# Properties

 Label 55.9.d.a.54.1 Level $55$ Weight $9$ Character 55.54 Self dual yes Analytic conductor $22.406$ Analytic rank $0$ Dimension $1$ CM discriminant -55 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,9,Mod(54,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("55.54");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 55.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: yes Analytic conductor: $$22.4058235534$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 54.1 Character $$\chi$$ $$=$$ 55.54

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-23.0000 q^{2} +273.000 q^{4} +625.000 q^{5} +1282.00 q^{7} -391.000 q^{8} +6561.00 q^{9} +O(q^{10})$$ $$q-23.0000 q^{2} +273.000 q^{4} +625.000 q^{5} +1282.00 q^{7} -391.000 q^{8} +6561.00 q^{9} -14375.0 q^{10} +14641.0 q^{11} -30878.0 q^{13} -29486.0 q^{14} -60895.0 q^{16} -5438.00 q^{17} -150903. q^{18} +170625. q^{20} -336743. q^{22} +390625. q^{25} +710194. q^{26} +349986. q^{28} +706562. q^{31} +1.50068e6 q^{32} +125074. q^{34} +801250. q^{35} +1.79115e6 q^{36} -244375. q^{40} +5.56688e6 q^{43} +3.99699e6 q^{44} +4.10062e6 q^{45} -4.12128e6 q^{49} -8.98438e6 q^{50} -8.42969e6 q^{52} +9.15062e6 q^{55} -501262. q^{56} -1.23874e7 q^{59} -1.62509e7 q^{62} +8.41120e6 q^{63} -1.89265e7 q^{64} -1.92988e7 q^{65} -1.48457e6 q^{68} -1.84288e7 q^{70} -3.48394e7 q^{71} -2.56535e6 q^{72} +5.63706e7 q^{73} +1.87698e7 q^{77} -3.80594e7 q^{80} +4.30467e7 q^{81} +9.00978e7 q^{83} -3.39875e6 q^{85} -1.28038e8 q^{86} -5.72463e6 q^{88} +1.25358e8 q^{89} -9.43144e7 q^{90} -3.95856e7 q^{91} +9.47894e7 q^{98} +9.60596e7 q^{99} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −23.0000 −1.43750 −0.718750 0.695269i $$-0.755285\pi$$
−0.718750 + 0.695269i $$0.755285\pi$$
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 273.000 1.06641
$$5$$ 625.000 1.00000
$$6$$ 0 0
$$7$$ 1282.00 0.533944 0.266972 0.963704i $$-0.413977\pi$$
0.266972 + 0.963704i $$0.413977\pi$$
$$8$$ −391.000 −0.0954590
$$9$$ 6561.00 1.00000
$$10$$ −14375.0 −1.43750
$$11$$ 14641.0 1.00000
$$12$$ 0 0
$$13$$ −30878.0 −1.08112 −0.540562 0.841304i $$-0.681788\pi$$
−0.540562 + 0.841304i $$0.681788\pi$$
$$14$$ −29486.0 −0.767545
$$15$$ 0 0
$$16$$ −60895.0 −0.929184
$$17$$ −5438.00 −0.0651094 −0.0325547 0.999470i $$-0.510364\pi$$
−0.0325547 + 0.999470i $$0.510364\pi$$
$$18$$ −150903. −1.43750
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 170625. 1.06641
$$21$$ 0 0
$$22$$ −336743. −1.43750
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 390625. 1.00000
$$26$$ 710194. 1.55412
$$27$$ 0 0
$$28$$ 349986. 0.569401
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 706562. 0.765074 0.382537 0.923940i $$-0.375050\pi$$
0.382537 + 0.923940i $$0.375050\pi$$
$$32$$ 1.50068e6 1.43116
$$33$$ 0 0
$$34$$ 125074. 0.0935947
$$35$$ 801250. 0.533944
$$36$$ 1.79115e6 1.06641
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ −244375. −0.0954590
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 5.56688e6 1.62831 0.814157 0.580645i $$-0.197199\pi$$
0.814157 + 0.580645i $$0.197199\pi$$
$$44$$ 3.99699e6 1.06641
$$45$$ 4.10062e6 1.00000
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −4.12128e6 −0.714904
$$50$$ −8.98438e6 −1.43750
$$51$$ 0 0
$$52$$ −8.42969e6 −1.15292
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 9.15062e6 1.00000
$$56$$ −501262. −0.0509698
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −1.23874e7 −1.02228 −0.511141 0.859497i $$-0.670777\pi$$
−0.511141 + 0.859497i $$0.670777\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ −1.62509e7 −1.09979
$$63$$ 8.41120e6 0.533944
$$64$$ −1.89265e7 −1.12811
$$65$$ −1.92988e7 −1.08112
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ −1.48457e6 −0.0694330
$$69$$ 0 0
$$70$$ −1.84288e7 −0.767545
$$71$$ −3.48394e7 −1.37100 −0.685499 0.728074i $$-0.740416\pi$$
−0.685499 + 0.728074i $$0.740416\pi$$
$$72$$ −2.56535e6 −0.0954590
$$73$$ 5.63706e7 1.98500 0.992501 0.122237i $$-0.0390068\pi$$
0.992501 + 0.122237i $$0.0390068\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.87698e7 0.533944
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ −3.80594e7 −0.929184
$$81$$ 4.30467e7 1.00000
$$82$$ 0 0
$$83$$ 9.00978e7 1.89846 0.949230 0.314582i $$-0.101864\pi$$
0.949230 + 0.314582i $$0.101864\pi$$
$$84$$ 0 0
$$85$$ −3.39875e6 −0.0651094
$$86$$ −1.28038e8 −2.34070
$$87$$ 0 0
$$88$$ −5.72463e6 −0.0954590
$$89$$ 1.25358e8 1.99798 0.998990 0.0449296i $$-0.0143064\pi$$
0.998990 + 0.0449296i $$0.0143064\pi$$
$$90$$ −9.43144e7 −1.43750
$$91$$ −3.95856e7 −0.577260
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 9.47894e7 1.02767
$$99$$ 9.60596e7 1.00000
$$100$$ 1.06641e8 1.06641
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 1.20733e7 0.103203
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −2.59593e8 −1.98042 −0.990211 0.139582i $$-0.955424\pi$$
−0.990211 + 0.139582i $$0.955424\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ −2.10464e8 −1.43750
$$111$$ 0 0
$$112$$ −7.80674e7 −0.496132
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −2.02591e8 −1.08112
$$118$$ 2.84909e8 1.46953
$$119$$ −6.97152e6 −0.0347648
$$120$$ 0 0
$$121$$ 2.14359e8 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 1.92891e8 0.815880
$$125$$ 2.44141e8 1.00000
$$126$$ −1.93458e8 −0.767545
$$127$$ 4.58024e8 1.76065 0.880326 0.474370i $$-0.157324\pi$$
0.880326 + 0.474370i $$0.157324\pi$$
$$128$$ 5.11362e7 0.190497
$$129$$ 0 0
$$130$$ 4.43871e8 1.55412
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 2.12626e6 0.00621527
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 2.18741e8 0.569401
$$141$$ 0 0
$$142$$ 8.01305e8 1.97081
$$143$$ −4.52085e8 −1.08112
$$144$$ −3.99532e8 −0.929184
$$145$$ 0 0
$$146$$ −1.29652e9 −2.85344
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ −3.56787e7 −0.0651094
$$154$$ −4.31705e8 −0.767545
$$155$$ 4.41601e8 0.765074
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 9.37926e8 1.43116
$$161$$ 0 0
$$162$$ −9.90075e8 −1.43750
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −2.07225e9 −2.72904
$$167$$ −1.49541e9 −1.92262 −0.961310 0.275470i $$-0.911166\pi$$
−0.961310 + 0.275470i $$0.911166\pi$$
$$168$$ 0 0
$$169$$ 1.37720e8 0.168830
$$170$$ 7.81712e7 0.0935947
$$171$$ 0 0
$$172$$ 1.51976e9 1.73644
$$173$$ −1.77796e9 −1.98490 −0.992449 0.122659i $$-0.960858\pi$$
−0.992449 + 0.122659i $$0.960858\pi$$
$$174$$ 0 0
$$175$$ 5.00781e8 0.533944
$$176$$ −8.91564e8 −0.929184
$$177$$ 0 0
$$178$$ −2.88323e9 −2.87210
$$179$$ 1.87027e9 1.82176 0.910881 0.412669i $$-0.135403\pi$$
0.910881 + 0.412669i $$0.135403\pi$$
$$180$$ 1.11947e9 1.06641
$$181$$ 4.99713e8 0.465593 0.232797 0.972525i $$-0.425212\pi$$
0.232797 + 0.972525i $$0.425212\pi$$
$$182$$ 9.10469e8 0.829812
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −7.96178e7 −0.0651094
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −9.57523e8 −0.719475 −0.359738 0.933054i $$-0.617134\pi$$
−0.359738 + 0.933054i $$0.617134\pi$$
$$192$$ 0 0
$$193$$ 2.62992e9 1.89545 0.947727 0.319082i $$-0.103374\pi$$
0.947727 + 0.319082i $$0.103374\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −1.12511e9 −0.762378
$$197$$ 1.94360e9 1.29045 0.645227 0.763991i $$-0.276763\pi$$
0.645227 + 0.763991i $$0.276763\pi$$
$$198$$ −2.20937e9 −1.43750
$$199$$ −8.78518e8 −0.560194 −0.280097 0.959972i $$-0.590367\pi$$
−0.280097 + 0.959972i $$0.590367\pi$$
$$200$$ −1.52734e8 −0.0954590
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 1.88032e9 1.00456
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 5.97063e9 2.84686
$$215$$ 3.47930e9 1.62831
$$216$$ 0 0
$$217$$ 9.05812e8 0.408507
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 2.49812e9 1.06641
$$221$$ 1.67915e8 0.0703913
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 1.92387e9 0.764160
$$225$$ 2.56289e9 1.00000
$$226$$ 0 0
$$227$$ −9.44174e8 −0.355589 −0.177795 0.984068i $$-0.556896\pi$$
−0.177795 + 0.984068i $$0.556896\pi$$
$$228$$ 0 0
$$229$$ −1.41584e8 −0.0514841 −0.0257420 0.999669i $$-0.508195\pi$$
−0.0257420 + 0.999669i $$0.508195\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.22732e9 −0.755716 −0.377858 0.925863i $$-0.623339\pi$$
−0.377858 + 0.925863i $$0.623339\pi$$
$$234$$ 4.65958e9 1.55412
$$235$$ 0 0
$$236$$ −3.38175e9 −1.09017
$$237$$ 0 0
$$238$$ 1.60345e8 0.0499744
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ −4.93025e9 −1.43750
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −2.57580e9 −0.714904
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −2.76266e8 −0.0730332
$$249$$ 0 0
$$250$$ −5.61523e9 −1.43750
$$251$$ −5.77805e9 −1.45575 −0.727874 0.685711i $$-0.759491\pi$$
−0.727874 + 0.685711i $$0.759491\pi$$
$$252$$ 2.29626e9 0.569401
$$253$$ 0 0
$$254$$ −1.05346e10 −2.53094
$$255$$ 0 0
$$256$$ 3.66906e9 0.854270
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −5.26856e9 −1.15292
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 4.13013e9 0.863258 0.431629 0.902051i $$-0.357939\pi$$
0.431629 + 0.902051i $$0.357939\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −9.70172e9 −1.85285 −0.926424 0.376482i $$-0.877134\pi$$
−0.926424 + 0.376482i $$0.877134\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 3.31147e8 0.0604986
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 5.71914e9 1.00000
$$276$$ 0 0
$$277$$ −1.15289e10 −1.95825 −0.979125 0.203260i $$-0.934846\pi$$
−0.979125 + 0.203260i $$0.934846\pi$$
$$278$$ 0 0
$$279$$ 4.63575e9 0.765074
$$280$$ −3.13289e8 −0.0509698
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 5.58002e9 0.869942 0.434971 0.900444i $$-0.356759\pi$$
0.434971 + 0.900444i $$0.356759\pi$$
$$284$$ −9.51114e9 −1.46204
$$285$$ 0 0
$$286$$ 1.03980e10 1.55412
$$287$$ 0 0
$$288$$ 9.84597e9 1.43116
$$289$$ −6.94619e9 −0.995761
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 1.53892e10 2.11682
$$293$$ 1.46797e10 1.99180 0.995902 0.0904403i $$-0.0288274\pi$$
0.995902 + 0.0904403i $$0.0288274\pi$$
$$294$$ 0 0
$$295$$ −7.74210e9 −1.02228
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 7.13674e9 0.869429
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 8.20611e8 0.0935947
$$307$$ −1.61270e10 −1.81551 −0.907756 0.419500i $$-0.862206\pi$$
−0.907756 + 0.419500i $$0.862206\pi$$
$$308$$ 5.12415e9 0.569401
$$309$$ 0 0
$$310$$ −1.01568e10 −1.09979
$$311$$ −1.76962e10 −1.89164 −0.945822 0.324685i $$-0.894742\pi$$
−0.945822 + 0.324685i $$0.894742\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 5.25700e9 0.533944
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −1.18291e10 −1.12811
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.17518e10 1.06641
$$325$$ −1.20617e10 −1.08112
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2.49727e9 0.208043 0.104021 0.994575i $$-0.466829\pi$$
0.104021 + 0.994575i $$0.466829\pi$$
$$332$$ 2.45967e10 2.02453
$$333$$ 0 0
$$334$$ 3.43943e10 2.76376
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −2.10737e10 −1.63388 −0.816941 0.576721i $$-0.804332\pi$$
−0.816941 + 0.576721i $$0.804332\pi$$
$$338$$ −3.16756e9 −0.242694
$$339$$ 0 0
$$340$$ −9.27859e8 −0.0694330
$$341$$ 1.03448e10 0.765074
$$342$$ 0 0
$$343$$ −1.26740e10 −0.915663
$$344$$ −2.17665e9 −0.155437
$$345$$ 0 0
$$346$$ 4.08931e10 2.85329
$$347$$ −2.38583e10 −1.64559 −0.822796 0.568336i $$-0.807587\pi$$
−0.822796 + 0.568336i $$0.807587\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ −1.15180e10 −0.767545
$$351$$ 0 0
$$352$$ 2.19715e10 1.43116
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ −2.17746e10 −1.37100
$$356$$ 3.42227e10 2.13066
$$357$$ 0 0
$$358$$ −4.30162e10 −2.61878
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ −1.60334e9 −0.0954590
$$361$$ 1.69836e10 1.00000
$$362$$ −1.14934e10 −0.669290
$$363$$ 0 0
$$364$$ −1.08069e10 −0.615594
$$365$$ 3.52316e10 1.98500
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 7.39178e9 0.381868 0.190934 0.981603i $$-0.438848\pi$$
0.190934 + 0.981603i $$0.438848\pi$$
$$374$$ 1.83121e9 0.0935947
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −4.12608e10 −1.99977 −0.999887 0.0150510i $$-0.995209\pi$$
−0.999887 + 0.0150510i $$0.995209\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 2.20230e10 1.03425
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 1.17311e10 0.533944
$$386$$ −6.04882e10 −2.72472
$$387$$ 3.65243e10 1.62831
$$388$$ 0 0
$$389$$ 3.96070e10 1.72971 0.864855 0.502022i $$-0.167411\pi$$
0.864855 + 0.502022i $$0.167411\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 1.61142e9 0.0682440
$$393$$ 0 0
$$394$$ −4.47028e10 −1.85503
$$395$$ 0 0
$$396$$ 2.62243e10 1.06641
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 2.02059e10 0.805279
$$399$$ 0 0
$$400$$ −2.37871e10 −0.929184
$$401$$ 3.23408e10 1.25076 0.625380 0.780321i $$-0.284944\pi$$
0.625380 + 0.780321i $$0.284944\pi$$
$$402$$ 0 0
$$403$$ −2.18172e10 −0.827140
$$404$$ 0 0
$$405$$ 2.69042e10 1.00000
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −1.58806e10 −0.545841
$$414$$ 0 0
$$415$$ 5.63111e10 1.89846
$$416$$ −4.63380e10 −1.54726
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −3.73828e10 −1.21287 −0.606437 0.795132i $$-0.707402\pi$$
−0.606437 + 0.795132i $$0.707402\pi$$
$$420$$ 0 0
$$421$$ 4.96899e10 1.58176 0.790879 0.611973i $$-0.209624\pi$$
0.790879 + 0.611973i $$0.209624\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.12422e9 −0.0651094
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −7.08688e10 −2.11193
$$429$$ 0 0
$$430$$ −8.00239e10 −2.34070
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ −2.08337e10 −0.587229
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ −3.57789e9 −0.0954590
$$441$$ −2.70397e10 −0.714904
$$442$$ −3.86203e9 −0.101188
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 0 0
$$445$$ 7.83486e10 1.99798
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −2.42638e10 −0.602348
$$449$$ −6.73426e10 −1.65693 −0.828465 0.560040i $$-0.810786\pi$$
−0.828465 + 0.560040i $$0.810786\pi$$
$$450$$ −5.89465e10 −1.43750
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 2.17160e10 0.511159
$$455$$ −2.47410e10 −0.577260
$$456$$ 0 0
$$457$$ −3.09729e10 −0.710096 −0.355048 0.934848i $$-0.615536\pi$$
−0.355048 + 0.934848i $$0.615536\pi$$
$$458$$ 3.25644e9 0.0740083
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 5.12284e10 1.08634
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ −5.53072e10 −1.15292
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.84346e9 0.0975860
$$473$$ 8.15047e10 1.62831
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1.90322e9 −0.0370734
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 5.85200e10 1.06641
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 5.92434e10 1.02767
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 6.00373e10 1.00000
$$496$$ −4.30261e10 −0.710895
$$497$$ −4.46641e10 −0.732036
$$498$$ 0 0
$$499$$ 9.34532e10 1.50727 0.753637 0.657291i $$-0.228298\pi$$
0.753637 + 0.657291i $$0.228298\pi$$
$$500$$ 6.66504e10 1.06641
$$501$$ 0 0
$$502$$ 1.32895e11 2.09264
$$503$$ 1.72848e10 0.270018 0.135009 0.990844i $$-0.456894\pi$$
0.135009 + 0.990844i $$0.456894\pi$$
$$504$$ −3.28878e9 −0.0509698
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 1.25041e11 1.87757
$$509$$ 1.14873e11 1.71138 0.855690 0.517489i $$-0.173133\pi$$
0.855690 + 0.517489i $$0.173133\pi$$
$$510$$ 0 0
$$511$$ 7.22671e10 1.05988
$$512$$ −9.74793e10 −1.41851
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 7.54581e9 0.103203
$$521$$ −8.53007e10 −1.15771 −0.578857 0.815429i $$-0.696501\pi$$
−0.578857 + 0.815429i $$0.696501\pi$$
$$522$$ 0 0
$$523$$ 8.41573e10 1.12483 0.562413 0.826857i $$-0.309873\pi$$
0.562413 + 0.826857i $$0.309873\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −9.49930e10 −1.24093
$$527$$ −3.84228e9 −0.0498135
$$528$$ 0 0
$$529$$ 7.83110e10 1.00000
$$530$$ 0 0
$$531$$ −8.12735e10 −1.02228
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −1.62245e11 −1.98042
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 2.23140e11 2.66347
$$539$$ −6.03396e10 −0.714904
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ −8.16070e9 −0.0931820
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2.38099e10 0.265955 0.132978 0.991119i $$-0.457546\pi$$
0.132978 + 0.991119i $$0.457546\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ −1.31540e11 −1.43750
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 2.65164e11 2.81498
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1.21370e11 −1.26093 −0.630466 0.776217i $$-0.717136\pi$$
−0.630466 + 0.776217i $$0.717136\pi$$
$$558$$ −1.06622e11 −1.09979
$$559$$ −1.71894e11 −1.76041
$$560$$ −4.87921e10 −0.496132
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −2.00486e11 −1.99549 −0.997746 0.0671044i $$-0.978624\pi$$
−0.997746 + 0.0671044i $$0.978624\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −1.28341e11 −1.25054
$$567$$ 5.51859e10 0.533944
$$568$$ 1.36222e10 0.130874
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ −1.23419e11 −1.15292
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −1.24177e11 −1.12811
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 1.59762e11 1.43141
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 1.15505e11 1.01367
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −2.20409e10 −0.189486
$$585$$ −1.26619e11 −1.08112
$$586$$ −3.37633e11 −2.86322
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 1.78068e11 1.46953
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 2.29888e11 1.85908 0.929538 0.368726i $$-0.120206\pi$$
0.929538 + 0.368726i $$0.120206\pi$$
$$594$$ 0 0
$$595$$ −4.35720e9 −0.0347648
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −2.51941e11 −1.95701 −0.978503 0.206234i $$-0.933879\pi$$
−0.978503 + 0.206234i $$0.933879\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ −1.64145e11 −1.24980
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 1.33974e11 1.00000
$$606$$ 0 0
$$607$$ −1.40814e11 −1.03727 −0.518634 0.854996i $$-0.673559\pi$$
−0.518634 + 0.854996i $$0.673559\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −9.74029e9 −0.0694330
$$613$$ −2.63116e11 −1.86340 −0.931700 0.363228i $$-0.881674\pi$$
−0.931700 + 0.363228i $$0.881674\pi$$
$$614$$ 3.70920e11 2.60980
$$615$$ 0 0
$$616$$ −7.33898e9 −0.0509698
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ −2.87787e11 −1.96024 −0.980120 0.198406i $$-0.936424\pi$$
−0.980120 + 0.198406i $$0.936424\pi$$
$$620$$ 1.20557e11 0.815880
$$621$$ 0 0
$$622$$ 4.07014e11 2.71924
$$623$$ 1.60709e11 1.06681
$$624$$ 0 0
$$625$$ 1.52588e11 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ −1.20911e11 −0.767545
$$631$$ −2.33288e11 −1.47155 −0.735774 0.677227i $$-0.763181\pi$$
−0.735774 + 0.677227i $$0.763181\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 2.86265e11 1.76065
$$636$$ 0 0
$$637$$ 1.27257e11 0.772900
$$638$$ 0 0
$$639$$ −2.28581e11 −1.37100
$$640$$ 3.19601e10 0.190497
$$641$$ 1.96069e11 1.16138 0.580692 0.814123i $$-0.302782\pi$$
0.580692 + 0.814123i $$0.302782\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ −1.68313e10 −0.0954590
$$649$$ −1.81363e11 −1.02228
$$650$$ 2.77420e11 1.55412
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 3.69847e11 1.98500
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 7.85325e10 0.411380 0.205690 0.978617i $$-0.434056\pi$$
0.205690 + 0.978617i $$0.434056\pi$$
$$662$$ −5.74371e10 −0.299062
$$663$$ 0 0
$$664$$ −3.52282e10 −0.181225
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −4.08246e11 −2.05029
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −4.09942e11 −1.99831 −0.999153 0.0411501i $$-0.986898\pi$$
−0.999153 + 0.0411501i $$0.986898\pi$$
$$674$$ 4.84694e11 2.34870
$$675$$ 0 0
$$676$$ 3.75976e10 0.180042
$$677$$ 3.81542e11 1.81630 0.908151 0.418644i $$-0.137494\pi$$
0.908151 + 0.418644i $$0.137494\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 1.32891e9 0.00621527
$$681$$ 0 0
$$682$$ −2.37930e11 −1.09979
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 2.91501e11 1.31627
$$687$$ 0 0
$$688$$ −3.38995e11 −1.51300
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −9.91396e10 −0.434846 −0.217423 0.976078i $$-0.569765\pi$$
−0.217423 + 0.976078i $$0.569765\pi$$
$$692$$ −4.85384e11 −2.11671
$$693$$ 1.23148e11 0.533944
$$694$$ 5.48742e11 2.36554
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 1.36713e11 0.569401
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −2.77104e11 −1.12811
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 3.95304e11 1.56440 0.782198 0.623030i $$-0.214099\pi$$
0.782198 + 0.623030i $$0.214099\pi$$
$$710$$ 5.00816e11 1.97081
$$711$$ 0 0
$$712$$ −4.90149e10 −0.190725
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −2.82553e11 −1.08112
$$716$$ 5.10583e11 1.94274
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −2.65761e11 −0.994433 −0.497216 0.867627i $$-0.665645\pi$$
−0.497216 + 0.867627i $$0.665645\pi$$
$$720$$ −2.49708e11 −0.929184
$$721$$ 0 0
$$722$$ −3.90622e11 −1.43750
$$723$$ 0 0
$$724$$ 1.36422e11 0.496511
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 1.54780e10 0.0551047
$$729$$ 2.82430e11 1.00000
$$730$$ −8.10327e11 −2.85344
$$731$$ −3.02727e10 −0.106019
$$732$$ 0 0
$$733$$ −5.78092e10 −0.200254 −0.100127 0.994975i $$-0.531925\pi$$
−0.100127 + 0.994975i $$0.531925\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −5.62478e11 −1.84566 −0.922828 0.385213i $$-0.874128\pi$$
−0.922828 + 0.385213i $$0.874128\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −1.70011e11 −0.548936
$$747$$ 5.91131e11 1.89846
$$748$$ −2.17356e10 −0.0694330
$$749$$ −3.32798e11 −1.05743
$$750$$ 0 0
$$751$$ 1.72938e11 0.543665 0.271833 0.962345i $$-0.412370\pi$$
0.271833 + 0.962345i $$0.412370\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 9.48998e11 2.87467
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2.61404e11 −0.767253
$$765$$ −2.22992e10 −0.0651094
$$766$$ 0 0
$$767$$ 3.82497e11 1.10521
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ −2.69815e11 −0.767545
$$771$$ 0 0
$$772$$ 7.17968e11 2.02132
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ −8.40059e11 −2.34070
$$775$$ 2.76001e11 0.765074
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −9.10960e11 −2.48646
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −5.10083e11 −1.37100
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 2.50965e11 0.664277
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −4.07899e10 −0.106330 −0.0531648 0.998586i $$-0.516931\pi$$
−0.0531648 + 0.998586i $$0.516931\pi$$
$$788$$ 5.30603e11 1.37615
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −3.75593e10 −0.0954590
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −2.39835e11 −0.597394
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 5.86204e11 1.43116
$$801$$ 8.22472e11 1.99798
$$802$$ −7.43839e11 −1.79797
$$803$$ 8.25321e11 1.98500
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 5.01796e11 1.18901
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ −6.18797e11 −1.43750
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −2.59721e11 −0.577260
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 3.65254e11 0.784647
$$827$$ −9.34829e11 −1.99853 −0.999264 0.0383566i $$-0.987788\pi$$
−0.999264 + 0.0383566i $$0.987788\pi$$
$$828$$ 0 0
$$829$$ −9.43477e11 −1.99762 −0.998811 0.0487573i $$-0.984474\pi$$
−0.998811 + 0.0487573i $$0.984474\pi$$
$$830$$ −1.29516e12 −2.72904
$$831$$ 0 0
$$832$$ 5.84414e11 1.21963
$$833$$ 2.24115e10 0.0465469
$$834$$ 0 0
$$835$$ −9.34629e11 −1.92262
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 8.59804e11 1.74351
$$839$$ −4.75598e11 −0.959824 −0.479912 0.877317i $$-0.659331\pi$$
−0.479912 + 0.877317i $$0.659331\pi$$
$$840$$ 0 0
$$841$$ 5.00246e11 1.00000
$$842$$ −1.14287e12 −2.27378
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 8.60751e10 0.168830
$$846$$ 0 0
$$847$$ 2.74808e11 0.533944
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 4.88570e10 0.0935947
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −9.31522e11 −1.75953 −0.879766 0.475408i $$-0.842301\pi$$
−0.879766 + 0.475408i $$0.842301\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 1.01501e11 0.189049
$$857$$ −9.72243e11 −1.80240 −0.901201 0.433402i $$-0.857313\pi$$
−0.901201 + 0.433402i $$0.857313\pi$$
$$858$$ 0 0
$$859$$ 5.59509e11 1.02762 0.513812 0.857903i $$-0.328233\pi$$
0.513812 + 0.857903i $$0.328233\pi$$
$$860$$ 9.49849e11 1.73644
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ −1.11123e12 −1.98490
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 2.47287e11 0.435634
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3.12988e11 0.533944
$$876$$ 0 0
$$877$$ 1.01344e12 1.71316 0.856580 0.516014i $$-0.172585\pi$$
0.856580 + 0.516014i $$0.172585\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ −5.57227e11 −0.929184
$$881$$ 1.16134e12 1.92777 0.963883 0.266325i $$-0.0858093\pi$$
0.963883 + 0.266325i $$0.0858093\pi$$
$$882$$ 6.21913e11 1.02767
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 4.58407e10 0.0750658
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −3.94108e11 −0.636679 −0.318340 0.947977i $$-0.603125\pi$$
−0.318340 + 0.947977i $$0.603125\pi$$
$$888$$ 0 0
$$889$$ 5.87187e11 0.940090
$$890$$ −1.80202e12 −2.87210
$$891$$ 6.30247e11 1.00000
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 1.16892e12 1.82176
$$896$$ 6.55565e10 0.101715
$$897$$ 0 0
$$898$$ 1.54888e12 2.38184
$$899$$ 0 0
$$900$$ 6.99669e11 1.06641
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 3.12321e11 0.465593
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ −2.57759e11 −0.379202
$$909$$ 0 0
$$910$$ 5.69043e11 0.829812
$$911$$ 5.12323e11 0.743824 0.371912 0.928268i $$-0.378702\pi$$
0.371912 + 0.928268i $$0.378702\pi$$
$$912$$ 0 0
$$913$$ 1.31912e12 1.89846
$$914$$ 7.12377e11 1.02076
$$915$$ 0 0
$$916$$ −3.86525e10 −0.0549029
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 1.07577e12 1.48222
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −4.16048e11 −0.558574 −0.279287 0.960208i $$-0.590098\pi$$
−0.279287 + 0.960208i $$0.590098\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −6.08058e11 −0.805901
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −4.97611e10 −0.0651094
$$936$$ 7.92129e10 0.103203
$$937$$ 5.90180e11 0.765643 0.382822 0.923822i $$-0.374952\pi$$
0.382822 + 0.923822i $$0.374952\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 7.54328e11 0.949888
$$945$$ 0 0
$$946$$ −1.87461e12 −2.34070
$$947$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$948$$ 0 0
$$949$$ −1.74061e12 −2.14603
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 2.72586e9 0.00331861
$$953$$ −2.00269e11 −0.242796 −0.121398 0.992604i $$-0.538738\pi$$
−0.121398 + 0.992604i $$0.538738\pi$$
$$954$$ 0 0
$$955$$ −5.98452e11 −0.719475
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −3.53661e11 −0.414662
$$962$$ 0 0
$$963$$ −1.70319e12 −1.98042
$$964$$ 0 0
$$965$$ 1.64370e12 1.89545
$$966$$ 0 0
$$967$$ 7.29707e11 0.834531 0.417266 0.908785i $$-0.362988\pi$$
0.417266 + 0.908785i $$0.362988\pi$$
$$968$$ −8.38143e10 −0.0954590
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −1.22257e12 −1.37530 −0.687650 0.726042i $$-0.741358\pi$$
−0.687650 + 0.726042i $$0.741358\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 1.83536e12 1.99798
$$980$$ −7.03193e11 −0.762378
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 1.21475e12 1.29045
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ −1.38086e12 −1.43750
$$991$$ 1.28116e12 1.32834 0.664168 0.747583i $$-0.268786\pi$$
0.664168 + 0.747583i $$0.268786\pi$$
$$992$$ 1.06032e12 1.09494
$$993$$ 0 0
$$994$$ 1.02727e12 1.05230
$$995$$ −5.49074e11 −0.560194
$$996$$ 0 0
$$997$$ −1.96549e12 −1.98925 −0.994627 0.103520i $$-0.966990\pi$$
−0.994627 + 0.103520i $$0.966990\pi$$
$$998$$ −2.14942e12 −2.16671
$$999$$ 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 55.9.d.a.54.1 1
5.4 even 2 55.9.d.b.54.1 yes 1
11.10 odd 2 55.9.d.b.54.1 yes 1
55.54 odd 2 CM 55.9.d.a.54.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
55.9.d.a.54.1 1 1.1 even 1 trivial
55.9.d.a.54.1 1 55.54 odd 2 CM
55.9.d.b.54.1 yes 1 5.4 even 2
55.9.d.b.54.1 yes 1 11.10 odd 2