Properties

 Label 3840.2.k.m.1921.2 Level $3840$ Weight $2$ Character 3840.1921 Analytic conductor $30.663$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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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 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} -4.00000i q^{11} +2.00000i q^{13} -1.00000 q^{15} +2.00000 q^{17} -4.00000i q^{19} -1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +4.00000 q^{33} -10.0000i q^{37} -2.00000 q^{39} -10.0000 q^{41} +4.00000i q^{43} -1.00000i q^{45} +8.00000 q^{47} -7.00000 q^{49} +2.00000i q^{51} -10.0000i q^{53} +4.00000 q^{55} +4.00000 q^{57} -4.00000i q^{59} +2.00000i q^{61} -2.00000 q^{65} -12.0000i q^{67} +8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} +1.00000 q^{81} -12.0000i q^{83} +2.00000i q^{85} -2.00000 q^{87} +6.00000 q^{89} +4.00000 q^{95} +2.00000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 2q^{15} + 4q^{17} - 2q^{25} + 8q^{33} - 4q^{39} - 20q^{41} + 16q^{47} - 14q^{49} + 8q^{55} + 8q^{57} - 4q^{65} + 16q^{71} - 20q^{73} + 2q^{81} - 4q^{87} + 12q^{89} + 8q^{95} + 4q^{97} + O(q^{100})$$

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
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ 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
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ − 4.00000i − 0.917663i −0.888523 0.458831i $$-0.848268\pi$$
0.888523 0.458831i $$-0.151732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 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
$$29$$ 2.00000i 0.371391i 0.982607 + 0.185695i $$0.0594537\pi$$
−0.982607 + 0.185695i $$0.940546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ − 1.00000i − 0.149071i
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 2.00000i 0.280056i
$$52$$ 0 0
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ − 4.00000i − 0.520756i −0.965507 0.260378i $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ 0 0
$$61$$ 2.00000i 0.256074i 0.991769 + 0.128037i $$0.0408676\pi$$
−0.991769 + 0.128037i $$0.959132\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ − 1.00000i − 0.115470i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 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
$$85$$ 2.00000i 0.216930i
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ 6.00000i 0.597022i 0.954406 + 0.298511i $$0.0964900\pi$$
−0.954406 + 0.298511i $$0.903510\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ − 14.0000i − 1.34096i −0.741929 0.670478i $$-0.766089\pi$$
0.741929 0.670478i $$-0.233911\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ − 10.0000i − 0.901670i
$$124$$ 0 0
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 12.0000i 1.04844i 0.851581 + 0.524222i $$0.175644\pi$$
−0.851581 + 0.524222i $$0.824356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ − 4.00000i − 0.339276i −0.985506 0.169638i $$-0.945740\pi$$
0.985506 0.169638i $$-0.0542598\pi$$
$$140$$ 0 0
$$141$$ 8.00000i 0.673722i
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ 0 0
$$147$$ − 7.00000i − 0.577350i
$$148$$ 0 0
$$149$$ 22.0000i 1.80231i 0.433497 + 0.901155i $$0.357280\pi$$
−0.433497 + 0.901155i $$0.642720\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ 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
$$165$$ 4.00000i 0.311400i
$$166$$ 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
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 0 0
$$179$$ − 20.0000i − 1.49487i −0.664335 0.747435i $$-0.731285\pi$$
0.664335 0.747435i $$-0.268715\pi$$
$$180$$ 0 0
$$181$$ − 10.0000i − 0.743294i −0.928374 0.371647i $$-0.878793\pi$$
0.928374 0.371647i $$-0.121207\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ 10.0000 0.735215
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 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
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ − 10.0000i − 0.698430i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ − 20.0000i − 1.37686i −0.725304 0.688428i $$-0.758301\pi$$
0.725304 0.688428i $$-0.241699\pi$$
$$212$$ 0 0
$$213$$ 8.00000i 0.548151i
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ − 10.0000i − 0.675737i
$$220$$ 0 0
$$221$$ 4.00000i 0.269069i
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 0 0
$$229$$ 6.00000i 0.396491i 0.980152 + 0.198246i $$0.0635244\pi$$
−0.980152 + 0.198246i $$0.936476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 8.00000i 0.521862i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ − 7.00000i − 0.447214i
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 12.0000i 0.757433i 0.925513 + 0.378717i $$0.123635\pi$$
−0.925513 + 0.378717i $$0.876365\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −2.00000 −0.125245
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ − 2.00000i − 0.123797i
$$262$$ 0 0
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ 10.0000 0.614295
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ 0 0
$$269$$ − 14.0000i − 0.853595i −0.904347 0.426798i $$-0.859642\pi$$
0.904347 0.426798i $$-0.140358\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 4.00000i 0.241209i
$$276$$ 0 0
$$277$$ 6.00000i 0.360505i 0.983620 + 0.180253i $$0.0576915\pi$$
−0.983620 + 0.180253i $$0.942309\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ − 12.0000i − 0.713326i −0.934233 0.356663i $$-0.883914\pi$$
0.934233 0.356663i $$-0.116086\pi$$
$$284$$ 0 0
$$285$$ 4.00000i 0.236940i
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 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
$$295$$ 4.00000 0.232889
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −6.00000 −0.344691
$$304$$ 0 0
$$305$$ −2.00000 −0.114520
$$306$$ 0 0
$$307$$ − 28.0000i − 1.59804i −0.601302 0.799022i $$-0.705351\pi$$
0.601302 0.799022i $$-0.294649\pi$$
$$308$$ 0 0
$$309$$ 16.0000i 0.910208i
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ 0 0
$$325$$ − 2.00000i − 0.110940i
$$326$$ 0 0
$$327$$ 14.0000 0.774202
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000i 0.659580i 0.944054 + 0.329790i $$0.106978\pi$$
−0.944054 + 0.329790i $$0.893022\pi$$
$$332$$ 0 0
$$333$$ 10.0000i 0.547997i
$$334$$ 0 0
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 2.00000i 0.108625i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 28.0000i − 1.50312i −0.659665 0.751559i $$-0.729302\pi$$
0.659665 0.751559i $$-0.270698\pi$$
$$348$$ 0 0
$$349$$ 2.00000i 0.107058i 0.998566 + 0.0535288i $$0.0170469\pi$$
−0.998566 + 0.0535288i $$0.982953\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 8.00000i 0.424596i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ − 10.0000i − 0.523424i
$$366$$ 0 0
$$367$$ −24.0000 −1.25279 −0.626395 0.779506i $$-0.715470\pi$$
−0.626395 + 0.779506i $$0.715470\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 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
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ − 20.0000i − 1.02733i −0.857991 0.513665i $$-0.828287\pi$$
0.857991 0.513665i $$-0.171713\pi$$
$$380$$ 0 0
$$381$$ − 8.00000i − 0.409852i
$$382$$ 0 0
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ 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
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 1.00000i 0.0496904i
$$406$$ 0 0
$$407$$ −40.0000 −1.98273
$$408$$ 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
$$419$$ − 4.00000i − 0.195413i −0.995215 0.0977064i $$-0.968849\pi$$
0.995215 0.0977064i $$-0.0311506\pi$$
$$420$$ 0 0
$$421$$ − 26.0000i − 1.26716i −0.773676 0.633581i $$-0.781584\pi$$
0.773676 0.633581i $$-0.218416\pi$$
$$422$$ 0 0
$$423$$ −8.00000 −0.388973
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 8.00000i 0.386244i
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ − 2.00000i − 0.0958927i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ 7.00000 0.333333
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 0 0
$$445$$ 6.00000i 0.284427i
$$446$$ 0 0
$$447$$ −22.0000 −1.04056
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ 40.0000i 1.88353i
$$452$$ 0 0
$$453$$ 8.00000i 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 0 0
$$459$$ − 2.00000i − 0.0933520i
$$460$$ 0 0
$$461$$ 18.0000i 0.838344i 0.907907 + 0.419172i $$0.137680\pi$$
−0.907907 + 0.419172i $$0.862320\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 28.0000i − 1.29569i −0.761774 0.647843i $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 4.00000i 0.183533i
$$476$$ 0 0
$$477$$ 10.0000i 0.457869i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2.00000i 0.0908153i
$$486$$ 0 0
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 28.0000i 1.26362i 0.775122 + 0.631811i $$0.217688\pi$$
−0.775122 + 0.631811i $$0.782312\pi$$
$$492$$ 0 0
$$493$$ 4.00000i 0.180151i
$$494$$ 0 0
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 4.00000i − 0.179065i −0.995984 0.0895323i $$-0.971463\pi$$
0.995984 0.0895323i $$-0.0285372\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 32.0000 1.42681 0.713405 0.700752i $$-0.247152\pi$$
0.713405 + 0.700752i $$0.247152\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 0 0
$$509$$ 34.0000i 1.50702i 0.657434 + 0.753512i $$0.271642\pi$$
−0.657434 + 0.753512i $$0.728358\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ 16.0000i 0.705044i
$$516$$ 0 0
$$517$$ − 32.0000i − 1.40736i
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 4.00000i 0.173585i
$$532$$ 0 0
$$533$$ − 20.0000i − 0.866296i
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 0 0
$$537$$ 20.0000 0.863064
$$538$$ 0 0
$$539$$ 28.0000i 1.20605i
$$540$$ 0 0
$$541$$ − 30.0000i − 1.28980i −0.764267 0.644900i $$-0.776899\pi$$
0.764267 0.644900i $$-0.223101\pi$$
$$542$$ 0 0
$$543$$ 10.0000 0.429141
$$544$$ 0 0
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ 0 0
$$549$$ − 2.00000i − 0.0853579i
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 10.0000i 0.424476i
$$556$$ 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
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 2.00000i 0.0841406i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ − 4.00000i − 0.167395i −0.996491 0.0836974i $$-0.973327\pi$$
0.996491 0.0836974i $$-0.0266729\pi$$
$$572$$ 0 0
$$573$$ 16.0000i 0.668410i
$$574$$ 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
$$579$$ 2.00000i 0.0831172i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −40.0000 −1.65663
$$584$$ 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
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.00000i 0.327418i
$$598$$ 0 0
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ − 5.00000i − 0.203279i
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 16.0000i 0.647291i
$$612$$ 0 0
$$613$$ 22.0000i 0.888572i 0.895885 + 0.444286i $$0.146543\pi$$
−0.895885 + 0.444286i $$0.853457\pi$$
$$614$$ 0 0
$$615$$ 10.0000 0.403239
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ − 4.00000i − 0.160774i −0.996764 0.0803868i $$-0.974384\pi$$
0.996764 0.0803868i $$-0.0256155\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ − 16.0000i − 0.638978i
$$628$$ 0 0
$$629$$ − 20.0000i − 0.797452i
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 0 0
$$635$$ − 8.00000i − 0.317470i
$$636$$ 0 0
$$637$$ − 14.0000i − 0.554700i
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ 36.0000i 1.41970i 0.704352 + 0.709851i $$0.251238\pi$$
−0.704352 + 0.709851i $$0.748762\pi$$
$$644$$ 0 0
$$645$$ − 4.00000i − 0.157500i
$$646$$ 0 0
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 46.0000i − 1.80012i −0.435767 0.900060i $$-0.643523\pi$$
0.435767 0.900060i $$-0.356477\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 0 0
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ − 20.0000i − 0.779089i −0.921008 0.389545i $$-0.872632\pi$$
0.921008 0.389545i $$-0.127368\pi$$
$$660$$ 0 0
$$661$$ 22.0000i 0.855701i 0.903850 + 0.427850i $$0.140729\pi$$
−0.903850 + 0.427850i $$0.859271\pi$$
$$662$$ 0 0
$$663$$ −4.00000 −0.155347
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 8.00000i 0.309298i
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ 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
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 0 0
$$685$$ 6.00000i 0.229248i
$$686$$ 0 0
$$687$$ −6.00000 −0.228914
$$688$$ 0 0
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ 44.0000i 1.67384i 0.547326 + 0.836919i $$0.315646\pi$$
−0.547326 + 0.836919i $$0.684354\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.00000 0.151729
$$696$$ 0 0
$$697$$ −20.0000 −0.757554
$$698$$ 0 0
$$699$$ 6.00000i 0.226941i
$$700$$ 0 0
$$701$$ 2.00000i 0.0755390i 0.999286 + 0.0377695i $$0.0120253\pi$$
−0.999286 + 0.0377695i $$0.987975\pi$$
$$702$$ 0 0
$$703$$ −40.0000 −1.50863
$$704$$ 0 0
$$705$$ −8.00000 −0.301297
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ − 26.0000i − 0.976450i −0.872718 0.488225i $$-0.837644\pi$$
0.872718 0.488225i $$-0.162356\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8.00000i 0.299183i
$$716$$ 0 0
$$717$$ − 16.0000i − 0.597531i
$$718$$ 0 0
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ − 14.0000i − 0.520666i
$$724$$ 0 0
$$725$$ − 2.00000i − 0.0742781i
$$726$$ 0 0
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 8.00000i 0.295891i
$$732$$ 0 0
$$733$$ − 14.0000i − 0.517102i −0.965998 0.258551i $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ 0 0
$$735$$ 7.00000 0.258199
$$736$$ 0 0
$$737$$ −48.0000 −1.76810
$$738$$ 0 0
$$739$$ 44.0000i 1.61857i 0.587419 + 0.809283i $$0.300144\pi$$
−0.587419 + 0.809283i $$0.699856\pi$$
$$740$$ 0 0
$$741$$ 8.00000i 0.293887i
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ −22.0000 −0.806018
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ 8.00000i 0.291150i
$$756$$ 0 0
$$757$$ − 26.0000i − 0.944986i −0.881334 0.472493i $$-0.843354\pi$$
0.881334 0.472493i $$-0.156646\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ − 2.00000i − 0.0723102i
$$766$$ 0 0
$$767$$ 8.00000 0.288863
$$768$$ 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
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 40.0000i 1.43315i
$$780$$ 0 0
$$781$$ − 32.0000i − 1.14505i
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ 14.0000 0.499681
$$786$$ 0 0
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ 0 0
$$789$$ − 16.0000i − 0.569615i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ 10.0000i 0.354663i
$$796$$ 0 0
$$797$$ 2.00000i 0.0708436i 0.999372 + 0.0354218i $$0.0112775\pi$$
−0.999372 + 0.0354218i $$0.988723\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 40.0000i 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 14.0000 0.492823
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ 12.0000i 0.421377i 0.977553 + 0.210688i $$0.0675706\pi$$
−0.977553 + 0.210688i $$0.932429\pi$$
$$812$$ 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
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 54.0000i 1.88461i 0.334751 + 0.942306i $$0.391348\pi$$
−0.334751 + 0.942306i $$0.608652\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ 0 0
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ 0 0
$$829$$ − 30.0000i − 1.04194i −0.853574 0.520972i $$-0.825570\pi$$
0.853574 0.520972i $$-0.174430\pi$$
$$830$$ 0 0
$$831$$ −6.00000 −0.208138
$$832$$ 0 0
$$833$$ −14.0000 −0.485071
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ 25.0000 0.862069
$$842$$ 0 0
$$843$$ 6.00000i 0.206651i
$$844$$ 0 0
$$845$$ 9.00000i 0.309609i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 6.00000i 0.205436i 0.994711 + 0.102718i $$0.0327539\pi$$
−0.994711 + 0.102718i $$0.967246\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ 0 0
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 0 0
$$859$$ − 20.0000i − 0.682391i −0.939992 0.341196i $$-0.889168\pi$$
0.939992 0.341196i $$-0.110832\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −56.0000 −1.90626 −0.953131 0.302558i $$-0.902160\pi$$
−0.953131 + 0.302558i $$0.902160\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 30.0000i − 1.01303i −0.862232 0.506514i $$-0.830934\pi$$
0.862232 0.506514i $$-0.169066\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ 0 0
$$883$$ − 44.0000i − 1.48072i −0.672212 0.740359i $$-0.734656\pi$$
0.672212 0.740359i $$-0.265344\pi$$
$$884$$ 0 0
$$885$$ 4.00000i 0.134459i
$$886$$ 0 0
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ − 32.0000i − 1.07084i
$$894$$ 0 0
$$895$$ 20.0000 0.668526
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ − 20.0000i − 0.666297i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ − 12.0000i − 0.398453i −0.979953 0.199227i $$-0.936157\pi$$
0.979953 0.199227i $$-0.0638430\pi$$
$$908$$ 0 0
$$909$$ − 6.00000i − 0.199007i
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 0 0
$$915$$ − 2.00000i − 0.0661180i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ 0 0
$$923$$ 16.0000i 0.526646i
$$924$$ 0 0
$$925$$ 10.0000i 0.328798i
$$926$$ 0 0
$$927$$ −16.0000 −0.525509
$$928$$ 0 0
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ 28.0000i 0.917663i
$$932$$ 0 0
$$933$$ 24.0000i 0.785725i
$$934$$ 0 0
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ 54.0000 1.76410 0.882052 0.471153i $$-0.156162\pi$$
0.882052 + 0.471153i $$0.156162\pi$$
$$938$$ 0 0
$$939$$ − 26.0000i − 0.848478i
$$940$$ 0 0
$$941$$ 50.0000i 1.62995i 0.579494 + 0.814977i $$0.303250\pi$$
−0.579494 + 0.814977i $$0.696750\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 0 0
$$949$$ − 20.0000i − 0.649227i
$$950$$ 0 0
$$951$$ −2.00000 −0.0648544
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 0 0
$$955$$ 16.0000i 0.517748i
$$956$$ 0 0
$$957$$ 8.00000i 0.258603i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 0 0
$$965$$ 2.00000i 0.0643823i
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 60.0000i 1.92549i 0.270408 + 0.962746i $$0.412841\pi$$
−0.270408 + 0.962746i $$0.587159\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 2.00000 0.0640513
$$976$$ 0 0
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ 0 0
$$979$$ − 24.0000i − 0.767043i
$$980$$ 0 0
$$981$$ 14.0000i 0.446986i
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ −12.0000 −0.380808
$$994$$ 0 0
$$995$$ 8.00000i 0.253617i
$$996$$ 0 0
$$997$$ 54.0000i 1.71020i 0.518465 + 0.855099i $$0.326503\pi$$
−0.518465 + 0.855099i $$0.673497\pi$$
$$998$$ 0 0
$$999$$ −10.0000 −0.316386
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.m.1921.2 2
4.3 odd 2 3840.2.k.r.1921.1 2
8.3 odd 2 3840.2.k.r.1921.2 2
8.5 even 2 inner 3840.2.k.m.1921.1 2
16.3 odd 4 240.2.a.d.1.1 1
16.5 even 4 960.2.a.l.1.1 1
16.11 odd 4 960.2.a.a.1.1 1
16.13 even 4 15.2.a.a.1.1 1
48.5 odd 4 2880.2.a.y.1.1 1
48.11 even 4 2880.2.a.bc.1.1 1
48.29 odd 4 45.2.a.a.1.1 1
48.35 even 4 720.2.a.c.1.1 1
80.3 even 4 1200.2.f.h.49.2 2
80.13 odd 4 75.2.b.b.49.2 2
80.19 odd 4 1200.2.a.e.1.1 1
80.27 even 4 4800.2.f.c.3649.2 2
80.29 even 4 75.2.a.b.1.1 1
80.37 odd 4 4800.2.f.bf.3649.1 2
80.43 even 4 4800.2.f.c.3649.1 2
80.53 odd 4 4800.2.f.bf.3649.2 2
80.59 odd 4 4800.2.a.bz.1.1 1
80.67 even 4 1200.2.f.h.49.1 2
80.69 even 4 4800.2.a.t.1.1 1
80.77 odd 4 75.2.b.b.49.1 2
112.13 odd 4 735.2.a.c.1.1 1
112.45 odd 12 735.2.i.d.226.1 2
112.61 odd 12 735.2.i.d.361.1 2
112.93 even 12 735.2.i.e.361.1 2
112.109 even 12 735.2.i.e.226.1 2
144.13 even 12 405.2.e.f.136.1 2
144.29 odd 12 405.2.e.c.271.1 2
144.61 even 12 405.2.e.f.271.1 2
144.77 odd 12 405.2.e.c.136.1 2
176.109 odd 4 1815.2.a.d.1.1 1
208.77 even 4 2535.2.a.j.1.1 1
240.29 odd 4 225.2.a.b.1.1 1
240.77 even 4 225.2.b.b.199.2 2
240.83 odd 4 3600.2.f.e.2449.1 2
240.173 even 4 225.2.b.b.199.1 2
240.179 even 4 3600.2.a.u.1.1 1
240.227 odd 4 3600.2.f.e.2449.2 2
272.237 even 4 4335.2.a.c.1.1 1
304.189 odd 4 5415.2.a.j.1.1 1
336.125 even 4 2205.2.a.i.1.1 1
368.45 odd 4 7935.2.a.d.1.1 1
528.461 even 4 5445.2.a.c.1.1 1
560.349 odd 4 3675.2.a.j.1.1 1
624.77 odd 4 7605.2.a.g.1.1 1
880.109 odd 4 9075.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 16.13 even 4
45.2.a.a.1.1 1 48.29 odd 4
75.2.a.b.1.1 1 80.29 even 4
75.2.b.b.49.1 2 80.77 odd 4
75.2.b.b.49.2 2 80.13 odd 4
225.2.a.b.1.1 1 240.29 odd 4
225.2.b.b.199.1 2 240.173 even 4
225.2.b.b.199.2 2 240.77 even 4
240.2.a.d.1.1 1 16.3 odd 4
405.2.e.c.136.1 2 144.77 odd 12
405.2.e.c.271.1 2 144.29 odd 12
405.2.e.f.136.1 2 144.13 even 12
405.2.e.f.271.1 2 144.61 even 12
720.2.a.c.1.1 1 48.35 even 4
735.2.a.c.1.1 1 112.13 odd 4
735.2.i.d.226.1 2 112.45 odd 12
735.2.i.d.361.1 2 112.61 odd 12
735.2.i.e.226.1 2 112.109 even 12
735.2.i.e.361.1 2 112.93 even 12
960.2.a.a.1.1 1 16.11 odd 4
960.2.a.l.1.1 1 16.5 even 4
1200.2.a.e.1.1 1 80.19 odd 4
1200.2.f.h.49.1 2 80.67 even 4
1200.2.f.h.49.2 2 80.3 even 4
1815.2.a.d.1.1 1 176.109 odd 4
2205.2.a.i.1.1 1 336.125 even 4
2535.2.a.j.1.1 1 208.77 even 4
2880.2.a.y.1.1 1 48.5 odd 4
2880.2.a.bc.1.1 1 48.11 even 4
3600.2.a.u.1.1 1 240.179 even 4
3600.2.f.e.2449.1 2 240.83 odd 4
3600.2.f.e.2449.2 2 240.227 odd 4
3675.2.a.j.1.1 1 560.349 odd 4
3840.2.k.m.1921.1 2 8.5 even 2 inner
3840.2.k.m.1921.2 2 1.1 even 1 trivial
3840.2.k.r.1921.1 2 4.3 odd 2
3840.2.k.r.1921.2 2 8.3 odd 2
4335.2.a.c.1.1 1 272.237 even 4
4800.2.a.t.1.1 1 80.69 even 4
4800.2.a.bz.1.1 1 80.59 odd 4
4800.2.f.c.3649.1 2 80.43 even 4
4800.2.f.c.3649.2 2 80.27 even 4
4800.2.f.bf.3649.1 2 80.37 odd 4
4800.2.f.bf.3649.2 2 80.53 odd 4
5415.2.a.j.1.1 1 304.189 odd 4
5445.2.a.c.1.1 1 528.461 even 4
7605.2.a.g.1.1 1 624.77 odd 4
7935.2.a.d.1.1 1 368.45 odd 4
9075.2.a.g.1.1 1 880.109 odd 4