# Properties

 Label 1152.2.d.a.577.2 Level $1152$ Weight $2$ Character 1152.577 Analytic conductor $9.199$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(577,1152)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1152.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$9.19876631285$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 577.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1152.577 Dual form 1152.2.d.a.577.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{7} +O(q^{10})$$ $$q-4.00000 q^{7} +4.00000i q^{11} -4.00000i q^{13} +2.00000 q^{17} -4.00000i q^{19} +8.00000 q^{23} +5.00000 q^{25} -8.00000i q^{29} +4.00000 q^{31} -4.00000i q^{37} +6.00000 q^{41} -4.00000i q^{43} -8.00000 q^{47} +9.00000 q^{49} +8.00000i q^{53} -12.0000i q^{59} -12.0000i q^{61} +12.0000i q^{67} -8.00000 q^{71} +6.00000 q^{73} -16.0000i q^{77} +4.00000 q^{79} +4.00000i q^{83} -6.00000 q^{89} +16.0000i q^{91} -2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7}+O(q^{10})$$ 2 * q - 8 * q^7 $$2 q - 8 q^{7} + 4 q^{17} + 16 q^{23} + 10 q^{25} + 8 q^{31} + 12 q^{41} - 16 q^{47} + 18 q^{49} - 16 q^{71} + 12 q^{73} + 8 q^{79} - 12 q^{89} - 4 q^{97}+O(q^{100})$$ 2 * q - 8 * q^7 + 4 * q^17 + 16 * q^23 + 10 * q^25 + 8 * q^31 + 12 * q^41 - 16 * q^47 + 18 * q^49 - 16 * q^71 + 12 * q^73 + 8 * q^79 - 12 * q^89 - 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.00000i 1.20605i 0.797724 + 0.603023i $$0.206037\pi$$
−0.797724 + 0.603023i $$0.793963\pi$$
$$12$$ 0 0
$$13$$ − 4.00000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ 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
$$19$$ − 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
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 8.00000i − 1.48556i −0.669534 0.742781i $$-0.733506\pi$$
0.669534 0.742781i $$-0.266494\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 4.00000i − 0.657596i −0.944400 0.328798i $$-0.893356\pi$$
0.944400 0.328798i $$-0.106644\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 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
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 8.00000i 1.09888i 0.835532 + 0.549442i $$0.185160\pi$$
−0.835532 + 0.549442i $$0.814840\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 12.0000i − 1.56227i −0.624364 0.781133i $$-0.714642\pi$$
0.624364 0.781133i $$-0.285358\pi$$
$$60$$ 0 0
$$61$$ − 12.0000i − 1.53644i −0.640184 0.768221i $$-0.721142\pi$$
0.640184 0.768221i $$-0.278858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 16.0000i − 1.82337i
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 16.0000i 1.67726i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 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
$$101$$ 8.00000i 0.796030i 0.917379 + 0.398015i $$0.130301\pi$$
−0.917379 + 0.398015i $$0.869699\pi$$
$$102$$ 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
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ − 4.00000i − 0.383131i −0.981480 0.191565i $$-0.938644\pi$$
0.981480 0.191565i $$-0.0613564\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 12.0000i − 1.04844i −0.851581 0.524222i $$-0.824356\pi$$
0.851581 0.524222i $$-0.175644\pi$$
$$132$$ 0 0
$$133$$ 16.0000i 1.38738i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ − 20.0000i − 1.69638i −0.529694 0.848189i $$-0.677693\pi$$
0.529694 0.848189i $$-0.322307\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 16.0000 1.33799
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.00000i 0.319235i 0.987179 + 0.159617i $$0.0510260\pi$$
−0.987179 + 0.159617i $$0.948974\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −32.0000 −2.52195
$$162$$ 0 0
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ −20.0000 −1.51186
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4.00000i 0.298974i 0.988764 + 0.149487i $$0.0477622\pi$$
−0.988764 + 0.149487i $$0.952238\pi$$
$$180$$ 0 0
$$181$$ 4.00000i 0.297318i 0.988889 + 0.148659i $$0.0474956\pi$$
−0.988889 + 0.148659i $$0.952504\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 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
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 24.0000i 1.70993i 0.518686 + 0.854965i $$0.326421\pi$$
−0.518686 + 0.854965i $$0.673579\pi$$
$$198$$ 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
$$203$$ 32.0000i 2.24596i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ − 4.00000i − 0.275371i −0.990476 0.137686i $$-0.956034\pi$$
0.990476 0.137686i $$-0.0439664\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 8.00000i − 0.538138i
$$222$$ 0 0
$$223$$ −20.0000 −1.33930 −0.669650 0.742677i $$-0.733556\pi$$
−0.669650 + 0.742677i $$0.733556\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 0 0
$$229$$ 4.00000i 0.264327i 0.991228 + 0.132164i $$0.0421925\pi$$
−0.991228 + 0.132164i $$0.957808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 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
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −16.0000 −1.01806
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 12.0000i − 0.757433i −0.925513 0.378717i $$-0.876365\pi$$
0.925513 0.378717i $$-0.123635\pi$$
$$252$$ 0 0
$$253$$ 32.0000i 2.01182i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ 16.0000i 0.994192i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 24.0000i 1.46331i 0.681677 + 0.731653i $$0.261251\pi$$
−0.681677 + 0.731653i $$0.738749\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 20.0000i 1.20605i
$$276$$ 0 0
$$277$$ 20.0000i 1.20168i 0.799368 + 0.600842i $$0.205168\pi$$
−0.799368 + 0.600842i $$0.794832\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 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
$$293$$ − 24.0000i − 1.40209i −0.713115 0.701047i $$-0.752716\pi$$
0.713115 0.701047i $$-0.247284\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 32.0000i − 1.85061i
$$300$$ 0 0
$$301$$ 16.0000i 0.922225i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −32.0000 −1.81455 −0.907277 0.420534i $$-0.861843\pi$$
−0.907277 + 0.420534i $$0.861843\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.00000i 0.449325i 0.974437 + 0.224662i $$0.0721279\pi$$
−0.974437 + 0.224662i $$0.927872\pi$$
$$318$$ 0 0
$$319$$ 32.0000 1.79166
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ 0 0
$$325$$ − 20.0000i − 1.10940i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ − 4.00000i − 0.219860i −0.993939 0.109930i $$-0.964937\pi$$
0.993939 0.109930i $$-0.0350627\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.0000i 0.866449i
$$342$$ 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 $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ − 28.0000i − 1.49881i −0.662114 0.749403i $$-0.730341\pi$$
0.662114 0.749403i $$-0.269659\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 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
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 36.0000 1.87918 0.939592 0.342296i $$-0.111204\pi$$
0.939592 + 0.342296i $$0.111204\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 32.0000i − 1.66136i
$$372$$ 0 0
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −32.0000 −1.64808
$$378$$ 0 0
$$379$$ − 4.00000i − 0.205466i −0.994709 0.102733i $$-0.967241\pi$$
0.994709 0.102733i $$-0.0327588\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 32.0000i − 1.62246i −0.584724 0.811232i $$-0.698797\pi$$
0.584724 0.811232i $$-0.301203\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 20.0000i 1.00377i 0.864934 + 0.501886i $$0.167360\pi$$
−0.864934 + 0.501886i $$0.832640\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 34.0000 1.69788 0.848939 0.528490i $$-0.177242\pi$$
0.848939 + 0.528490i $$0.177242\pi$$
$$402$$ 0 0
$$403$$ − 16.0000i − 0.797017i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 16.0000 0.793091
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 48.0000i 2.36193i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 20.0000i 0.977064i 0.872546 + 0.488532i $$0.162467\pi$$
−0.872546 + 0.488532i $$0.837533\pi$$
$$420$$ 0 0
$$421$$ 20.0000i 0.974740i 0.873195 + 0.487370i $$0.162044\pi$$
−0.873195 + 0.487370i $$0.837956\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 10.0000 0.485071
$$426$$ 0 0
$$427$$ 48.0000i 2.32288i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 32.0000i − 1.53077i
$$438$$ 0 0
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 36.0000i 1.71041i 0.518289 + 0.855206i $$0.326569\pi$$
−0.518289 + 0.855206i $$0.673431\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ 24.0000i 1.13012i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ 0 0
$$469$$ − 48.0000i − 2.21643i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ − 20.0000i − 0.917663i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 12.0000i − 0.541552i −0.962642 0.270776i $$-0.912720\pi$$
0.962642 0.270776i $$-0.0872803\pi$$
$$492$$ 0 0
$$493$$ − 16.0000i − 0.720604i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 32.0000 1.43540
$$498$$ 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
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 24.0000i − 1.06378i −0.846813 0.531891i $$-0.821482\pi$$
0.846813 0.531891i $$-0.178518\pi$$
$$510$$ 0 0
$$511$$ −24.0000 −1.06170
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 32.0000i − 1.40736i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.00000 0.348485
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 24.0000i − 1.03956i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 36.0000i 1.55063i
$$540$$ 0 0
$$541$$ 28.0000i 1.20381i 0.798566 + 0.601907i $$0.205592\pi$$
−0.798566 + 0.601907i $$0.794408\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 36.0000i − 1.53925i −0.638497 0.769624i $$-0.720443\pi$$
0.638497 0.769624i $$-0.279557\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −32.0000 −1.36325
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 16.0000i 0.677942i 0.940797 + 0.338971i $$0.110079\pi$$
−0.940797 + 0.338971i $$0.889921\pi$$
$$558$$ 0 0
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ − 4.00000i − 0.167395i −0.996491 0.0836974i $$-0.973327\pi$$
0.996491 0.0836974i $$-0.0266729\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 40.0000 1.66812
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 16.0000i − 0.663792i
$$582$$ 0 0
$$583$$ −32.0000 −1.32530
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 4.00000i 0.165098i 0.996587 + 0.0825488i $$0.0263060\pi$$
−0.996587 + 0.0825488i $$0.973694\pi$$
$$588$$ 0 0
$$589$$ − 16.0000i − 0.659269i
$$590$$ 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
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 32.0000i 1.29458i
$$612$$ 0 0
$$613$$ − 36.0000i − 1.45403i −0.686624 0.727013i $$-0.740908\pi$$
0.686624 0.727013i $$-0.259092\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 10.0000 0.402585 0.201292 0.979531i $$-0.435486\pi$$
0.201292 + 0.979531i $$0.435486\pi$$
$$618$$ 0 0
$$619$$ 28.0000i 1.12542i 0.826656 + 0.562708i $$0.190240\pi$$
−0.826656 + 0.562708i $$0.809760\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 24.0000 0.961540
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 8.00000i − 0.318981i
$$630$$ 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
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 36.0000i − 1.42637i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ − 36.0000i − 1.41970i −0.704352 0.709851i $$-0.748762\pi$$
0.704352 0.709851i $$-0.251238\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 16.0000i 0.626128i 0.949732 + 0.313064i $$0.101356\pi$$
−0.949732 + 0.313064i $$0.898644\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 36.0000i 1.40236i 0.712984 + 0.701180i $$0.247343\pi$$
−0.712984 + 0.701180i $$0.752657\pi$$
$$660$$ 0 0
$$661$$ − 4.00000i − 0.155582i −0.996970 0.0777910i $$-0.975213\pi$$
0.996970 0.0777910i $$-0.0247867\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 64.0000i − 2.47809i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 48.0000 1.85302
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 16.0000i 0.614930i 0.951559 + 0.307465i $$0.0994807\pi$$
−0.951559 + 0.307465i $$0.900519\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 44.0000i − 1.68361i −0.539779 0.841807i $$-0.681492\pi$$
0.539779 0.841807i $$-0.318508\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 32.0000 1.21910
$$690$$ 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
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 8.00000i 0.302156i 0.988522 + 0.151078i $$0.0482744\pi$$
−0.988522 + 0.151078i $$0.951726\pi$$
$$702$$ 0 0
$$703$$ −16.0000 −0.603451
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 32.0000i − 1.20348i
$$708$$ 0 0
$$709$$ 20.0000i 0.751116i 0.926799 + 0.375558i $$0.122549\pi$$
−0.926799 + 0.375558i $$0.877451\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 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
$$725$$ − 40.0000i − 1.48556i
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 8.00000i − 0.295891i
$$732$$ 0 0
$$733$$ 28.0000i 1.03420i 0.855924 + 0.517102i $$0.172989\pi$$
−0.855924 + 0.517102i $$0.827011\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −48.0000 −1.76810
$$738$$ 0 0
$$739$$ 28.0000i 1.03000i 0.857191 + 0.514998i $$0.172207\pi$$
−0.857191 + 0.514998i $$0.827793\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 16.0000i − 0.584627i
$$750$$ 0 0
$$751$$ −44.0000 −1.60558 −0.802791 0.596260i $$-0.796653\pi$$
−0.802791 + 0.596260i $$0.796653\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 12.0000i − 0.436147i −0.975932 0.218074i $$-0.930023\pi$$
0.975932 0.218074i $$-0.0699773\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ 16.0000i 0.579239i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −48.0000 −1.73318
$$768$$ 0 0
$$769$$ 46.0000 1.65880 0.829401 0.558653i $$-0.188682\pi$$
0.829401 + 0.558653i $$0.188682\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 8.00000i − 0.287740i −0.989597 0.143870i $$-0.954045\pi$$
0.989597 0.143870i $$-0.0459547\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 24.0000i − 0.859889i
$$780$$ 0 0
$$781$$ − 32.0000i − 1.14505i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 12.0000i 0.427754i 0.976861 + 0.213877i $$0.0686091\pi$$
−0.976861 + 0.213877i $$0.931391\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −56.0000 −1.99113
$$792$$ 0 0
$$793$$ −48.0000 −1.70453
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 16.0000i − 0.566749i −0.959009 0.283375i $$-0.908546\pi$$
0.959009 0.283375i $$-0.0914540\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 24.0000i 0.846942i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ 44.0000i 1.54505i 0.634985 + 0.772524i $$0.281006\pi$$
−0.634985 + 0.772524i $$0.718994\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 24.0000i 0.837606i 0.908077 + 0.418803i $$0.137550\pi$$
−0.908077 + 0.418803i $$0.862450\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 4.00000i 0.139094i 0.997579 + 0.0695468i $$0.0221553\pi$$
−0.997579 + 0.0695468i $$0.977845\pi$$
$$828$$ 0 0
$$829$$ − 4.00000i − 0.138926i −0.997585 0.0694629i $$-0.977871\pi$$
0.997585 0.0694629i $$-0.0221285\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −35.0000 −1.20690
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 20.0000 0.687208
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 32.0000i − 1.09695i
$$852$$ 0 0
$$853$$ 44.0000i 1.50653i 0.657716 + 0.753266i $$0.271523\pi$$
−0.657716 + 0.753266i $$0.728477\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 0 0
$$859$$ 28.0000i 0.955348i 0.878537 + 0.477674i $$0.158520\pi$$
−0.878537 + 0.477674i $$0.841480\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 16.0000 0.544646 0.272323 0.962206i $$-0.412208\pi$$
0.272323 + 0.962206i $$0.412208\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 16.0000i 0.542763i
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20.0000i 0.675352i 0.941262 + 0.337676i $$0.109641\pi$$
−0.941262 + 0.337676i $$0.890359\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ 28.0000i 0.942275i 0.882060 + 0.471138i $$0.156156\pi$$
−0.882060 + 0.471138i $$0.843844\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 32.0000i 1.07084i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 32.0000i − 1.06726i
$$900$$ 0 0
$$901$$ 16.0000i 0.533037i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 4.00000i − 0.132818i −0.997792 0.0664089i $$-0.978846\pi$$
0.997792 0.0664089i $$-0.0211542\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 48.0000i 1.58510i
$$918$$ 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
$$923$$ 32.0000i 1.05329i
$$924$$ 0 0
$$925$$ − 20.0000i − 0.657596i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ − 36.0000i − 1.17985i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 32.0000i − 1.04317i −0.853199 0.521585i $$-0.825341\pi$$
0.853199 0.521585i $$-0.174659\pi$$
$$942$$ 0 0
$$943$$ 48.0000 1.56310
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ 0 0
$$949$$ − 24.0000i − 0.779073i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −42.0000 −1.36051 −0.680257 0.732974i $$-0.738132\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −12.0000 −0.385894 −0.192947 0.981209i $$-0.561805\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 20.0000i 0.641831i 0.947108 + 0.320915i $$0.103990\pi$$
−0.947108 + 0.320915i $$0.896010\pi$$
$$972$$ 0 0
$$973$$ 80.0000i 2.56468i
$$974$$ 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
$$979$$ − 24.0000i − 0.767043i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 32.0000i − 1.01754i
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 44.0000i 1.39349i 0.717317 + 0.696747i $$0.245370\pi$$
−0.717317 + 0.696747i $$0.754630\pi$$
$$998$$ 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 1152.2.d.a.577.2 2
3.2 odd 2 384.2.d.a.193.1 2
4.3 odd 2 1152.2.d.f.577.1 2
8.3 odd 2 1152.2.d.f.577.2 2
8.5 even 2 inner 1152.2.d.a.577.1 2
12.11 even 2 384.2.d.b.193.2 yes 2
16.3 odd 4 2304.2.a.f.1.1 1
16.5 even 4 2304.2.a.j.1.1 1
16.11 odd 4 2304.2.a.g.1.1 1
16.13 even 4 2304.2.a.k.1.1 1
24.5 odd 2 384.2.d.a.193.2 yes 2
24.11 even 2 384.2.d.b.193.1 yes 2
48.5 odd 4 768.2.a.c.1.1 1
48.11 even 4 768.2.a.f.1.1 1
48.29 odd 4 768.2.a.g.1.1 1
48.35 even 4 768.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.d.a.193.1 2 3.2 odd 2
384.2.d.a.193.2 yes 2 24.5 odd 2
384.2.d.b.193.1 yes 2 24.11 even 2
384.2.d.b.193.2 yes 2 12.11 even 2
768.2.a.b.1.1 1 48.35 even 4
768.2.a.c.1.1 1 48.5 odd 4
768.2.a.f.1.1 1 48.11 even 4
768.2.a.g.1.1 1 48.29 odd 4
1152.2.d.a.577.1 2 8.5 even 2 inner
1152.2.d.a.577.2 2 1.1 even 1 trivial
1152.2.d.f.577.1 2 4.3 odd 2
1152.2.d.f.577.2 2 8.3 odd 2
2304.2.a.f.1.1 1 16.3 odd 4
2304.2.a.g.1.1 1 16.11 odd 4
2304.2.a.j.1.1 1 16.5 even 4
2304.2.a.k.1.1 1 16.13 even 4