# Properties

 Label 2304.2.d.r.1153.1 Level $2304$ Weight $2$ Character 2304.1153 Analytic conductor $18.398$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1153.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.1153 Dual form 2304.2.d.r.1153.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000i q^{5} +4.00000 q^{7} +O(q^{10})$$ $$q-2.00000i q^{5} +4.00000 q^{7} +2.00000i q^{11} -2.00000i q^{13} +2.00000 q^{17} -2.00000i q^{19} +4.00000 q^{23} +1.00000 q^{25} -6.00000i q^{29} -8.00000i q^{35} +10.0000i q^{37} -6.00000 q^{41} +6.00000i q^{43} +8.00000 q^{47} +9.00000 q^{49} +6.00000i q^{53} +4.00000 q^{55} -14.0000i q^{59} -2.00000i q^{61} -4.00000 q^{65} -10.0000i q^{67} +12.0000 q^{71} -14.0000 q^{73} +8.00000i q^{77} -8.00000 q^{79} -6.00000i q^{83} -4.00000i q^{85} -2.00000 q^{89} -8.00000i q^{91} -4.00000 q^{95} -2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{7} + O(q^{10})$$ $$2q + 8q^{7} + 4q^{17} + 8q^{23} + 2q^{25} - 12q^{41} + 16q^{47} + 18q^{49} + 8q^{55} - 8q^{65} + 24q^{71} - 28q^{73} - 16q^{79} - 4q^{89} - 8q^{95} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ − 2.00000i − 0.894427i −0.894427 0.447214i $$-0.852416\pi$$
0.894427 0.447214i $$-0.147584\pi$$
$$6$$ 0 0
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.00000i 0.603023i 0.953463 + 0.301511i $$0.0974911\pi$$
−0.953463 + 0.301511i $$0.902509\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ − 2.00000i − 0.458831i −0.973329 0.229416i $$-0.926318\pi$$
0.973329 0.229416i $$-0.0736815\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 6.00000i − 1.11417i −0.830455 0.557086i $$-0.811919\pi$$
0.830455 0.557086i $$-0.188081\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 8.00000i − 1.35225i
$$36$$ 0 0
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ 0 0
$$39$$ 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
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 14.0000i − 1.82264i −0.411693 0.911322i $$-0.635063\pi$$
0.411693 0.911322i $$-0.364937\pi$$
$$60$$ 0 0
$$61$$ − 2.00000i − 0.256074i −0.991769 0.128037i $$-0.959132\pi$$
0.991769 0.128037i $$-0.0408676\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ − 10.0000i − 1.22169i −0.791748 0.610847i $$-0.790829\pi$$
0.791748 0.610847i $$-0.209171\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.00000i 0.911685i
$$78$$ 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
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ − 4.00000i − 0.433861i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ − 8.00000i − 0.838628i
$$92$$ 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 $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000i 0.597022i 0.954406 + 0.298511i $$0.0964900\pi$$
−0.954406 + 0.298511i $$0.903510\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.00000i 0.193347i 0.995316 + 0.0966736i $$0.0308203\pi$$
−0.995316 + 0.0966736i $$0.969180\pi$$
$$108$$ 0 0
$$109$$ 6.00000i 0.574696i 0.957826 + 0.287348i $$0.0927736\pi$$
−0.957826 + 0.287348i $$0.907226\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ − 8.00000i − 0.746004i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ 7.00000 0.636364
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 12.0000i − 1.07331i
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 6.00000i − 0.524222i −0.965038 0.262111i $$-0.915581\pi$$
0.965038 0.262111i $$-0.0844187\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ − 10.0000i − 0.848189i −0.905618 0.424094i $$-0.860592\pi$$
0.905618 0.424094i $$-0.139408\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 18.0000i − 1.47462i −0.675556 0.737309i $$-0.736096\pi$$
0.675556 0.737309i $$-0.263904\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ − 2.00000i − 0.156652i −0.996928 0.0783260i $$-0.975042\pi$$
0.996928 0.0783260i $$-0.0249575\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −20.0000 −1.54765 −0.773823 0.633402i $$-0.781658\pi$$
−0.773823 + 0.633402i $$0.781658\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 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
$$179$$ − 6.00000i − 0.448461i −0.974536 0.224231i $$-0.928013\pi$$
0.974536 0.224231i $$-0.0719869\pi$$
$$180$$ 0 0
$$181$$ 2.00000i 0.148659i 0.997234 + 0.0743294i $$0.0236816\pi$$
−0.997234 + 0.0743294i $$0.976318\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 20.0000 1.47043
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ 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 $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.0000i 0.997459i 0.866758 + 0.498729i $$0.166200\pi$$
−0.866758 + 0.498729i $$0.833800\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 24.0000i − 1.68447i
$$204$$ 0 0
$$205$$ 12.0000i 0.838116i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 22.0000i 1.51454i 0.653101 + 0.757271i $$0.273468\pi$$
−0.653101 + 0.757271i $$0.726532\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 12.0000 0.818393
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 4.00000i − 0.269069i
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ 0 0
$$229$$ − 14.0000i − 0.925146i −0.886581 0.462573i $$-0.846926\pi$$
0.886581 0.462573i $$-0.153074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ − 16.0000i − 1.04372i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 18.0000i − 1.14998i
$$246$$ 0 0
$$247$$ −4.00000 −0.254514
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 18.0000i 1.13615i 0.822977 + 0.568075i $$0.192312\pi$$
−0.822977 + 0.568075i $$0.807688\pi$$
$$252$$ 0 0
$$253$$ 8.00000i 0.502956i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 40.0000i 2.48548i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ 12.0000 0.737154
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 10.0000i 0.609711i 0.952399 + 0.304855i $$0.0986081\pi$$
−0.952399 + 0.304855i $$0.901392\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2.00000i 0.120605i
$$276$$ 0 0
$$277$$ − 6.00000i − 0.360505i −0.983620 0.180253i $$-0.942309\pi$$
0.983620 0.180253i $$-0.0576915\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ 6.00000i 0.356663i 0.983970 + 0.178331i $$0.0570699\pi$$
−0.983970 + 0.178331i $$0.942930\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −24.0000 −1.41668
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ 0 0
$$295$$ −28.0000 −1.63022
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 8.00000i − 0.462652i
$$300$$ 0 0
$$301$$ 24.0000i 1.38334i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −4.00000 −0.229039
$$306$$ 0 0
$$307$$ − 18.0000i − 1.02731i −0.857996 0.513657i $$-0.828290\pi$$
0.857996 0.513657i $$-0.171710\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −28.0000 −1.58773 −0.793867 0.608091i $$-0.791935\pi$$
−0.793867 + 0.608091i $$0.791935\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 4.00000i − 0.222566i
$$324$$ 0 0
$$325$$ − 2.00000i − 0.110940i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ 14.0000i 0.769510i 0.923019 + 0.384755i $$0.125714\pi$$
−0.923019 + 0.384755i $$0.874286\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −20.0000 −1.09272
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.0000i 0.966291i 0.875540 + 0.483145i $$0.160506\pi$$
−0.875540 + 0.483145i $$0.839494\pi$$
$$348$$ 0 0
$$349$$ − 10.0000i − 0.535288i −0.963518 0.267644i $$-0.913755\pi$$
0.963518 0.267644i $$-0.0862451\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ − 24.0000i − 1.27379i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ 15.0000 0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 28.0000i 1.46559i
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 24.0000i 1.24602i
$$372$$ 0 0
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ − 2.00000i − 0.102733i −0.998680 0.0513665i $$-0.983642\pi$$
0.998680 0.0513665i $$-0.0163577\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 16.0000 0.815436
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 10.0000i − 0.507020i −0.967333 0.253510i $$-0.918415\pi$$
0.967333 0.253510i $$-0.0815851\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 16.0000i 0.805047i
$$396$$ 0 0
$$397$$ 6.00000i 0.301131i 0.988600 + 0.150566i $$0.0481095\pi$$
−0.988600 + 0.150566i $$0.951890\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −20.0000 −0.991363
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 56.0000i − 2.75558i
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 26.0000i 1.27018i 0.772437 + 0.635092i $$0.219038\pi$$
−0.772437 + 0.635092i $$0.780962\pi$$
$$420$$ 0 0
$$421$$ 34.0000i 1.65706i 0.559946 + 0.828529i $$0.310822\pi$$
−0.559946 + 0.828529i $$0.689178\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ − 8.00000i − 0.387147i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ 0 0
$$433$$ 30.0000 1.44171 0.720854 0.693087i $$-0.243750\pi$$
0.720854 + 0.693087i $$0.243750\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 8.00000i − 0.382692i
$$438$$ 0 0
$$439$$ −36.0000 −1.71819 −0.859093 0.511819i $$-0.828972\pi$$
−0.859093 + 0.511819i $$0.828972\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 6.00000i − 0.285069i −0.989790 0.142534i $$-0.954475\pi$$
0.989790 0.142534i $$-0.0455251\pi$$
$$444$$ 0 0
$$445$$ 4.00000i 0.189618i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ 0 0
$$451$$ − 12.0000i − 0.565058i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −16.0000 −0.750092
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.0000i 0.465746i 0.972507 + 0.232873i $$0.0748127\pi$$
−0.972507 + 0.232873i $$0.925187\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 14.0000i − 0.647843i −0.946084 0.323921i $$-0.894999\pi$$
0.946084 0.323921i $$-0.105001\pi$$
$$468$$ 0 0
$$469$$ − 40.0000i − 1.84703i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −12.0000 −0.551761
$$474$$ 0 0
$$475$$ − 2.00000i − 0.0917663i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 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
$$485$$ 4.00000i 0.181631i
$$486$$ 0 0
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 10.0000i 0.451294i 0.974209 + 0.225647i $$0.0724495\pi$$
−0.974209 + 0.225647i $$0.927550\pi$$
$$492$$ 0 0
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 48.0000 2.15309
$$498$$ 0 0
$$499$$ 22.0000i 0.984855i 0.870353 + 0.492428i $$0.163890\pi$$
−0.870353 + 0.492428i $$0.836110\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 14.0000i − 0.620539i −0.950649 0.310270i $$-0.899581\pi$$
0.950649 0.310270i $$-0.100419\pi$$
$$510$$ 0 0
$$511$$ −56.0000 −2.47729
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 8.00000i − 0.352522i
$$516$$ 0 0
$$517$$ 16.0000i 0.703679i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ 14.0000i 0.612177i 0.952003 + 0.306089i $$0.0990204\pi$$
−0.952003 + 0.306089i $$0.900980\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 4.00000 0.172935
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 18.0000i 0.775315i
$$540$$ 0 0
$$541$$ − 34.0000i − 1.46177i −0.682498 0.730887i $$-0.739107\pi$$
0.682498 0.730887i $$-0.260893\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 12.0000 0.514024
$$546$$ 0 0
$$547$$ 38.0000i 1.62476i 0.583127 + 0.812381i $$0.301829\pi$$
−0.583127 + 0.812381i $$0.698171\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ −32.0000 −1.36078
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ 0 0
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 18.0000i 0.758610i 0.925272 + 0.379305i $$0.123837\pi$$
−0.925272 + 0.379305i $$0.876163\pi$$
$$564$$ 0 0
$$565$$ 4.00000i 0.168281i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ 38.0000i 1.59025i 0.606445 + 0.795125i $$0.292595\pi$$
−0.606445 + 0.795125i $$0.707405\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 24.0000i − 0.995688i
$$582$$ 0 0
$$583$$ −12.0000 −0.496989
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 34.0000i 1.40333i 0.712507 + 0.701665i $$0.247560\pi$$
−0.712507 + 0.701665i $$0.752440\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ − 16.0000i − 0.655936i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ −30.0000 −1.22373 −0.611863 0.790964i $$-0.709580\pi$$
−0.611863 + 0.790964i $$0.709580\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 14.0000i − 0.569181i
$$606$$ 0 0
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 16.0000i − 0.647291i
$$612$$ 0 0
$$613$$ 34.0000i 1.37325i 0.727013 + 0.686624i $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ 0 0
$$619$$ 46.0000i 1.84890i 0.381308 + 0.924448i $$0.375474\pi$$
−0.381308 + 0.924448i $$0.624526\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −8.00000 −0.320513
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 20.0000i 0.797452i
$$630$$ 0 0
$$631$$ 44.0000 1.75161 0.875806 0.482663i $$-0.160330\pi$$
0.875806 + 0.482663i $$0.160330\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 32.0000i − 1.26988i
$$636$$ 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
$$643$$ − 42.0000i − 1.65632i −0.560493 0.828159i $$-0.689388\pi$$
0.560493 0.828159i $$-0.310612\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 28.0000 1.09910
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 42.0000i 1.64359i 0.569785 + 0.821794i $$0.307026\pi$$
−0.569785 + 0.821794i $$0.692974\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 6.00000i − 0.233727i −0.993148 0.116863i $$-0.962716\pi$$
0.993148 0.116863i $$-0.0372840\pi$$
$$660$$ 0 0
$$661$$ 34.0000i 1.32245i 0.750189 + 0.661223i $$0.229962\pi$$
−0.750189 + 0.661223i $$0.770038\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −16.0000 −0.620453
$$666$$ 0 0
$$667$$ − 24.0000i − 0.929284i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 22.0000i 0.845529i 0.906240 + 0.422764i $$0.138940\pi$$
−0.906240 + 0.422764i $$0.861060\pi$$
$$678$$ 0 0
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 42.0000i 1.60709i 0.595247 + 0.803543i $$0.297054\pi$$
−0.595247 + 0.803543i $$0.702946\pi$$
$$684$$ 0 0
$$685$$ − 20.0000i − 0.764161i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 6.00000i 0.228251i 0.993466 + 0.114125i $$0.0364066\pi$$
−0.993466 + 0.114125i $$0.963593\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −20.0000 −0.758643
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.00000i 0.0755390i 0.999286 + 0.0377695i $$0.0120253\pi$$
−0.999286 + 0.0377695i $$0.987975\pi$$
$$702$$ 0 0
$$703$$ 20.0000 0.754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 24.0000i 0.902613i
$$708$$ 0 0
$$709$$ − 22.0000i − 0.826227i −0.910679 0.413114i $$-0.864441\pi$$
0.910679 0.413114i $$-0.135559\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ − 8.00000i − 0.299183i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 6.00000i − 0.222834i
$$726$$ 0 0
$$727$$ 12.0000 0.445055 0.222528 0.974926i $$-0.428569\pi$$
0.222528 + 0.974926i $$0.428569\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 12.0000i 0.443836i
$$732$$ 0 0
$$733$$ 6.00000i 0.221615i 0.993842 + 0.110808i $$0.0353437\pi$$
−0.993842 + 0.110808i $$0.964656\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 20.0000 0.736709
$$738$$ 0 0
$$739$$ − 18.0000i − 0.662141i −0.943606 0.331070i $$-0.892590\pi$$
0.943606 0.331070i $$-0.107410\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 44.0000 1.61420 0.807102 0.590412i $$-0.201035\pi$$
0.807102 + 0.590412i $$0.201035\pi$$
$$744$$ 0 0
$$745$$ −36.0000 −1.31894
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 8.00000i 0.292314i
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 8.00000i 0.291150i
$$756$$ 0 0
$$757$$ − 46.0000i − 1.67190i −0.548807 0.835949i $$-0.684918\pi$$
0.548807 0.835949i $$-0.315082\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ 24.0000i 0.868858i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −28.0000 −1.01102
$$768$$ 0 0
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 54.0000i 1.94225i 0.238581 + 0.971123i $$0.423318\pi$$
−0.238581 + 0.971123i $$0.576682\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.0000i 0.429945i
$$780$$ 0 0
$$781$$ 24.0000i 0.858788i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −36.0000 −1.28490
$$786$$ 0 0
$$787$$ 22.0000i 0.784215i 0.919919 + 0.392108i $$0.128254\pi$$
−0.919919 + 0.392108i $$0.871746\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −8.00000 −0.284447
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 18.0000i 0.637593i 0.947823 + 0.318796i $$0.103279\pi$$
−0.947823 + 0.318796i $$0.896721\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 28.0000i − 0.988099i
$$804$$ 0 0
$$805$$ − 32.0000i − 1.12785i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ − 18.0000i − 0.632065i −0.948748 0.316033i $$-0.897649\pi$$
0.948748 0.316033i $$-0.102351\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 12.0000 0.419827
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 10.0000i − 0.349002i −0.984657 0.174501i $$-0.944169\pi$$
0.984657 0.174501i $$-0.0558313\pi$$
$$822$$ 0 0
$$823$$ 28.0000 0.976019 0.488009 0.872838i $$-0.337723\pi$$
0.488009 + 0.872838i $$0.337723\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 22.0000i − 0.765015i −0.923952 0.382507i $$-0.875061\pi$$
0.923952 0.382507i $$-0.124939\pi$$
$$828$$ 0 0
$$829$$ 14.0000i 0.486240i 0.969996 + 0.243120i $$0.0781709\pi$$
−0.969996 + 0.243120i $$0.921829\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 40.0000i 1.38426i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 18.0000i − 0.619219i
$$846$$ 0 0
$$847$$ 28.0000 0.962091
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 40.0000i 1.37118i
$$852$$ 0 0
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ − 50.0000i − 1.70598i −0.521929 0.852989i $$-0.674787\pi$$
0.521929 0.852989i $$-0.325213\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 32.0000 1.08929 0.544646 0.838666i $$-0.316664\pi$$
0.544646 + 0.838666i $$0.316664\pi$$
$$864$$ 0 0
$$865$$ 36.0000 1.22404
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 16.0000i − 0.542763i
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 48.0000i − 1.62270i
$$876$$ 0 0
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 46.0000 1.54978 0.774890 0.632096i $$-0.217805\pi$$
0.774890 + 0.632096i $$0.217805\pi$$
$$882$$ 0 0
$$883$$ − 34.0000i − 1.14419i −0.820187 0.572096i $$-0.806131\pi$$
0.820187 0.572096i $$-0.193869\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 0 0
$$889$$ 64.0000 2.14649
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 16.0000i − 0.535420i
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000i 0.399778i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 4.00000 0.132964
$$906$$ 0 0
$$907$$ 38.0000i 1.26177i 0.775877 + 0.630885i $$0.217308\pi$$
−0.775877 + 0.630885i $$0.782692\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 24.0000i − 0.792550i
$$918$$ 0 0
$$919$$ −36.0000 −1.18753 −0.593765 0.804638i $$-0.702359\pi$$
−0.593765 + 0.804638i $$0.702359\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 24.0000i − 0.789970i
$$924$$ 0 0
$$925$$ 10.0000i 0.328798i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ − 18.0000i − 0.589926i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ −46.0000 −1.50275 −0.751377 0.659873i $$-0.770610\pi$$
−0.751377 + 0.659873i $$0.770610\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 38.0000i − 1.23876i −0.785090 0.619382i $$-0.787383\pi$$
0.785090 0.619382i $$-0.212617\pi$$
$$942$$ 0 0
$$943$$ −24.0000 −0.781548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 14.0000i − 0.454939i −0.973785 0.227469i $$-0.926955\pi$$
0.973785 0.227469i $$-0.0730452\pi$$
$$948$$ 0 0
$$949$$ 28.0000i 0.908918i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 58.0000 1.87880 0.939402 0.342817i $$-0.111381\pi$$
0.939402 + 0.342817i $$0.111381\pi$$
$$954$$ 0 0
$$955$$ − 32.0000i − 1.03550i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 40.0000 1.29167
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 4.00000i 0.128765i
$$966$$ 0 0
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 38.0000i − 1.21948i −0.792602 0.609739i $$-0.791274\pi$$
0.792602 0.609739i $$-0.208726\pi$$
$$972$$ 0 0
$$973$$ − 40.0000i − 1.28234i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ − 4.00000i − 0.127841i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 20.0000 0.637901 0.318950 0.947771i $$-0.396670\pi$$
0.318950 + 0.947771i $$0.396670\pi$$
$$984$$ 0 0
$$985$$ 28.0000 0.892154
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 24.0000i 0.763156i
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 8.00000i − 0.253617i
$$996$$ 0 0
$$997$$ − 54.0000i − 1.71020i −0.518465 0.855099i $$-0.673497\pi$$
0.518465 0.855099i $$-0.326503\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2304.2.d.r.1153.1 2
3.2 odd 2 256.2.b.c.129.1 2
4.3 odd 2 2304.2.d.b.1153.1 2
8.3 odd 2 2304.2.d.b.1153.2 2
8.5 even 2 inner 2304.2.d.r.1153.2 2
12.11 even 2 256.2.b.a.129.2 2
16.3 odd 4 1152.2.a.h.1.1 1
16.5 even 4 1152.2.a.m.1.1 1
16.11 odd 4 1152.2.a.r.1.1 1
16.13 even 4 1152.2.a.c.1.1 1
24.5 odd 2 256.2.b.c.129.2 2
24.11 even 2 256.2.b.a.129.1 2
48.5 odd 4 128.2.a.a.1.1 1
48.11 even 4 128.2.a.c.1.1 yes 1
48.29 odd 4 128.2.a.d.1.1 yes 1
48.35 even 4 128.2.a.b.1.1 yes 1
96.5 odd 8 1024.2.e.i.257.2 4
96.11 even 8 1024.2.e.m.257.2 4
96.29 odd 8 1024.2.e.i.769.2 4
96.35 even 8 1024.2.e.m.769.1 4
96.53 odd 8 1024.2.e.i.257.1 4
96.59 even 8 1024.2.e.m.257.1 4
96.77 odd 8 1024.2.e.i.769.1 4
96.83 even 8 1024.2.e.m.769.2 4
240.29 odd 4 3200.2.a.h.1.1 1
240.53 even 4 3200.2.c.l.2049.1 2
240.59 even 4 3200.2.a.e.1.1 1
240.77 even 4 3200.2.c.e.2049.1 2
240.83 odd 4 3200.2.c.k.2049.1 2
240.107 odd 4 3200.2.c.f.2049.1 2
240.149 odd 4 3200.2.a.x.1.1 1
240.173 even 4 3200.2.c.e.2049.2 2
240.179 even 4 3200.2.a.u.1.1 1
240.197 even 4 3200.2.c.l.2049.2 2
240.203 odd 4 3200.2.c.f.2049.2 2
240.227 odd 4 3200.2.c.k.2049.2 2
336.83 odd 4 6272.2.a.g.1.1 1
336.125 even 4 6272.2.a.a.1.1 1
336.251 odd 4 6272.2.a.b.1.1 1
336.293 even 4 6272.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.a.a.1.1 1 48.5 odd 4
128.2.a.b.1.1 yes 1 48.35 even 4
128.2.a.c.1.1 yes 1 48.11 even 4
128.2.a.d.1.1 yes 1 48.29 odd 4
256.2.b.a.129.1 2 24.11 even 2
256.2.b.a.129.2 2 12.11 even 2
256.2.b.c.129.1 2 3.2 odd 2
256.2.b.c.129.2 2 24.5 odd 2
1024.2.e.i.257.1 4 96.53 odd 8
1024.2.e.i.257.2 4 96.5 odd 8
1024.2.e.i.769.1 4 96.77 odd 8
1024.2.e.i.769.2 4 96.29 odd 8
1024.2.e.m.257.1 4 96.59 even 8
1024.2.e.m.257.2 4 96.11 even 8
1024.2.e.m.769.1 4 96.35 even 8
1024.2.e.m.769.2 4 96.83 even 8
1152.2.a.c.1.1 1 16.13 even 4
1152.2.a.h.1.1 1 16.3 odd 4
1152.2.a.m.1.1 1 16.5 even 4
1152.2.a.r.1.1 1 16.11 odd 4
2304.2.d.b.1153.1 2 4.3 odd 2
2304.2.d.b.1153.2 2 8.3 odd 2
2304.2.d.r.1153.1 2 1.1 even 1 trivial
2304.2.d.r.1153.2 2 8.5 even 2 inner
3200.2.a.e.1.1 1 240.59 even 4
3200.2.a.h.1.1 1 240.29 odd 4
3200.2.a.u.1.1 1 240.179 even 4
3200.2.a.x.1.1 1 240.149 odd 4
3200.2.c.e.2049.1 2 240.77 even 4
3200.2.c.e.2049.2 2 240.173 even 4
3200.2.c.f.2049.1 2 240.107 odd 4
3200.2.c.f.2049.2 2 240.203 odd 4
3200.2.c.k.2049.1 2 240.83 odd 4
3200.2.c.k.2049.2 2 240.227 odd 4
3200.2.c.l.2049.1 2 240.53 even 4
3200.2.c.l.2049.2 2 240.197 even 4
6272.2.a.a.1.1 1 336.125 even 4
6272.2.a.b.1.1 1 336.251 odd 4
6272.2.a.g.1.1 1 336.83 odd 4
6272.2.a.h.1.1 1 336.293 even 4