − 18.0000i − 0.757266i
$$566$$ 0 0
$$567$$ −4.00000 −0.167984
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 20.0000i 0.836974i 0.908223 + 0.418487i $$0.137439\pi$$
−0.908223 + 0.418487i $$0.862561\pi$$
$$572$$ 0 0
$$573$$ − 24.0000i − 1.00261i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ 22.0000i 0.914289i
$$580$$ 0 0
$$581$$ 48.0000i 1.99138i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 2.00000 0.0826898
$$586$$ 0 0
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 0 0
$$589$$ − 32.0000i − 1.31854i
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 0 0
$$595$$ − 24.0000i − 0.983904i
$$596$$ 0 0
$$597$$ − 8.00000i − 0.327418i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ 0 0
$$605$$ 11.0000i 0.447214i
$$606$$ 0 0
$$607$$ 4.00000 0.162355 0.0811775 0.996700i $$-0.474132\pi$$
0.0811775 + 0.996700i $$0.474132\pi$$
$$608$$ 0 0
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 0 0
$$615$$ 6.00000 0.241943
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ 44.0000i 1.76851i 0.467005 + 0.884255i $$0.345333\pi$$
−0.467005 + 0.884255i $$0.654667\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 72.0000 2.88462
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 12.0000i − 0.478471i
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ −20.0000 −0.794929
$$634$$ 0 0
$$635$$ − 20.0000i − 0.793676i
$$636$$ 0 0
$$637$$ 18.0000i 0.713186i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 0 0
$$645$$ − 4.00000i − 0.157500i
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ − 32.0000i − 1.25418i
$$652$$ 0 0
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ − 48.0000i − 1.86981i −0.354892 0.934907i $$-0.615482\pi$$
0.354892 0.934907i $$-0.384518\pi$$
$$660$$ 0 0
$$661$$ − 14.0000i − 0.544537i −0.962221 0.272268i $$-0.912226\pi$$
0.962221 0.272268i $$-0.0877739\pi$$
$$662$$ 0 0
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ 16.0000 0.620453
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 20.0000i 0.773245i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ 0 0
$$675$$ − 1.00000i − 0.0384900i
$$676$$ 0 0
$$677$$ 6.00000i 0.230599i 0.993331 + 0.115299i $$0.0367827\pi$$
−0.993331 + 0.115299i $$0.963217\pi$$
$$678$$ 0 0
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 0 0
$$685$$ − 6.00000i − 0.229248i
$$686$$ 0 0
$$687$$ 10.0000 0.381524
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ − 44.0000i − 1.67384i −0.547326 0.836919i $$-0.684354\pi$$
0.547326 0.836919i $$-0.315646\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.00000 0.151729
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ 0 0
$$699$$ − 18.0000i − 0.680823i
$$700$$ 0 0
$$701$$ − 6.00000i − 0.226617i −0.993560 0.113308i $$-0.963855\pi$$
0.993560 0.113308i $$-0.0361448\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 72.0000i 2.70784i
$$708$$ 0 0
$$709$$ − 38.0000i − 1.42712i −0.700594 0.713560i $$-0.747082\pi$$
0.700594 0.713560i $$-0.252918\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 24.0000i 0.896296i
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 0 0
$$723$$ − 2.00000i − 0.0743808i
$$724$$ 0 0
$$725$$ 6.00000i 0.222834i
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ − 24.0000i − 0.887672i
$$732$$ 0 0
$$733$$ − 22.0000i − 0.812589i −0.913742 0.406294i $$-0.866821\pi$$
0.913742 0.406294i $$-0.133179\pi$$
$$734$$ 0 0
$$735$$ 9.00000 0.331970
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 52.0000i 1.91285i 0.291977 + 0.956425i $$0.405687\pi$$
−0.291977 + 0.956425i $$0.594313\pi$$
$$740$$ 0 0
$$741$$ 8.00000i 0.293887i
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ 48.0000i 1.75388i
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ −24.0000 −0.874609
$$754$$ 0 0
$$755$$ 8.00000i 0.291150i
$$756$$ 0 0
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ 40.0000i 1.44810i
$$764$$ 0 0
$$765$$ − 6.00000i − 0.216930i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 18.0000i 0.648254i
$$772$$ 0 0
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 0 0
$$775$$ 8.00000 0.287368
$$776$$ 0 0
$$777$$ 8.00000 0.286998
$$778$$ 0 0
$$779$$ 24.0000i 0.859889i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 72.0000 2.56003
$$792$$ 0 0
$$793$$ 20.0000 0.710221
$$794$$ 0 0
$$795$$ 6.00000i 0.212798i
$$796$$ 0 0
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.00000 −0.211210
$$808$$ 0 0
$$809$$ 54.0000 1.89854 0.949269 0.314464i $$-0.101825\pi$$
0.949269 + 0.314464i $$0.101825\pi$$
$$810$$ 0 0
$$811$$ − 4.00000i − 0.140459i −0.997531 0.0702295i $$-0.977627\pi$$
0.997531 0.0702295i $$-0.0223732\pi$$
$$812$$ 0 0
$$813$$ − 16.0000i − 0.561144i
$$814$$ 0 0
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 0 0
$$819$$ 8.00000i 0.279543i
$$820$$ 0 0
$$821$$ − 18.0000i − 0.628204i −0.949389 0.314102i $$-0.898297\pi$$
0.949389 0.314102i $$-0.101703\pi$$
$$822$$ 0 0
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ 0 0
$$829$$ 38.0000i 1.31979i 0.751356 + 0.659897i $$0.229400\pi$$
−0.751356 + 0.659897i $$0.770600\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ 54.0000 1.87099
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 8.00000i − 0.276520i
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ 18.0000i 0.619953i
$$844$$ 0 0
$$845$$ 9.00000i 0.309609i
$$846$$ 0 0
$$847$$ −44.0000 −1.51186
$$848$$ 0 0
$$849$$ −28.0000 −0.960958
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 46.0000i 1.57501i 0.616308 + 0.787505i $$0.288628\pi$$
−0.616308 + 0.787505i $$0.711372\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ − 4.00000i − 0.136478i −0.997669 0.0682391i $$-0.978262\pi$$
0.997669 0.0682391i $$-0.0217381\pi$$
$$860$$ 0 0
$$861$$ 24.0000i 0.817918i
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 4.00000i 0.135225i
$$876$$ 0 0
$$877$$ 2.00000i 0.0675352i 0.999430 + 0.0337676i $$0.0107506\pi$$
−0.999430 + 0.0337676i $$0.989249\pi$$
$$878$$ 0 0
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ 0 0
$$883$$ 4.00000i 0.134611i 0.997732 + 0.0673054i $$0.0214402\pi$$
−0.997732 + 0.0673054i $$0.978560\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 80.0000 2.68311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 48.0000i 1.60089i
$$900$$ 0 0
$$901$$ 36.0000i 1.19933i
$$902$$ 0 0
$$903$$ 16.0000 0.532447
$$904$$ 0 0
$$905$$ 14.0000 0.465376
$$906$$ 0 0
$$907$$ 44.0000i 1.46100i 0.682915 + 0.730498i $$0.260712\pi$$
−0.682915 + 0.730498i $$0.739288\pi$$
$$908$$ 0 0
$$909$$ 18.0000i 0.597022i
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ − 10.0000i − 0.330590i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000i 0.0657596i
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ 36.0000i 1.17985i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ 0 0
$$939$$ 2.00000i 0.0652675i
$$940$$ 0 0
$$941$$ 18.0000i 0.586783i 0.955992 + 0.293392i $$0.0947840\pi$$
−0.955992 + 0.293392i $$0.905216\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 4.00000 0.130120
$$946$$ 0 0
$$947$$ − 36.0000i − 1.16984i −0.811090 0.584921i $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ 0 0
$$949$$ − 4.00000i − 0.129845i
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ 24.0000i 0.776622i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 0 0
$$965$$ − 22.0000i − 0.708205i
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 0 0
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 24.0000i 0.770197i 0.922876 + 0.385098i $$0.125832\pi$$
−0.922876 + 0.385098i $$0.874168\pi$$
$$972$$ 0 0
$$973$$ 16.0000i 0.512936i
$$974$$ 0 0
$$975$$ −2.00000 −0.0640513
$$976$$ 0 0
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.0000i 0.319275i
$$982$$ 0 0
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ −28.0000 −0.888553
$$994$$ 0 0
$$995$$ 8.00000i 0.253617i
$$996$$ 0 0
$$997$$ − 26.0000i − 0.823428i −0.911313 0.411714i $$-0.864930\pi$$
0.911313 0.411714i $$-0.135070\pi$$
$$998$$ 0 0
$$999$$ 2.00000 0.0632772
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3840.2.k.f.1921.1 2
4.3 odd 2 3840.2.k.y.1921.2 2
8.3 odd 2 3840.2.k.y.1921.1 2
8.5 even 2 inner 3840.2.k.f.1921.2 2
16.3 odd 4 960.2.a.e.1.1 1
16.5 even 4 240.2.a.b.1.1 1
16.11 odd 4 30.2.a.a.1.1 1
16.13 even 4 960.2.a.p.1.1 1
48.5 odd 4 720.2.a.j.1.1 1
48.11 even 4 90.2.a.c.1.1 1
48.29 odd 4 2880.2.a.q.1.1 1
48.35 even 4 2880.2.a.a.1.1 1
80.3 even 4 4800.2.f.p.3649.1 2
80.13 odd 4 4800.2.f.w.3649.2 2
80.19 odd 4 4800.2.a.cq.1.1 1
80.27 even 4 150.2.c.a.49.1 2
80.29 even 4 4800.2.a.d.1.1 1
80.37 odd 4 1200.2.f.e.49.2 2
80.43 even 4 150.2.c.a.49.2 2
80.53 odd 4 1200.2.f.e.49.1 2
80.59 odd 4 150.2.a.b.1.1 1
80.67 even 4 4800.2.f.p.3649.2 2
80.69 even 4 1200.2.a.k.1.1 1
80.77 odd 4 4800.2.f.w.3649.1 2
112.11 odd 12 1470.2.i.o.961.1 2
112.27 even 4 1470.2.a.d.1.1 1
112.59 even 12 1470.2.i.q.961.1 2
112.75 even 12 1470.2.i.q.361.1 2
112.107 odd 12 1470.2.i.o.361.1 2
144.11 even 12 810.2.e.b.271.1 2
144.43 odd 12 810.2.e.l.271.1 2
144.59 even 12 810.2.e.b.541.1 2
144.139 odd 12 810.2.e.l.541.1 2
176.43 even 4 3630.2.a.w.1.1 1
208.155 odd 4 5070.2.a.w.1.1 1
208.187 even 4 5070.2.b.k.1351.2 2
208.203 even 4 5070.2.b.k.1351.1 2
240.53 even 4 3600.2.f.i.2449.1 2
240.59 even 4 450.2.a.d.1.1 1
240.107 odd 4 450.2.c.b.199.2 2
240.149 odd 4 3600.2.a.f.1.1 1
240.197 even 4 3600.2.f.i.2449.2 2
240.203 odd 4 450.2.c.b.199.1 2
272.203 odd 4 8670.2.a.g.1.1 1
336.251 odd 4 4410.2.a.z.1.1 1
560.139 even 4 7350.2.a.ct.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 1 16.11 odd 4
90.2.a.c.1.1 1 48.11 even 4
150.2.a.b.1.1 1 80.59 odd 4
150.2.c.a.49.1 2 80.27 even 4
150.2.c.a.49.2 2 80.43 even 4
240.2.a.b.1.1 1 16.5 even 4
450.2.a.d.1.1 1 240.59 even 4
450.2.c.b.199.1 2 240.203 odd 4
450.2.c.b.199.2 2 240.107 odd 4
720.2.a.j.1.1 1 48.5 odd 4
810.2.e.b.271.1 2 144.11 even 12
810.2.e.b.541.1 2 144.59 even 12
810.2.e.l.271.1 2 144.43 odd 12
810.2.e.l.541.1 2 144.139 odd 12
960.2.a.e.1.1 1 16.3 odd 4
960.2.a.p.1.1 1 16.13 even 4
1200.2.a.k.1.1 1 80.69 even 4
1200.2.f.e.49.1 2 80.53 odd 4
1200.2.f.e.49.2 2 80.37 odd 4
1470.2.a.d.1.1 1 112.27 even 4
1470.2.i.o.361.1 2 112.107 odd 12
1470.2.i.o.961.1 2 112.11 odd 12
1470.2.i.q.361.1 2 112.75 even 12
1470.2.i.q.961.1 2 112.59 even 12
2880.2.a.a.1.1 1 48.35 even 4
2880.2.a.q.1.1 1 48.29 odd 4
3600.2.a.f.1.1 1 240.149 odd 4
3600.2.f.i.2449.1 2 240.53 even 4
3600.2.f.i.2449.2 2 240.197 even 4
3630.2.a.w.1.1 1 176.43 even 4
3840.2.k.f.1921.1 2 1.1 even 1 trivial
3840.2.k.f.1921.2 2 8.5 even 2 inner
3840.2.k.y.1921.1 2 8.3 odd 2
3840.2.k.y.1921.2 2 4.3 odd 2
4410.2.a.z.1.1 1 336.251 odd 4
4800.2.a.d.1.1 1 80.29 even 4
4800.2.a.cq.1.1 1 80.19 odd 4
4800.2.f.p.3649.1 2 80.3 even 4
4800.2.f.p.3649.2 2 80.67 even 4
4800.2.f.w.3649.1 2 80.77 odd 4
4800.2.f.w.3649.2 2 80.13 odd 4
5070.2.a.w.1.1 1 208.155 odd 4
5070.2.b.k.1351.1 2 208.203 even 4
5070.2.b.k.1351.2 2 208.187 even 4
7350.2.a.ct.1.1 1 560.139 even 4
8670.2.a.g.1.1 1 272.203 odd 4