# Properties

 Label 9576.2.a.bf.1.1 Level $9576$ Weight $2$ Character 9576.1 Self dual yes Analytic conductor $76.465$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9576.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: $$76.4647449756$$ Analytic rank: $$1$$ 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: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 9576.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.73205 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q-2.73205 q^{5} -1.00000 q^{7} -3.46410 q^{11} +4.00000 q^{13} -6.73205 q^{17} +1.00000 q^{19} +7.46410 q^{23} +2.46410 q^{25} -0.732051 q^{29} +3.46410 q^{31} +2.73205 q^{35} -0.535898 q^{37} -4.53590 q^{41} +4.92820 q^{43} +11.6603 q^{47} +1.00000 q^{49} +2.19615 q^{53} +9.46410 q^{55} -8.00000 q^{59} +11.4641 q^{61} -10.9282 q^{65} -4.00000 q^{67} -4.19615 q^{71} +7.46410 q^{73} +3.46410 q^{77} +1.46410 q^{79} -10.1962 q^{83} +18.3923 q^{85} -10.3923 q^{89} -4.00000 q^{91} -2.73205 q^{95} +6.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^7 $$2 q - 2 q^{5} - 2 q^{7} + 8 q^{13} - 10 q^{17} + 2 q^{19} + 8 q^{23} - 2 q^{25} + 2 q^{29} + 2 q^{35} - 8 q^{37} - 16 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} - 6 q^{53} + 12 q^{55} - 16 q^{59} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 2 q^{71} + 8 q^{73} - 4 q^{79} - 10 q^{83} + 16 q^{85} - 8 q^{91} - 2 q^{95} + 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^7 + 8 * q^13 - 10 * q^17 + 2 * q^19 + 8 * q^23 - 2 * q^25 + 2 * q^29 + 2 * q^35 - 8 * q^37 - 16 * q^41 - 4 * q^43 + 6 * q^47 + 2 * q^49 - 6 * q^53 + 12 * q^55 - 16 * q^59 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 2 * q^71 + 8 * q^73 - 4 * q^79 - 10 * q^83 + 16 * q^85 - 8 * q^91 - 2 * q^95 + 12 * 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −2.73205 −1.22181 −0.610905 0.791704i $$-0.709194\pi$$
−0.610905 + 0.791704i $$0.709194\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −6.73205 −1.63276 −0.816381 0.577514i $$-0.804023\pi$$
−0.816381 + 0.577514i $$0.804023\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 7.46410 1.55637 0.778186 0.628033i $$-0.216140\pi$$
0.778186 + 0.628033i $$0.216140\pi$$
$$24$$ 0 0
$$25$$ 2.46410 0.492820
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −0.732051 −0.135938 −0.0679692 0.997687i $$-0.521652\pi$$
−0.0679692 + 0.997687i $$0.521652\pi$$
$$30$$ 0 0
$$31$$ 3.46410 0.622171 0.311086 0.950382i $$-0.399307\pi$$
0.311086 + 0.950382i $$0.399307\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.73205 0.461801
$$36$$ 0 0
$$37$$ −0.535898 −0.0881012 −0.0440506 0.999029i $$-0.514026\pi$$
−0.0440506 + 0.999029i $$0.514026\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.53590 −0.708388 −0.354194 0.935172i $$-0.615245\pi$$
−0.354194 + 0.935172i $$0.615245\pi$$
$$42$$ 0 0
$$43$$ 4.92820 0.751544 0.375772 0.926712i $$-0.377378\pi$$
0.375772 + 0.926712i $$0.377378\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 11.6603 1.70082 0.850411 0.526118i $$-0.176353\pi$$
0.850411 + 0.526118i $$0.176353\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 2.19615 0.301665 0.150832 0.988559i $$-0.451805\pi$$
0.150832 + 0.988559i $$0.451805\pi$$
$$54$$ 0 0
$$55$$ 9.46410 1.27614
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 11.4641 1.46783 0.733914 0.679243i $$-0.237692\pi$$
0.733914 + 0.679243i $$0.237692\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −10.9282 −1.35548
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.19615 −0.497992 −0.248996 0.968505i $$-0.580101\pi$$
−0.248996 + 0.968505i $$0.580101\pi$$
$$72$$ 0 0
$$73$$ 7.46410 0.873607 0.436804 0.899557i $$-0.356111\pi$$
0.436804 + 0.899557i $$0.356111\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.46410 0.394771
$$78$$ 0 0
$$79$$ 1.46410 0.164724 0.0823622 0.996602i $$-0.473754\pi$$
0.0823622 + 0.996602i $$0.473754\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −10.1962 −1.11917 −0.559587 0.828772i $$-0.689040\pi$$
−0.559587 + 0.828772i $$0.689040\pi$$
$$84$$ 0 0
$$85$$ 18.3923 1.99493
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −10.3923 −1.10158 −0.550791 0.834643i $$-0.685674\pi$$
−0.550791 + 0.834643i $$0.685674\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −2.73205 −0.280302
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 16.5885 1.65061 0.825307 0.564685i $$-0.191002\pi$$
0.825307 + 0.564685i $$0.191002\pi$$
$$102$$ 0 0
$$103$$ −6.92820 −0.682656 −0.341328 0.939944i $$-0.610877\pi$$
−0.341328 + 0.939944i $$0.610877\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.19615 0.792352 0.396176 0.918175i $$-0.370337\pi$$
0.396176 + 0.918175i $$0.370337\pi$$
$$108$$ 0 0
$$109$$ −11.8564 −1.13564 −0.567819 0.823154i $$-0.692213\pi$$
−0.567819 + 0.823154i $$0.692213\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −0.732051 −0.0688655 −0.0344328 0.999407i $$-0.510962\pi$$
−0.0344328 + 0.999407i $$0.510962\pi$$
$$114$$ 0 0
$$115$$ −20.3923 −1.90159
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6.73205 0.617126
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 6.92820 0.619677
$$126$$ 0 0
$$127$$ −10.9282 −0.969721 −0.484861 0.874591i $$-0.661130\pi$$
−0.484861 + 0.874591i $$0.661130\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2.19615 0.191879 0.0959394 0.995387i $$-0.469415\pi$$
0.0959394 + 0.995387i $$0.469415\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2.92820 −0.250173 −0.125087 0.992146i $$-0.539921\pi$$
−0.125087 + 0.992146i $$0.539921\pi$$
$$138$$ 0 0
$$139$$ 9.46410 0.802735 0.401367 0.915917i $$-0.368535\pi$$
0.401367 + 0.915917i $$0.368535\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −13.8564 −1.15873
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2.53590 0.207749 0.103874 0.994590i $$-0.466876\pi$$
0.103874 + 0.994590i $$0.466876\pi$$
$$150$$ 0 0
$$151$$ −14.5359 −1.18291 −0.591457 0.806336i $$-0.701447\pi$$
−0.591457 + 0.806336i $$0.701447\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −9.46410 −0.760175
$$156$$ 0 0
$$157$$ −6.39230 −0.510161 −0.255081 0.966920i $$-0.582102\pi$$
−0.255081 + 0.966920i $$0.582102\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −7.46410 −0.588254
$$162$$ 0 0
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −16.3923 −1.26847 −0.634237 0.773138i $$-0.718686\pi$$
−0.634237 + 0.773138i $$0.718686\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −10.3923 −0.790112 −0.395056 0.918657i $$-0.629275\pi$$
−0.395056 + 0.918657i $$0.629275\pi$$
$$174$$ 0 0
$$175$$ −2.46410 −0.186269
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 19.5167 1.45874 0.729372 0.684117i $$-0.239812\pi$$
0.729372 + 0.684117i $$0.239812\pi$$
$$180$$ 0 0
$$181$$ 3.85641 0.286644 0.143322 0.989676i $$-0.454221\pi$$
0.143322 + 0.989676i $$0.454221\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.46410 0.107643
$$186$$ 0 0
$$187$$ 23.3205 1.70536
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2.39230 −0.173101 −0.0865506 0.996247i $$-0.527584\pi$$
−0.0865506 + 0.996247i $$0.527584\pi$$
$$192$$ 0 0
$$193$$ −17.3205 −1.24676 −0.623379 0.781920i $$-0.714240\pi$$
−0.623379 + 0.781920i $$0.714240\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.92820 0.493614 0.246807 0.969065i $$-0.420619\pi$$
0.246807 + 0.969065i $$0.420619\pi$$
$$198$$ 0 0
$$199$$ 24.7846 1.75693 0.878467 0.477803i $$-0.158567\pi$$
0.878467 + 0.477803i $$0.158567\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0.732051 0.0513799
$$204$$ 0 0
$$205$$ 12.3923 0.865516
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −3.46410 −0.239617
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −13.4641 −0.918244
$$216$$ 0 0
$$217$$ −3.46410 −0.235159
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −26.9282 −1.81139
$$222$$ 0 0
$$223$$ −21.3205 −1.42773 −0.713863 0.700285i $$-0.753056\pi$$
−0.713863 + 0.700285i $$0.753056\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −6.53590 −0.433803 −0.216901 0.976194i $$-0.569595\pi$$
−0.216901 + 0.976194i $$0.569595\pi$$
$$228$$ 0 0
$$229$$ −12.9282 −0.854320 −0.427160 0.904176i $$-0.640486\pi$$
−0.427160 + 0.904176i $$0.640486\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −21.4641 −1.40616 −0.703080 0.711111i $$-0.748192\pi$$
−0.703080 + 0.711111i $$0.748192\pi$$
$$234$$ 0 0
$$235$$ −31.8564 −2.07808
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 7.85641 0.508189 0.254094 0.967179i $$-0.418223\pi$$
0.254094 + 0.967179i $$0.418223\pi$$
$$240$$ 0 0
$$241$$ 28.9282 1.86343 0.931715 0.363191i $$-0.118313\pi$$
0.931715 + 0.363191i $$0.118313\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −2.73205 −0.174544
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −7.26795 −0.458749 −0.229374 0.973338i $$-0.573668\pi$$
−0.229374 + 0.973338i $$0.573668\pi$$
$$252$$ 0 0
$$253$$ −25.8564 −1.62558
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 27.8564 1.73763 0.868817 0.495133i $$-0.164881\pi$$
0.868817 + 0.495133i $$0.164881\pi$$
$$258$$ 0 0
$$259$$ 0.535898 0.0332991
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −3.07180 −0.189415 −0.0947076 0.995505i $$-0.530192\pi$$
−0.0947076 + 0.995505i $$0.530192\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0.928203 0.0565935 0.0282968 0.999600i $$-0.490992\pi$$
0.0282968 + 0.999600i $$0.490992\pi$$
$$270$$ 0 0
$$271$$ −11.3205 −0.687672 −0.343836 0.939030i $$-0.611726\pi$$
−0.343836 + 0.939030i $$0.611726\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −8.53590 −0.514734
$$276$$ 0 0
$$277$$ −26.2487 −1.57713 −0.788566 0.614950i $$-0.789176\pi$$
−0.788566 + 0.614950i $$0.789176\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −15.2679 −0.910809 −0.455405 0.890285i $$-0.650505\pi$$
−0.455405 + 0.890285i $$0.650505\pi$$
$$282$$ 0 0
$$283$$ −5.85641 −0.348127 −0.174064 0.984734i $$-0.555690\pi$$
−0.174064 + 0.984734i $$0.555690\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 4.53590 0.267746
$$288$$ 0 0
$$289$$ 28.3205 1.66591
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −27.4641 −1.60447 −0.802235 0.597008i $$-0.796356\pi$$
−0.802235 + 0.597008i $$0.796356\pi$$
$$294$$ 0 0
$$295$$ 21.8564 1.27253
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 29.8564 1.72664
$$300$$ 0 0
$$301$$ −4.92820 −0.284057
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −31.3205 −1.79341
$$306$$ 0 0
$$307$$ 13.3205 0.760242 0.380121 0.924937i $$-0.375882\pi$$
0.380121 + 0.924937i $$0.375882\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −28.7321 −1.62925 −0.814623 0.579991i $$-0.803056\pi$$
−0.814623 + 0.579991i $$0.803056\pi$$
$$312$$ 0 0
$$313$$ −13.3205 −0.752920 −0.376460 0.926433i $$-0.622859\pi$$
−0.376460 + 0.926433i $$0.622859\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 11.2679 0.632871 0.316436 0.948614i $$-0.397514\pi$$
0.316436 + 0.948614i $$0.397514\pi$$
$$318$$ 0 0
$$319$$ 2.53590 0.141983
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6.73205 −0.374581
$$324$$ 0 0
$$325$$ 9.85641 0.546735
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −11.6603 −0.642851
$$330$$ 0 0
$$331$$ 4.39230 0.241423 0.120711 0.992688i $$-0.461482\pi$$
0.120711 + 0.992688i $$0.461482\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 10.9282 0.597072
$$336$$ 0 0
$$337$$ 3.07180 0.167331 0.0836657 0.996494i $$-0.473337\pi$$
0.0836657 + 0.996494i $$0.473337\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −35.8564 −1.92487 −0.962436 0.271507i $$-0.912478\pi$$
−0.962436 + 0.271507i $$0.912478\pi$$
$$348$$ 0 0
$$349$$ 23.8564 1.27700 0.638502 0.769620i $$-0.279554\pi$$
0.638502 + 0.769620i $$0.279554\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 7.80385 0.415357 0.207678 0.978197i $$-0.433409\pi$$
0.207678 + 0.978197i $$0.433409\pi$$
$$354$$ 0 0
$$355$$ 11.4641 0.608451
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4.53590 −0.239396 −0.119698 0.992810i $$-0.538193\pi$$
−0.119698 + 0.992810i $$0.538193\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −20.3923 −1.06738
$$366$$ 0 0
$$367$$ −25.8564 −1.34969 −0.674847 0.737958i $$-0.735790\pi$$
−0.674847 + 0.737958i $$0.735790\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.19615 −0.114019
$$372$$ 0 0
$$373$$ 25.7128 1.33136 0.665679 0.746238i $$-0.268142\pi$$
0.665679 + 0.746238i $$0.268142\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −2.92820 −0.150810
$$378$$ 0 0
$$379$$ −30.2487 −1.55377 −0.776886 0.629641i $$-0.783202\pi$$
−0.776886 + 0.629641i $$0.783202\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −21.4641 −1.09676 −0.548382 0.836228i $$-0.684756\pi$$
−0.548382 + 0.836228i $$0.684756\pi$$
$$384$$ 0 0
$$385$$ −9.46410 −0.482335
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −2.53590 −0.128575 −0.0642876 0.997931i $$-0.520477\pi$$
−0.0642876 + 0.997931i $$0.520477\pi$$
$$390$$ 0 0
$$391$$ −50.2487 −2.54119
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 16.0526 0.801627 0.400813 0.916160i $$-0.368728\pi$$
0.400813 + 0.916160i $$0.368728\pi$$
$$402$$ 0 0
$$403$$ 13.8564 0.690237
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.85641 0.0920187
$$408$$ 0 0
$$409$$ 35.7128 1.76588 0.882942 0.469481i $$-0.155559\pi$$
0.882942 + 0.469481i $$0.155559\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 27.8564 1.36742
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 10.1962 0.498115 0.249057 0.968489i $$-0.419879\pi$$
0.249057 + 0.968489i $$0.419879\pi$$
$$420$$ 0 0
$$421$$ −25.3205 −1.23405 −0.617023 0.786945i $$-0.711661\pi$$
−0.617023 + 0.786945i $$0.711661\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −16.5885 −0.804658
$$426$$ 0 0
$$427$$ −11.4641 −0.554787
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 22.0526 1.06223 0.531117 0.847298i $$-0.321772\pi$$
0.531117 + 0.847298i $$0.321772\pi$$
$$432$$ 0 0
$$433$$ −4.92820 −0.236834 −0.118417 0.992964i $$-0.537782\pi$$
−0.118417 + 0.992964i $$0.537782\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7.46410 0.357056
$$438$$ 0 0
$$439$$ 11.4641 0.547152 0.273576 0.961850i $$-0.411794\pi$$
0.273576 + 0.961850i $$0.411794\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 19.8564 0.943406 0.471703 0.881757i $$-0.343639\pi$$
0.471703 + 0.881757i $$0.343639\pi$$
$$444$$ 0 0
$$445$$ 28.3923 1.34592
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −34.5885 −1.63233 −0.816165 0.577819i $$-0.803904\pi$$
−0.816165 + 0.577819i $$0.803904\pi$$
$$450$$ 0 0
$$451$$ 15.7128 0.739887
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 10.9282 0.512322
$$456$$ 0 0
$$457$$ −8.39230 −0.392575 −0.196288 0.980546i $$-0.562889\pi$$
−0.196288 + 0.980546i $$0.562889\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 3.51666 0.163787 0.0818936 0.996641i $$-0.473903\pi$$
0.0818936 + 0.996641i $$0.473903\pi$$
$$462$$ 0 0
$$463$$ 13.8564 0.643962 0.321981 0.946746i $$-0.395651\pi$$
0.321981 + 0.946746i $$0.395651\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 3.26795 0.151223 0.0756113 0.997137i $$-0.475909\pi$$
0.0756113 + 0.997137i $$0.475909\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −17.0718 −0.784962
$$474$$ 0 0
$$475$$ 2.46410 0.113061
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −32.7321 −1.49557 −0.747783 0.663943i $$-0.768882\pi$$
−0.747783 + 0.663943i $$0.768882\pi$$
$$480$$ 0 0
$$481$$ −2.14359 −0.0977395
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −16.3923 −0.744336
$$486$$ 0 0
$$487$$ −33.8564 −1.53418 −0.767090 0.641539i $$-0.778296\pi$$
−0.767090 + 0.641539i $$0.778296\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −19.8564 −0.896107 −0.448054 0.894007i $$-0.647883\pi$$
−0.448054 + 0.894007i $$0.647883\pi$$
$$492$$ 0 0
$$493$$ 4.92820 0.221955
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4.19615 0.188223
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −11.6603 −0.519905 −0.259953 0.965621i $$-0.583707\pi$$
−0.259953 + 0.965621i $$0.583707\pi$$
$$504$$ 0 0
$$505$$ −45.3205 −2.01674
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −26.7846 −1.18721 −0.593603 0.804758i $$-0.702295\pi$$
−0.593603 + 0.804758i $$0.702295\pi$$
$$510$$ 0 0
$$511$$ −7.46410 −0.330192
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 18.9282 0.834076
$$516$$ 0 0
$$517$$ −40.3923 −1.77645
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −37.7128 −1.65223 −0.826114 0.563503i $$-0.809453\pi$$
−0.826114 + 0.563503i $$0.809453\pi$$
$$522$$ 0 0
$$523$$ −7.46410 −0.326382 −0.163191 0.986594i $$-0.552179\pi$$
−0.163191 + 0.986594i $$0.552179\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −23.3205 −1.01586
$$528$$ 0 0
$$529$$ 32.7128 1.42230
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −18.1436 −0.785886
$$534$$ 0 0
$$535$$ −22.3923 −0.968104
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −3.46410 −0.149209
$$540$$ 0 0
$$541$$ 4.14359 0.178147 0.0890735 0.996025i $$-0.471609\pi$$
0.0890735 + 0.996025i $$0.471609\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 32.3923 1.38753
$$546$$ 0 0
$$547$$ −36.1051 −1.54374 −0.771872 0.635778i $$-0.780679\pi$$
−0.771872 + 0.635778i $$0.780679\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −0.732051 −0.0311864
$$552$$ 0 0
$$553$$ −1.46410 −0.0622599
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 6.53590 0.276935 0.138467 0.990367i $$-0.455782\pi$$
0.138467 + 0.990367i $$0.455782\pi$$
$$558$$ 0 0
$$559$$ 19.7128 0.833763
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −21.4641 −0.904604 −0.452302 0.891865i $$-0.649397\pi$$
−0.452302 + 0.891865i $$0.649397\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −16.0526 −0.672958 −0.336479 0.941691i $$-0.609236\pi$$
−0.336479 + 0.941691i $$0.609236\pi$$
$$570$$ 0 0
$$571$$ −30.9282 −1.29431 −0.647153 0.762361i $$-0.724040\pi$$
−0.647153 + 0.762361i $$0.724040\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 18.3923 0.767012
$$576$$ 0 0
$$577$$ 10.6795 0.444593 0.222297 0.974979i $$-0.428645\pi$$
0.222297 + 0.974979i $$0.428645\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 10.1962 0.423008
$$582$$ 0 0
$$583$$ −7.60770 −0.315079
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −16.0526 −0.662560 −0.331280 0.943532i $$-0.607480\pi$$
−0.331280 + 0.943532i $$0.607480\pi$$
$$588$$ 0 0
$$589$$ 3.46410 0.142736
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 20.9808 0.861577 0.430788 0.902453i $$-0.358236\pi$$
0.430788 + 0.902453i $$0.358236\pi$$
$$594$$ 0 0
$$595$$ −18.3923 −0.754011
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −18.0526 −0.737608 −0.368804 0.929507i $$-0.620233\pi$$
−0.368804 + 0.929507i $$0.620233\pi$$
$$600$$ 0 0
$$601$$ −28.6410 −1.16829 −0.584146 0.811649i $$-0.698570\pi$$
−0.584146 + 0.811649i $$0.698570\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −2.73205 −0.111074
$$606$$ 0 0
$$607$$ 16.7846 0.681266 0.340633 0.940196i $$-0.389359\pi$$
0.340633 + 0.940196i $$0.389359\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 46.6410 1.88689
$$612$$ 0 0
$$613$$ 28.3923 1.14675 0.573377 0.819292i $$-0.305633\pi$$
0.573377 + 0.819292i $$0.305633\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 19.7128 0.793608 0.396804 0.917903i $$-0.370119\pi$$
0.396804 + 0.917903i $$0.370119\pi$$
$$618$$ 0 0
$$619$$ −6.53590 −0.262700 −0.131350 0.991336i $$-0.541931\pi$$
−0.131350 + 0.991336i $$0.541931\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 10.3923 0.416359
$$624$$ 0 0
$$625$$ −31.2487 −1.24995
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 3.60770 0.143848
$$630$$ 0 0
$$631$$ −7.07180 −0.281524 −0.140762 0.990043i $$-0.544955\pi$$
−0.140762 + 0.990043i $$0.544955\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 29.8564 1.18482
$$636$$ 0 0
$$637$$ 4.00000 0.158486
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −17.1244 −0.676371 −0.338186 0.941079i $$-0.609813\pi$$
−0.338186 + 0.941079i $$0.609813\pi$$
$$642$$ 0 0
$$643$$ −41.5692 −1.63933 −0.819665 0.572843i $$-0.805840\pi$$
−0.819665 + 0.572843i $$0.805840\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −5.12436 −0.201459 −0.100730 0.994914i $$-0.532118\pi$$
−0.100730 + 0.994914i $$0.532118\pi$$
$$648$$ 0 0
$$649$$ 27.7128 1.08782
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 26.9282 1.05378 0.526891 0.849933i $$-0.323358\pi$$
0.526891 + 0.849933i $$0.323358\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 13.2679 0.516846 0.258423 0.966032i $$-0.416797\pi$$
0.258423 + 0.966032i $$0.416797\pi$$
$$660$$ 0 0
$$661$$ −42.6410 −1.65854 −0.829272 0.558846i $$-0.811244\pi$$
−0.829272 + 0.558846i $$0.811244\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.73205 0.105944
$$666$$ 0 0
$$667$$ −5.46410 −0.211571
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −39.7128 −1.53310
$$672$$ 0 0
$$673$$ 16.2487 0.626342 0.313171 0.949697i $$-0.398609\pi$$
0.313171 + 0.949697i $$0.398609\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 8.92820 0.343139 0.171569 0.985172i $$-0.445116\pi$$
0.171569 + 0.985172i $$0.445116\pi$$
$$678$$ 0 0
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 18.7321 0.716762 0.358381 0.933575i $$-0.383329\pi$$
0.358381 + 0.933575i $$0.383329\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 8.78461 0.334667
$$690$$ 0 0
$$691$$ −39.3205 −1.49582 −0.747911 0.663799i $$-0.768943\pi$$
−0.747911 + 0.663799i $$0.768943\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −25.8564 −0.980789
$$696$$ 0 0
$$697$$ 30.5359 1.15663
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −12.0000 −0.453234 −0.226617 0.973984i $$-0.572767\pi$$
−0.226617 + 0.973984i $$0.572767\pi$$
$$702$$ 0 0
$$703$$ −0.535898 −0.0202118
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −16.5885 −0.623873
$$708$$ 0 0
$$709$$ −34.2487 −1.28624 −0.643119 0.765767i $$-0.722360\pi$$
−0.643119 + 0.765767i $$0.722360\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 25.8564 0.968330
$$714$$ 0 0
$$715$$ 37.8564 1.41575
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 38.5885 1.43911 0.719553 0.694437i $$-0.244347\pi$$
0.719553 + 0.694437i $$0.244347\pi$$
$$720$$ 0 0
$$721$$ 6.92820 0.258020
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −1.80385 −0.0669932
$$726$$ 0 0
$$727$$ 12.3923 0.459605 0.229803 0.973237i $$-0.426192\pi$$
0.229803 + 0.973237i $$0.426192\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −33.1769 −1.22709
$$732$$ 0 0
$$733$$ 0.535898 0.0197939 0.00989693 0.999951i $$-0.496850\pi$$
0.00989693 + 0.999951i $$0.496850\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 13.8564 0.510407
$$738$$ 0 0
$$739$$ 40.9282 1.50557 0.752784 0.658267i $$-0.228710\pi$$
0.752784 + 0.658267i $$0.228710\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −52.3013 −1.91875 −0.959374 0.282138i $$-0.908956\pi$$
−0.959374 + 0.282138i $$0.908956\pi$$
$$744$$ 0 0
$$745$$ −6.92820 −0.253830
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −8.19615 −0.299481
$$750$$ 0 0
$$751$$ 4.39230 0.160277 0.0801387 0.996784i $$-0.474464\pi$$
0.0801387 + 0.996784i $$0.474464\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 39.7128 1.44530
$$756$$ 0 0
$$757$$ 49.7128 1.80684 0.903421 0.428754i $$-0.141047\pi$$
0.903421 + 0.428754i $$0.141047\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10.7321 −0.389037 −0.194518 0.980899i $$-0.562314\pi$$
−0.194518 + 0.980899i $$0.562314\pi$$
$$762$$ 0 0
$$763$$ 11.8564 0.429231
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −32.0000 −1.15545
$$768$$ 0 0
$$769$$ 32.5359 1.17327 0.586637 0.809850i $$-0.300451\pi$$
0.586637 + 0.809850i $$0.300451\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 30.3923 1.09314 0.546568 0.837415i $$-0.315934\pi$$
0.546568 + 0.837415i $$0.315934\pi$$
$$774$$ 0 0
$$775$$ 8.53590 0.306619
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −4.53590 −0.162515
$$780$$ 0 0
$$781$$ 14.5359 0.520135
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 17.4641 0.623321
$$786$$ 0 0
$$787$$ 23.4641 0.836405 0.418202 0.908354i $$-0.362660\pi$$
0.418202 + 0.908354i $$0.362660\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0.732051 0.0260287
$$792$$ 0 0
$$793$$ 45.8564 1.62841
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 12.2487 0.433872 0.216936 0.976186i $$-0.430394\pi$$
0.216936 + 0.976186i $$0.430394\pi$$
$$798$$ 0 0
$$799$$ −78.4974 −2.77704
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −25.8564 −0.912453
$$804$$ 0 0
$$805$$ 20.3923 0.718734
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 25.0718 0.881477 0.440739 0.897635i $$-0.354717\pi$$
0.440739 + 0.897635i $$0.354717\pi$$
$$810$$ 0 0
$$811$$ 27.7128 0.973128 0.486564 0.873645i $$-0.338250\pi$$
0.486564 + 0.873645i $$0.338250\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 27.3205 0.956996
$$816$$ 0 0
$$817$$ 4.92820 0.172416
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −15.3205 −0.534689 −0.267345 0.963601i $$-0.586146\pi$$
−0.267345 + 0.963601i $$0.586146\pi$$
$$822$$ 0 0
$$823$$ −10.0000 −0.348578 −0.174289 0.984695i $$-0.555763\pi$$
−0.174289 + 0.984695i $$0.555763\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −0.483340 −0.0168074 −0.00840368 0.999965i $$-0.502675\pi$$
−0.00840368 + 0.999965i $$0.502675\pi$$
$$828$$ 0 0
$$829$$ 52.9282 1.83827 0.919136 0.393940i $$-0.128888\pi$$
0.919136 + 0.393940i $$0.128888\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −6.73205 −0.233252
$$834$$ 0 0
$$835$$ 44.7846 1.54984
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −8.78461 −0.303278 −0.151639 0.988436i $$-0.548455\pi$$
−0.151639 + 0.988436i $$0.548455\pi$$
$$840$$ 0 0
$$841$$ −28.4641 −0.981521
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −8.19615 −0.281956
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −4.00000 −0.137118
$$852$$ 0 0
$$853$$ −32.9282 −1.12744 −0.563720 0.825966i $$-0.690630\pi$$
−0.563720 + 0.825966i $$0.690630\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 6.39230 0.218357 0.109178 0.994022i $$-0.465178\pi$$
0.109178 + 0.994022i $$0.465178\pi$$
$$858$$ 0 0
$$859$$ 0.784610 0.0267705 0.0133853 0.999910i $$-0.495739\pi$$
0.0133853 + 0.999910i $$0.495739\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −22.0526 −0.750678 −0.375339 0.926888i $$-0.622474\pi$$
−0.375339 + 0.926888i $$0.622474\pi$$
$$864$$ 0 0
$$865$$ 28.3923 0.965367
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −5.07180 −0.172049
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −6.92820 −0.234216
$$876$$ 0 0
$$877$$ 54.7846 1.84994 0.924972 0.380034i $$-0.124088\pi$$
0.924972 + 0.380034i $$0.124088\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 27.5167 0.927060 0.463530 0.886081i $$-0.346583\pi$$
0.463530 + 0.886081i $$0.346583\pi$$
$$882$$ 0 0
$$883$$ 15.7128 0.528778 0.264389 0.964416i $$-0.414830\pi$$
0.264389 + 0.964416i $$0.414830\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −40.1051 −1.34660 −0.673299 0.739370i $$-0.735123\pi$$
−0.673299 + 0.739370i $$0.735123\pi$$
$$888$$ 0 0
$$889$$ 10.9282 0.366520
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 11.6603 0.390196
$$894$$ 0 0
$$895$$ −53.3205 −1.78231
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −2.53590 −0.0845769
$$900$$ 0 0
$$901$$ −14.7846 −0.492547
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −10.5359 −0.350225
$$906$$ 0 0
$$907$$ −30.6410 −1.01742 −0.508709 0.860938i $$-0.669877\pi$$
−0.508709 + 0.860938i $$0.669877\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −9.94744 −0.329573 −0.164787 0.986329i $$-0.552694\pi$$
−0.164787 + 0.986329i $$0.552694\pi$$
$$912$$ 0 0
$$913$$ 35.3205 1.16894
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −2.19615 −0.0725233
$$918$$ 0 0
$$919$$ −13.8564 −0.457081 −0.228540 0.973534i $$-0.573395\pi$$
−0.228540 + 0.973534i $$0.573395\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −16.7846 −0.552472
$$924$$ 0 0
$$925$$ −1.32051 −0.0434180
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 32.1962 1.05632 0.528161 0.849144i $$-0.322882\pi$$
0.528161 + 0.849144i $$0.322882\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −63.7128 −2.08363
$$936$$ 0 0
$$937$$ 12.1436 0.396714 0.198357 0.980130i $$-0.436439\pi$$
0.198357 + 0.980130i $$0.436439\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 28.6410 0.933670 0.466835 0.884344i $$-0.345394\pi$$
0.466835 + 0.884344i $$0.345394\pi$$
$$942$$ 0 0
$$943$$ −33.8564 −1.10252
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 33.0333 1.07344 0.536719 0.843761i $$-0.319663\pi$$
0.536719 + 0.843761i $$0.319663\pi$$
$$948$$ 0 0
$$949$$ 29.8564 0.969180
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −34.5885 −1.12043 −0.560215 0.828347i $$-0.689281\pi$$
−0.560215 + 0.828347i $$0.689281\pi$$
$$954$$ 0 0
$$955$$ 6.53590 0.211497
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 2.92820 0.0945566
$$960$$ 0 0
$$961$$ −19.0000 −0.612903
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 47.3205 1.52330
$$966$$ 0 0
$$967$$ 2.78461 0.0895470 0.0447735 0.998997i $$-0.485743\pi$$
0.0447735 + 0.998997i $$0.485743\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 54.9282 1.76273 0.881365 0.472436i $$-0.156625\pi$$
0.881365 + 0.472436i $$0.156625\pi$$
$$972$$ 0 0
$$973$$ −9.46410 −0.303405
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 21.1244 0.675828 0.337914 0.941177i $$-0.390279\pi$$
0.337914 + 0.941177i $$0.390279\pi$$
$$978$$ 0 0
$$979$$ 36.0000 1.15056
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −28.7846 −0.918086 −0.459043 0.888414i $$-0.651808\pi$$
−0.459043 + 0.888414i $$0.651808\pi$$
$$984$$ 0 0
$$985$$ −18.9282 −0.603103
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 36.7846 1.16968
$$990$$ 0 0
$$991$$ −11.2154 −0.356269 −0.178134 0.984006i $$-0.557006\pi$$
−0.178134 + 0.984006i $$0.557006\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −67.7128 −2.14664
$$996$$ 0 0
$$997$$ 30.1051 0.953439 0.476719 0.879056i $$-0.341826\pi$$
0.476719 + 0.879056i $$0.341826\pi$$
$$998$$ 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 9576.2.a.bf.1.1 2
3.2 odd 2 9576.2.a.br.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
9576.2.a.bf.1.1 2 1.1 even 1 trivial
9576.2.a.br.1.2 yes 2 3.2 odd 2