# Properties

 Label 768.2.a.j.1.1 Level $768$ Weight $2$ Character 768.1 Self dual yes Analytic conductor $6.133$ 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] = [768,2,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.a (trivial)

## 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: yes Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 192) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -3.46410 q^{5} -3.46410 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -3.46410 q^{5} -3.46410 q^{7} +1.00000 q^{9} +3.46410 q^{15} +6.00000 q^{17} -4.00000 q^{19} +3.46410 q^{21} +6.92820 q^{23} +7.00000 q^{25} -1.00000 q^{27} +3.46410 q^{29} -3.46410 q^{31} +12.0000 q^{35} +6.92820 q^{37} +6.00000 q^{41} -4.00000 q^{43} -3.46410 q^{45} -6.92820 q^{47} +5.00000 q^{49} -6.00000 q^{51} +3.46410 q^{53} +4.00000 q^{57} +12.0000 q^{59} -6.92820 q^{61} -3.46410 q^{63} +4.00000 q^{67} -6.92820 q^{69} -6.92820 q^{71} -2.00000 q^{73} -7.00000 q^{75} +10.3923 q^{79} +1.00000 q^{81} -20.7846 q^{85} -3.46410 q^{87} -6.00000 q^{89} +3.46410 q^{93} +13.8564 q^{95} -2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{19} + 14 q^{25} - 2 q^{27} + 24 q^{35} + 12 q^{41} - 8 q^{43} + 10 q^{49} - 12 q^{51} + 8 q^{57} + 24 q^{59} + 8 q^{67} - 4 q^{73} - 14 q^{75} + 2 q^{81} - 12 q^{89} - 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 + 12 * q^17 - 8 * q^19 + 14 * q^25 - 2 * q^27 + 24 * q^35 + 12 * q^41 - 8 * q^43 + 10 * q^49 - 12 * q^51 + 8 * q^57 + 24 * q^59 + 8 * q^67 - 4 * q^73 - 14 * q^75 + 2 * q^81 - 12 * q^89 - 4 * q^97

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −3.46410 −1.54919 −0.774597 0.632456i $$-0.782047\pi$$
−0.774597 + 0.632456i $$0.782047\pi$$
$$6$$ 0 0
$$7$$ −3.46410 −1.30931 −0.654654 0.755929i $$-0.727186\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 3.46410 0.894427
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 3.46410 0.755929
$$22$$ 0 0
$$23$$ 6.92820 1.44463 0.722315 0.691564i $$-0.243078\pi$$
0.722315 + 0.691564i $$0.243078\pi$$
$$24$$ 0 0
$$25$$ 7.00000 1.40000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 3.46410 0.643268 0.321634 0.946864i $$-0.395768\pi$$
0.321634 + 0.946864i $$0.395768\pi$$
$$30$$ 0 0
$$31$$ −3.46410 −0.622171 −0.311086 0.950382i $$-0.600693\pi$$
−0.311086 + 0.950382i $$0.600693\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 12.0000 2.02837
$$36$$ 0 0
$$37$$ 6.92820 1.13899 0.569495 0.821995i $$-0.307139\pi$$
0.569495 + 0.821995i $$0.307139\pi$$
$$38$$ 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
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ −3.46410 −0.516398
$$46$$ 0 0
$$47$$ −6.92820 −1.01058 −0.505291 0.862949i $$-0.668615\pi$$
−0.505291 + 0.862949i $$0.668615\pi$$
$$48$$ 0 0
$$49$$ 5.00000 0.714286
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ 3.46410 0.475831 0.237915 0.971286i $$-0.423536\pi$$
0.237915 + 0.971286i $$0.423536\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −6.92820 −0.887066 −0.443533 0.896258i $$-0.646275\pi$$
−0.443533 + 0.896258i $$0.646275\pi$$
$$62$$ 0 0
$$63$$ −3.46410 −0.436436
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −6.92820 −0.834058
$$70$$ 0 0
$$71$$ −6.92820 −0.822226 −0.411113 0.911584i $$-0.634860\pi$$
−0.411113 + 0.911584i $$0.634860\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ −7.00000 −0.808290
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.3923 1.16923 0.584613 0.811312i $$-0.301246\pi$$
0.584613 + 0.811312i $$0.301246\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −20.7846 −2.25441
$$86$$ 0 0
$$87$$ −3.46410 −0.371391
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 3.46410 0.359211
$$94$$ 0 0
$$95$$ 13.8564 1.42164
$$96$$ 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
$$101$$ 3.46410 0.344691 0.172345 0.985037i $$-0.444865\pi$$
0.172345 + 0.985037i $$0.444865\pi$$
$$102$$ 0 0
$$103$$ 17.3205 1.70664 0.853320 0.521387i $$-0.174585\pi$$
0.853320 + 0.521387i $$0.174585\pi$$
$$104$$ 0 0
$$105$$ −12.0000 −1.17108
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 13.8564 1.32720 0.663602 0.748086i $$-0.269027\pi$$
0.663602 + 0.748086i $$0.269027\pi$$
$$110$$ 0 0
$$111$$ −6.92820 −0.657596
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −24.0000 −2.23801
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −20.7846 −1.90532
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −6.00000 −0.541002
$$124$$ 0 0
$$125$$ −6.92820 −0.619677
$$126$$ 0 0
$$127$$ 3.46410 0.307389 0.153695 0.988118i $$-0.450883\pi$$
0.153695 + 0.988118i $$0.450883\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 13.8564 1.20150
$$134$$ 0 0
$$135$$ 3.46410 0.298142
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 6.92820 0.583460
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ 0 0
$$147$$ −5.00000 −0.412393
$$148$$ 0 0
$$149$$ 10.3923 0.851371 0.425685 0.904871i $$-0.360033\pi$$
0.425685 + 0.904871i $$0.360033\pi$$
$$150$$ 0 0
$$151$$ −3.46410 −0.281905 −0.140952 0.990016i $$-0.545016\pi$$
−0.140952 + 0.990016i $$0.545016\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 12.0000 0.963863
$$156$$ 0 0
$$157$$ 6.92820 0.552931 0.276465 0.961024i $$-0.410837\pi$$
0.276465 + 0.961024i $$0.410837\pi$$
$$158$$ 0 0
$$159$$ −3.46410 −0.274721
$$160$$ 0 0
$$161$$ −24.0000 −1.89146
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 13.8564 1.07224 0.536120 0.844141i $$-0.319889\pi$$
0.536120 + 0.844141i $$0.319889\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ −17.3205 −1.31685 −0.658427 0.752645i $$-0.728778\pi$$
−0.658427 + 0.752645i $$0.728778\pi$$
$$174$$ 0 0
$$175$$ −24.2487 −1.83303
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 13.8564 1.02994 0.514969 0.857209i $$-0.327803\pi$$
0.514969 + 0.857209i $$0.327803\pi$$
$$182$$ 0 0
$$183$$ 6.92820 0.512148
$$184$$ 0 0
$$185$$ −24.0000 −1.76452
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 3.46410 0.251976
$$190$$ 0 0
$$191$$ −13.8564 −1.00261 −0.501307 0.865269i $$-0.667147\pi$$
−0.501307 + 0.865269i $$0.667147\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −10.3923 −0.740421 −0.370211 0.928948i $$-0.620714\pi$$
−0.370211 + 0.928948i $$0.620714\pi$$
$$198$$ 0 0
$$199$$ −10.3923 −0.736691 −0.368345 0.929689i $$-0.620076\pi$$
−0.368345 + 0.929689i $$0.620076\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ −12.0000 −0.842235
$$204$$ 0 0
$$205$$ −20.7846 −1.45166
$$206$$ 0 0
$$207$$ 6.92820 0.481543
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ 6.92820 0.474713
$$214$$ 0 0
$$215$$ 13.8564 0.944999
$$216$$ 0 0
$$217$$ 12.0000 0.814613
$$218$$ 0 0
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 3.46410 0.231973 0.115987 0.993251i $$-0.462997\pi$$
0.115987 + 0.993251i $$0.462997\pi$$
$$224$$ 0 0
$$225$$ 7.00000 0.466667
$$226$$ 0 0
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 0 0
$$229$$ −27.7128 −1.83131 −0.915657 0.401960i $$-0.868329\pi$$
−0.915657 + 0.401960i $$0.868329\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 24.0000 1.56559
$$236$$ 0 0
$$237$$ −10.3923 −0.675053
$$238$$ 0 0
$$239$$ 27.7128 1.79259 0.896296 0.443455i $$-0.146248\pi$$
0.896296 + 0.443455i $$0.146248\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −17.3205 −1.10657
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 20.7846 1.30158
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ 3.46410 0.214423
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ 0 0
$$269$$ 31.1769 1.90089 0.950445 0.310893i $$-0.100628\pi$$
0.950445 + 0.310893i $$0.100628\pi$$
$$270$$ 0 0
$$271$$ 3.46410 0.210429 0.105215 0.994450i $$-0.466447\pi$$
0.105215 + 0.994450i $$0.466447\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 13.8564 0.832551 0.416275 0.909239i $$-0.363335\pi$$
0.416275 + 0.909239i $$0.363335\pi$$
$$278$$ 0 0
$$279$$ −3.46410 −0.207390
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ −13.8564 −0.820783
$$286$$ 0 0
$$287$$ −20.7846 −1.22688
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 0 0
$$293$$ 3.46410 0.202375 0.101187 0.994867i $$-0.467736\pi$$
0.101187 + 0.994867i $$0.467736\pi$$
$$294$$ 0 0
$$295$$ −41.5692 −2.42025
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 13.8564 0.798670
$$302$$ 0 0
$$303$$ −3.46410 −0.199007
$$304$$ 0 0
$$305$$ 24.0000 1.37424
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ −17.3205 −0.985329
$$310$$ 0 0
$$311$$ 13.8564 0.785725 0.392862 0.919597i $$-0.371485\pi$$
0.392862 + 0.919597i $$0.371485\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 0 0
$$315$$ 12.0000 0.676123
$$316$$ 0 0
$$317$$ 31.1769 1.75107 0.875535 0.483155i $$-0.160509\pi$$
0.875535 + 0.483155i $$0.160509\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −13.8564 −0.766261
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 6.92820 0.379663
$$334$$ 0 0
$$335$$ −13.8564 −0.757056
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 6.92820 0.374088
$$344$$ 0 0
$$345$$ 24.0000 1.29212
$$346$$ 0 0
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 0 0
$$349$$ −6.92820 −0.370858 −0.185429 0.982658i $$-0.559368\pi$$
−0.185429 + 0.982658i $$0.559368\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 24.0000 1.27379
$$356$$ 0 0
$$357$$ 20.7846 1.10004
$$358$$ 0 0
$$359$$ −6.92820 −0.365657 −0.182828 0.983145i $$-0.558525\pi$$
−0.182828 + 0.983145i $$0.558525\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 6.92820 0.362639
$$366$$ 0 0
$$367$$ 24.2487 1.26577 0.632886 0.774245i $$-0.281870\pi$$
0.632886 + 0.774245i $$0.281870\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ 20.7846 1.07619 0.538093 0.842885i $$-0.319145\pi$$
0.538093 + 0.842885i $$0.319145\pi$$
$$374$$ 0 0
$$375$$ 6.92820 0.357771
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −3.46410 −0.177471
$$382$$ 0 0
$$383$$ −13.8564 −0.708029 −0.354015 0.935240i $$-0.615184\pi$$
−0.354015 + 0.935240i $$0.615184\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 24.2487 1.22946 0.614729 0.788738i $$-0.289265\pi$$
0.614729 + 0.788738i $$0.289265\pi$$
$$390$$ 0 0
$$391$$ 41.5692 2.10225
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ −36.0000 −1.81136
$$396$$ 0 0
$$397$$ −34.6410 −1.73858 −0.869291 0.494300i $$-0.835424\pi$$
−0.869291 + 0.494300i $$0.835424\pi$$
$$398$$ 0 0
$$399$$ −13.8564 −0.693688
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −3.46410 −0.172133
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 0 0
$$413$$ −41.5692 −2.04549
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 20.0000 0.979404
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ −13.8564 −0.675320 −0.337660 0.941268i $$-0.609635\pi$$
−0.337660 + 0.941268i $$0.609635\pi$$
$$422$$ 0 0
$$423$$ −6.92820 −0.336861
$$424$$ 0 0
$$425$$ 42.0000 2.03730
$$426$$ 0 0
$$427$$ 24.0000 1.16144
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.92820 0.333720 0.166860 0.985981i $$-0.446637\pi$$
0.166860 + 0.985981i $$0.446637\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 12.0000 0.575356
$$436$$ 0 0
$$437$$ −27.7128 −1.32568
$$438$$ 0 0
$$439$$ 31.1769 1.48799 0.743996 0.668184i $$-0.232928\pi$$
0.743996 + 0.668184i $$0.232928\pi$$
$$440$$ 0 0
$$441$$ 5.00000 0.238095
$$442$$ 0 0
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ 20.7846 0.985285
$$446$$ 0 0
$$447$$ −10.3923 −0.491539
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 3.46410 0.162758
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −31.1769 −1.45205 −0.726027 0.687666i $$-0.758635\pi$$
−0.726027 + 0.687666i $$0.758635\pi$$
$$462$$ 0 0
$$463$$ −10.3923 −0.482971 −0.241486 0.970404i $$-0.577635\pi$$
−0.241486 + 0.970404i $$0.577635\pi$$
$$464$$ 0 0
$$465$$ −12.0000 −0.556487
$$466$$ 0 0
$$467$$ 24.0000 1.11059 0.555294 0.831654i $$-0.312606\pi$$
0.555294 + 0.831654i $$0.312606\pi$$
$$468$$ 0 0
$$469$$ −13.8564 −0.639829
$$470$$ 0 0
$$471$$ −6.92820 −0.319235
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −28.0000 −1.28473
$$476$$ 0 0
$$477$$ 3.46410 0.158610
$$478$$ 0 0
$$479$$ 20.7846 0.949673 0.474837 0.880074i $$-0.342507\pi$$
0.474837 + 0.880074i $$0.342507\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 24.0000 1.09204
$$484$$ 0 0
$$485$$ 6.92820 0.314594
$$486$$ 0 0
$$487$$ 10.3923 0.470920 0.235460 0.971884i $$-0.424340\pi$$
0.235460 + 0.971884i $$0.424340\pi$$
$$488$$ 0 0
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ 20.7846 0.936092
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 24.0000 1.07655
$$498$$ 0 0
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ −13.8564 −0.619059
$$502$$ 0 0
$$503$$ −34.6410 −1.54457 −0.772283 0.635278i $$-0.780885\pi$$
−0.772283 + 0.635278i $$0.780885\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 0 0
$$507$$ 13.0000 0.577350
$$508$$ 0 0
$$509$$ −24.2487 −1.07481 −0.537403 0.843326i $$-0.680594\pi$$
−0.537403 + 0.843326i $$0.680594\pi$$
$$510$$ 0 0
$$511$$ 6.92820 0.306486
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ −60.0000 −2.64392
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 17.3205 0.760286
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ 24.2487 1.05830
$$526$$ 0 0
$$527$$ −20.7846 −0.905392
$$528$$ 0 0
$$529$$ 25.0000 1.08696
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −41.5692 −1.79719
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 13.8564 0.595733 0.297867 0.954607i $$-0.403725\pi$$
0.297867 + 0.954607i $$0.403725\pi$$
$$542$$ 0 0
$$543$$ −13.8564 −0.594635
$$544$$ 0 0
$$545$$ −48.0000 −2.05609
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 0 0
$$549$$ −6.92820 −0.295689
$$550$$ 0 0
$$551$$ −13.8564 −0.590303
$$552$$ 0 0
$$553$$ −36.0000 −1.53088
$$554$$ 0 0
$$555$$ 24.0000 1.01874
$$556$$ 0 0
$$557$$ 10.3923 0.440336 0.220168 0.975462i $$-0.429339\pi$$
0.220168 + 0.975462i $$0.429339\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ 20.7846 0.874415
$$566$$ 0 0
$$567$$ −3.46410 −0.145479
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ 13.8564 0.578860
$$574$$ 0 0
$$575$$ 48.4974 2.02248
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 13.8564 0.570943
$$590$$ 0 0
$$591$$ 10.3923 0.427482
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 72.0000 2.95171
$$596$$ 0 0
$$597$$ 10.3923 0.425329
$$598$$ 0 0
$$599$$ −34.6410 −1.41539 −0.707697 0.706516i $$-0.750266\pi$$
−0.707697 + 0.706516i $$0.750266\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 38.1051 1.54919
$$606$$ 0 0
$$607$$ 3.46410 0.140604 0.0703018 0.997526i $$-0.477604\pi$$
0.0703018 + 0.997526i $$0.477604\pi$$
$$608$$ 0 0
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −34.6410 −1.39914 −0.699569 0.714565i $$-0.746625\pi$$
−0.699569 + 0.714565i $$0.746625\pi$$
$$614$$ 0 0
$$615$$ 20.7846 0.838116
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ −6.92820 −0.278019
$$622$$ 0 0
$$623$$ 20.7846 0.832718
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 41.5692 1.65747
$$630$$ 0 0
$$631$$ −17.3205 −0.689519 −0.344759 0.938691i $$-0.612039\pi$$
−0.344759 + 0.938691i $$0.612039\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ −12.0000 −0.476205
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −6.92820 −0.274075
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ 0 0
$$645$$ −13.8564 −0.545595
$$646$$ 0 0
$$647$$ 34.6410 1.36188 0.680939 0.732340i $$-0.261572\pi$$
0.680939 + 0.732340i $$0.261572\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −12.0000 −0.470317
$$652$$ 0 0
$$653$$ 10.3923 0.406682 0.203341 0.979108i $$-0.434820\pi$$
0.203341 + 0.979108i $$0.434820\pi$$
$$654$$ 0 0
$$655$$ −41.5692 −1.62424
$$656$$ 0 0
$$657$$ −2.00000 −0.0780274
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 6.92820 0.269476 0.134738 0.990881i $$-0.456981\pi$$
0.134738 + 0.990881i $$0.456981\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −48.0000 −1.86136
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 0 0
$$669$$ −3.46410 −0.133930
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ 0 0
$$675$$ −7.00000 −0.269430
$$676$$ 0 0
$$677$$ −3.46410 −0.133136 −0.0665681 0.997782i $$-0.521205\pi$$
−0.0665681 + 0.997782i $$0.521205\pi$$
$$678$$ 0 0
$$679$$ 6.92820 0.265880
$$680$$ 0 0
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ −20.7846 −0.794139
$$686$$ 0 0
$$687$$ 27.7128 1.05731
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 69.2820 2.62802
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ −10.3923 −0.392512 −0.196256 0.980553i $$-0.562878\pi$$
−0.196256 + 0.980553i $$0.562878\pi$$
$$702$$ 0 0
$$703$$ −27.7128 −1.04521
$$704$$ 0 0
$$705$$ −24.0000 −0.903892
$$706$$ 0 0
$$707$$ −12.0000 −0.451306
$$708$$ 0 0
$$709$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$710$$ 0 0
$$711$$ 10.3923 0.389742
$$712$$ 0 0
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −27.7128 −1.03495
$$718$$ 0 0
$$719$$ −20.7846 −0.775135 −0.387568 0.921841i $$-0.626685\pi$$
−0.387568 + 0.921841i $$0.626685\pi$$
$$720$$ 0 0
$$721$$ −60.0000 −2.23452
$$722$$ 0 0
$$723$$ 22.0000 0.818189
$$724$$ 0 0
$$725$$ 24.2487 0.900575
$$726$$ 0 0
$$727$$ 10.3923 0.385429 0.192715 0.981255i $$-0.438271\pi$$
0.192715 + 0.981255i $$0.438271\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ −27.7128 −1.02360 −0.511798 0.859106i $$-0.671020\pi$$
−0.511798 + 0.859106i $$0.671020\pi$$
$$734$$ 0 0
$$735$$ 17.3205 0.638877
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −27.7128 −1.01668 −0.508342 0.861155i $$-0.669742\pi$$
−0.508342 + 0.861155i $$0.669742\pi$$
$$744$$ 0 0
$$745$$ −36.0000 −1.31894
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −41.5692 −1.51891
$$750$$ 0 0
$$751$$ 24.2487 0.884848 0.442424 0.896806i $$-0.354119\pi$$
0.442424 + 0.896806i $$0.354119\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 12.0000 0.436725
$$756$$ 0 0
$$757$$ −27.7128 −1.00724 −0.503620 0.863925i $$-0.667999\pi$$
−0.503620 + 0.863925i $$0.667999\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ −48.0000 −1.73772
$$764$$ 0 0
$$765$$ −20.7846 −0.751469
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 0 0
$$773$$ −38.1051 −1.37055 −0.685273 0.728286i $$-0.740317\pi$$
−0.685273 + 0.728286i $$0.740317\pi$$
$$774$$ 0 0
$$775$$ −24.2487 −0.871039
$$776$$ 0 0
$$777$$ 24.0000 0.860995
$$778$$ 0 0
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −3.46410 −0.123797
$$784$$ 0 0
$$785$$ −24.0000 −0.856597
$$786$$ 0 0
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 20.7846 0.739016
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 12.0000 0.425596
$$796$$ 0 0
$$797$$ 24.2487 0.858933 0.429467 0.903083i $$-0.358702\pi$$
0.429467 + 0.903083i $$0.358702\pi$$
$$798$$ 0 0
$$799$$ −41.5692 −1.47061
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 83.1384 2.93024
$$806$$ 0 0
$$807$$ −31.1769 −1.09748
$$808$$ 0 0
$$809$$ 54.0000 1.89854 0.949269 0.314464i $$-0.101825\pi$$
0.949269 + 0.314464i $$0.101825\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ −3.46410 −0.121491
$$814$$ 0 0
$$815$$ −69.2820 −2.42684
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −24.2487 −0.846286 −0.423143 0.906063i $$-0.639073\pi$$
−0.423143 + 0.906063i $$0.639073\pi$$
$$822$$ 0 0
$$823$$ −31.1769 −1.08676 −0.543379 0.839487i $$-0.682856\pi$$
−0.543379 + 0.839487i $$0.682856\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −41.5692 −1.44376 −0.721879 0.692019i $$-0.756721\pi$$
−0.721879 + 0.692019i $$0.756721\pi$$
$$830$$ 0 0
$$831$$ −13.8564 −0.480673
$$832$$ 0 0
$$833$$ 30.0000 1.03944
$$834$$ 0 0
$$835$$ −48.0000 −1.66111
$$836$$ 0 0
$$837$$ 3.46410 0.119737
$$838$$ 0 0
$$839$$ 6.92820 0.239188 0.119594 0.992823i $$-0.461841\pi$$
0.119594 + 0.992823i $$0.461841\pi$$
$$840$$ 0 0
$$841$$ −17.0000 −0.586207
$$842$$ 0 0
$$843$$ 30.0000 1.03325
$$844$$ 0 0
$$845$$ 45.0333 1.54919
$$846$$ 0 0
$$847$$ 38.1051 1.30931
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 48.0000 1.64542
$$852$$ 0 0
$$853$$ −48.4974 −1.66052 −0.830260 0.557376i $$-0.811808\pi$$
−0.830260 + 0.557376i $$0.811808\pi$$
$$854$$ 0 0
$$855$$ 13.8564 0.473879
$$856$$ 0 0
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 20.7846 0.708338
$$862$$ 0 0
$$863$$ −13.8564 −0.471678 −0.235839 0.971792i $$-0.575784\pi$$
−0.235839 + 0.971792i $$0.575784\pi$$
$$864$$ 0 0
$$865$$ 60.0000 2.04006
$$866$$ 0 0
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 24.0000 0.811348
$$876$$ 0 0
$$877$$ −6.92820 −0.233949 −0.116974 0.993135i $$-0.537320\pi$$
−0.116974 + 0.993135i $$0.537320\pi$$
$$878$$ 0 0
$$879$$ −3.46410 −0.116841
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 0 0
$$885$$ 41.5692 1.39733
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ −12.0000 −0.402467
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 27.7128 0.927374
$$894$$ 0 0
$$895$$ 41.5692 1.38951
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ 20.7846 0.692436
$$902$$ 0 0
$$903$$ −13.8564 −0.461112
$$904$$ 0 0
$$905$$ −48.0000 −1.59557
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 0 0
$$909$$ 3.46410 0.114897
$$910$$ 0 0
$$911$$ 55.4256 1.83633 0.918166 0.396195i $$-0.129670\pi$$
0.918166 + 0.396195i $$0.129670\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −24.0000 −0.793416
$$916$$ 0 0
$$917$$ −41.5692 −1.37274
$$918$$ 0 0
$$919$$ 51.9615 1.71405 0.857026 0.515273i $$-0.172309\pi$$
0.857026 + 0.515273i $$0.172309\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 48.4974 1.59459
$$926$$ 0 0
$$927$$ 17.3205 0.568880
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −20.0000 −0.655474
$$932$$ 0 0
$$933$$ −13.8564 −0.453638
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 0 0
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ 24.2487 0.790485 0.395243 0.918577i $$-0.370660\pi$$
0.395243 + 0.918577i $$0.370660\pi$$
$$942$$ 0 0
$$943$$ 41.5692 1.35368
$$944$$ 0 0
$$945$$ −12.0000 −0.390360
$$946$$ 0 0
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −31.1769 −1.01098
$$952$$ 0 0
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 0 0
$$955$$ 48.0000 1.55324
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −20.7846 −0.671170
$$960$$ 0 0
$$961$$ −19.0000 −0.612903
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 0 0
$$965$$ −6.92820 −0.223027
$$966$$ 0 0
$$967$$ −38.1051 −1.22538 −0.612689 0.790324i $$-0.709912\pi$$
−0.612689 + 0.790324i $$0.709912\pi$$
$$968$$ 0 0
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ 0 0
$$973$$ 69.2820 2.22108
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 54.0000 1.72761 0.863807 0.503824i $$-0.168074\pi$$
0.863807 + 0.503824i $$0.168074\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 13.8564 0.442401
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 36.0000 1.14706
$$986$$ 0 0
$$987$$ −24.0000 −0.763928
$$988$$ 0 0
$$989$$ −27.7128 −0.881216
$$990$$ 0 0
$$991$$ 24.2487 0.770286 0.385143 0.922857i $$-0.374152\pi$$
0.385143 + 0.922857i $$0.374152\pi$$
$$992$$ 0 0
$$993$$ 20.0000 0.634681
$$994$$ 0 0
$$995$$ 36.0000 1.14128
$$996$$ 0 0
$$997$$ 20.7846 0.658255 0.329128 0.944285i $$-0.393245\pi$$
0.329128 + 0.944285i $$0.393245\pi$$
$$998$$ 0 0
$$999$$ −6.92820 −0.219199
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 768.2.a.j.1.1 2
3.2 odd 2 2304.2.a.s.1.2 2
4.3 odd 2 768.2.a.k.1.1 2
8.3 odd 2 inner 768.2.a.j.1.2 2
8.5 even 2 768.2.a.k.1.2 2
12.11 even 2 2304.2.a.u.1.2 2
16.3 odd 4 192.2.d.a.97.4 yes 4
16.5 even 4 192.2.d.a.97.3 yes 4
16.11 odd 4 192.2.d.a.97.1 4
16.13 even 4 192.2.d.a.97.2 yes 4
24.5 odd 2 2304.2.a.u.1.1 2
24.11 even 2 2304.2.a.s.1.1 2
48.5 odd 4 576.2.d.b.289.4 4
48.11 even 4 576.2.d.b.289.3 4
48.29 odd 4 576.2.d.b.289.2 4
48.35 even 4 576.2.d.b.289.1 4
80.3 even 4 4800.2.d.j.1249.3 4
80.13 odd 4 4800.2.d.o.1249.1 4
80.19 odd 4 4800.2.k.j.2401.2 4
80.27 even 4 4800.2.d.j.1249.2 4
80.29 even 4 4800.2.k.j.2401.3 4
80.37 odd 4 4800.2.d.o.1249.4 4
80.43 even 4 4800.2.d.o.1249.3 4
80.53 odd 4 4800.2.d.j.1249.1 4
80.59 odd 4 4800.2.k.j.2401.4 4
80.67 even 4 4800.2.d.o.1249.2 4
80.69 even 4 4800.2.k.j.2401.1 4
80.77 odd 4 4800.2.d.j.1249.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.d.a.97.1 4 16.11 odd 4
192.2.d.a.97.2 yes 4 16.13 even 4
192.2.d.a.97.3 yes 4 16.5 even 4
192.2.d.a.97.4 yes 4 16.3 odd 4
576.2.d.b.289.1 4 48.35 even 4
576.2.d.b.289.2 4 48.29 odd 4
576.2.d.b.289.3 4 48.11 even 4
576.2.d.b.289.4 4 48.5 odd 4
768.2.a.j.1.1 2 1.1 even 1 trivial
768.2.a.j.1.2 2 8.3 odd 2 inner
768.2.a.k.1.1 2 4.3 odd 2
768.2.a.k.1.2 2 8.5 even 2
2304.2.a.s.1.1 2 24.11 even 2
2304.2.a.s.1.2 2 3.2 odd 2
2304.2.a.u.1.1 2 24.5 odd 2
2304.2.a.u.1.2 2 12.11 even 2
4800.2.d.j.1249.1 4 80.53 odd 4
4800.2.d.j.1249.2 4 80.27 even 4
4800.2.d.j.1249.3 4 80.3 even 4
4800.2.d.j.1249.4 4 80.77 odd 4
4800.2.d.o.1249.1 4 80.13 odd 4
4800.2.d.o.1249.2 4 80.67 even 4
4800.2.d.o.1249.3 4 80.43 even 4
4800.2.d.o.1249.4 4 80.37 odd 4
4800.2.k.j.2401.1 4 80.69 even 4
4800.2.k.j.2401.2 4 80.19 odd 4
4800.2.k.j.2401.3 4 80.29 even 4
4800.2.k.j.2401.4 4 80.59 odd 4