# Properties

 Label 121.4.a.a.1.1 Level $121$ Weight $4$ Character 121.1 Self dual yes Analytic conductor $7.139$ Analytic rank $0$ Dimension $1$ CM discriminant -11 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,4,Mod(1,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 121.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: $$7.13923111069$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 121.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+8.00000 q^{3} -8.00000 q^{4} +18.0000 q^{5} +37.0000 q^{9} +O(q^{10})$$ $$q+8.00000 q^{3} -8.00000 q^{4} +18.0000 q^{5} +37.0000 q^{9} -64.0000 q^{12} +144.000 q^{15} +64.0000 q^{16} -144.000 q^{20} -108.000 q^{23} +199.000 q^{25} +80.0000 q^{27} +340.000 q^{31} -296.000 q^{36} -434.000 q^{37} +666.000 q^{45} -36.0000 q^{47} +512.000 q^{48} -343.000 q^{49} -738.000 q^{53} -720.000 q^{59} -1152.00 q^{60} -512.000 q^{64} -416.000 q^{67} -864.000 q^{69} +612.000 q^{71} +1592.00 q^{75} +1152.00 q^{80} -359.000 q^{81} +1674.00 q^{89} +864.000 q^{92} +2720.00 q^{93} -34.0000 q^{97} +O(q^{100})$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 8.00000 1.53960 0.769800 0.638285i $$-0.220356\pi$$
0.769800 + 0.638285i $$0.220356\pi$$
$$4$$ −8.00000 −1.00000
$$5$$ 18.0000 1.60997 0.804984 0.593296i $$-0.202174\pi$$
0.804984 + 0.593296i $$0.202174\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ 37.0000 1.37037
$$10$$ 0 0
$$11$$ 0 0
$$12$$ −64.0000 −1.53960
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 144.000 2.47871
$$16$$ 64.0000 1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −144.000 −1.60997
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −108.000 −0.979111 −0.489556 0.871972i $$-0.662841\pi$$
−0.489556 + 0.871972i $$0.662841\pi$$
$$24$$ 0 0
$$25$$ 199.000 1.59200
$$26$$ 0 0
$$27$$ 80.0000 0.570222
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 340.000 1.96986 0.984932 0.172940i $$-0.0553268\pi$$
0.984932 + 0.172940i $$0.0553268\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −296.000 −1.37037
$$37$$ −434.000 −1.92836 −0.964178 0.265257i $$-0.914543\pi$$
−0.964178 + 0.265257i $$0.914543\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 666.000 2.20625
$$46$$ 0 0
$$47$$ −36.0000 −0.111726 −0.0558632 0.998438i $$-0.517791\pi$$
−0.0558632 + 0.998438i $$0.517791\pi$$
$$48$$ 512.000 1.53960
$$49$$ −343.000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −738.000 −1.91268 −0.956341 0.292255i $$-0.905595\pi$$
−0.956341 + 0.292255i $$0.905595\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −720.000 −1.58875 −0.794373 0.607430i $$-0.792200\pi$$
−0.794373 + 0.607430i $$0.792200\pi$$
$$60$$ −1152.00 −2.47871
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −512.000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −416.000 −0.758545 −0.379272 0.925285i $$-0.623826\pi$$
−0.379272 + 0.925285i $$0.623826\pi$$
$$68$$ 0 0
$$69$$ −864.000 −1.50744
$$70$$ 0 0
$$71$$ 612.000 1.02297 0.511486 0.859292i $$-0.329095\pi$$
0.511486 + 0.859292i $$0.329095\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ 1592.00 2.45104
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 1152.00 1.60997
$$81$$ −359.000 −0.492455
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1674.00 1.99375 0.996874 0.0790026i $$-0.0251735\pi$$
0.996874 + 0.0790026i $$0.0251735\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 864.000 0.979111
$$93$$ 2720.00 3.03280
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −34.0000 −0.0355895 −0.0177947 0.999842i $$-0.505665\pi$$
−0.0177947 + 0.999842i $$0.505665\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −1592.00 −1.59200
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 1172.00 1.12117 0.560585 0.828097i $$-0.310576\pi$$
0.560585 + 0.828097i $$0.310576\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ −640.000 −0.570222
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ −3472.00 −2.96890
$$112$$ 0 0
$$113$$ 2142.00 1.78321 0.891604 0.452817i $$-0.149581\pi$$
0.891604 + 0.452817i $$0.149581\pi$$
$$114$$ 0 0
$$115$$ −1944.00 −1.57634
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −2720.00 −1.96986
$$125$$ 1332.00 0.953102
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 1440.00 0.918040
$$136$$ 0 0
$$137$$ 1206.00 0.752084 0.376042 0.926603i $$-0.377285\pi$$
0.376042 + 0.926603i $$0.377285\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −288.000 −0.172014
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 2368.00 1.37037
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2744.00 −1.53960
$$148$$ 3472.00 1.92836
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 6120.00 3.17142
$$156$$ 0 0
$$157$$ −1334.00 −0.678120 −0.339060 0.940765i $$-0.610109\pi$$
−0.339060 + 0.940765i $$0.610109\pi$$
$$158$$ 0 0
$$159$$ −5904.00 −2.94477
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 3728.00 1.79141 0.895704 0.444651i $$-0.146672\pi$$
0.895704 + 0.444651i $$0.146672\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −2197.00 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −5760.00 −2.44603
$$178$$ 0 0
$$179$$ −2016.00 −0.841804 −0.420902 0.907106i $$-0.638286\pi$$
−0.420902 + 0.907106i $$0.638286\pi$$
$$180$$ −5328.00 −2.20625
$$181$$ −2050.00 −0.841852 −0.420926 0.907095i $$-0.638295\pi$$
−0.420926 + 0.907095i $$0.638295\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −7812.00 −3.10459
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 288.000 0.111726
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 5220.00 1.97752 0.988759 0.149518i $$-0.0477722\pi$$
0.988759 + 0.149518i $$0.0477722\pi$$
$$192$$ −4096.00 −1.53960
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 2744.00 1.00000
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ 3940.00 1.40351 0.701757 0.712417i $$-0.252399\pi$$
0.701757 + 0.712417i $$0.252399\pi$$
$$200$$ 0 0
$$201$$ −3328.00 −1.16786
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −3996.00 −1.34174
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 5904.00 1.91268
$$213$$ 4896.00 1.57497
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 668.000 0.200595 0.100297 0.994958i $$-0.468021\pi$$
0.100297 + 0.994958i $$0.468021\pi$$
$$224$$ 0 0
$$225$$ 7363.00 2.18163
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 3310.00 0.955157 0.477579 0.878589i $$-0.341515\pi$$
0.477579 + 0.878589i $$0.341515\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ −648.000 −0.179876
$$236$$ 5760.00 1.58875
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 9216.00 2.47871
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ −5032.00 −1.32841
$$244$$ 0 0
$$245$$ −6174.00 −1.60997
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 648.000 0.162954 0.0814769 0.996675i $$-0.474036\pi$$
0.0814769 + 0.996675i $$0.474036\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 4096.00 1.00000
$$257$$ −8046.00 −1.95290 −0.976451 0.215740i $$-0.930784\pi$$
−0.976451 + 0.215740i $$0.930784\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −13284.0 −3.07936
$$266$$ 0 0
$$267$$ 13392.0 3.06958
$$268$$ 3328.00 0.758545
$$269$$ 2790.00 0.632377 0.316188 0.948696i $$-0.397597\pi$$
0.316188 + 0.948696i $$0.397597\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 6912.00 1.50744
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 0 0
$$279$$ 12580.0 2.69944
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ −4896.00 −1.02297
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4913.00 −1.00000
$$290$$ 0 0
$$291$$ −272.000 −0.0547935
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ −12960.0 −2.55783
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −12736.0 −2.45104
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 9376.00 1.72616
$$310$$ 0 0
$$311$$ 9468.00 1.72631 0.863153 0.504943i $$-0.168486\pi$$
0.863153 + 0.504943i $$0.168486\pi$$
$$312$$ 0 0
$$313$$ −10982.0 −1.98319 −0.991596 0.129370i $$-0.958705\pi$$
−0.991596 + 0.129370i $$0.958705\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5994.00 1.06201 0.531004 0.847369i $$-0.321815\pi$$
0.531004 + 0.847369i $$0.321815\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −9216.00 −1.60997
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 2872.00 0.492455
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 8120.00 1.34839 0.674193 0.738555i $$-0.264492\pi$$
0.674193 + 0.738555i $$0.264492\pi$$
$$332$$ 0 0
$$333$$ −16058.0 −2.64256
$$334$$ 0 0
$$335$$ −7488.00 −1.22123
$$336$$ 0 0
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ 0 0
$$339$$ 17136.0 2.74543
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −15552.0 −2.42693
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 8802.00 1.32715 0.663574 0.748111i $$-0.269039\pi$$
0.663574 + 0.748111i $$0.269039\pi$$
$$354$$ 0 0
$$355$$ 11016.0 1.64695
$$356$$ −13392.0 −1.99375
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −6859.00 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −9916.00 −1.41038 −0.705192 0.709016i $$-0.749139\pi$$
−0.705192 + 0.709016i $$0.749139\pi$$
$$368$$ −6912.00 −0.979111
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −21760.0 −3.03280
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ 10656.0 1.46740
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 12800.0 1.73481 0.867403 0.497605i $$-0.165787\pi$$
0.867403 + 0.497605i $$0.165787\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 14508.0 1.93557 0.967786 0.251774i $$-0.0810139\pi$$
0.967786 + 0.251774i $$0.0810139\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 272.000 0.0355895
$$389$$ 14130.0 1.84170 0.920848 0.389923i $$-0.127498\pi$$
0.920848 + 0.389923i $$0.127498\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2374.00 −0.300120 −0.150060 0.988677i $$-0.547947\pi$$
−0.150060 + 0.988677i $$0.547947\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 12736.0 1.59200
$$401$$ −9090.00 −1.13200 −0.566001 0.824404i $$-0.691510\pi$$
−0.566001 + 0.824404i $$0.691510\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −6462.00 −0.792838
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 9648.00 1.15791
$$412$$ −9376.00 −1.12117
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −16344.0 −1.90562 −0.952812 0.303560i $$-0.901825\pi$$
−0.952812 + 0.303560i $$0.901825\pi$$
$$420$$ 0 0
$$421$$ −11630.0 −1.34635 −0.673173 0.739485i $$-0.735069\pi$$
−0.673173 + 0.739485i $$0.735069\pi$$
$$422$$ 0 0
$$423$$ −1332.00 −0.153107
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 5120.00 0.570222
$$433$$ −13282.0 −1.47412 −0.737058 0.675830i $$-0.763786\pi$$
−0.737058 + 0.675830i $$0.763786\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ −12691.0 −1.37037
$$442$$ 0 0
$$443$$ −18648.0 −1.99998 −0.999992 0.00391276i $$-0.998755\pi$$
−0.999992 + 0.00391276i $$0.998755\pi$$
$$444$$ 27776.0 2.96890
$$445$$ 30132.0 3.20987
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6786.00 0.713254 0.356627 0.934247i $$-0.383927\pi$$
0.356627 + 0.934247i $$0.383927\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −17136.0 −1.78321
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 15552.0 1.57634
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 13268.0 1.33178 0.665892 0.746048i $$-0.268051\pi$$
0.665892 + 0.746048i $$0.268051\pi$$
$$464$$ 0 0
$$465$$ 48960.0 4.88272
$$466$$ 0 0
$$467$$ 4176.00 0.413795 0.206897 0.978363i $$-0.433663\pi$$
0.206897 + 0.978363i $$0.433663\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −10672.0 −1.04403
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −27306.0 −2.62108
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −612.000 −0.0572979
$$486$$ 0 0
$$487$$ 16684.0 1.55241 0.776206 0.630480i $$-0.217142\pi$$
0.776206 + 0.630480i $$0.217142\pi$$
$$488$$ 0 0
$$489$$ 29824.0 2.75805
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 21760.0 1.96986
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4120.00 0.369612 0.184806 0.982775i $$-0.440834\pi$$
0.184806 + 0.982775i $$0.440834\pi$$
$$500$$ −10656.0 −0.953102
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −17576.0 −1.53960
$$508$$ 0 0
$$509$$ −22410.0 −1.95148 −0.975742 0.218922i $$-0.929746\pi$$
−0.975742 + 0.218922i $$0.929746\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 21096.0 1.80505
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 20070.0 1.68768 0.843841 0.536593i $$-0.180289\pi$$
0.843841 + 0.536593i $$0.180289\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −503.000 −0.0413413
$$530$$ 0 0
$$531$$ −26640.0 −2.17717
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −16128.0 −1.29604
$$538$$ 0 0
$$539$$ 0 0
$$540$$ −11520.0 −0.918040
$$541$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$542$$ 0 0
$$543$$ −16400.0 −1.29612
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ −9648.00 −0.752084
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −62496.0 −4.77983
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 2304.00 0.172014
$$565$$ 38556.0 2.87091
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ 41760.0 3.04459
$$574$$ 0 0
$$575$$ −21492.0 −1.55874
$$576$$ −18944.0 −1.37037
$$577$$ −22466.0 −1.62092 −0.810461 0.585793i $$-0.800783\pi$$
−0.810461 + 0.585793i $$0.800783\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −26064.0 −1.83267 −0.916334 0.400414i $$-0.868866\pi$$
−0.916334 + 0.400414i $$0.868866\pi$$
$$588$$ 21952.0 1.53960
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −27776.0 −1.92836
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 31520.0 2.16085
$$598$$ 0 0
$$599$$ 18036.0 1.23027 0.615134 0.788422i $$-0.289102\pi$$
0.615134 + 0.788422i $$0.289102\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ −15392.0 −1.03949
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 3654.00 0.238419 0.119209 0.992869i $$-0.461964\pi$$
0.119209 + 0.992869i $$0.461964\pi$$
$$618$$ 0 0
$$619$$ 1856.00 0.120515 0.0602576 0.998183i $$-0.480808\pi$$
0.0602576 + 0.998183i $$0.480808\pi$$
$$620$$ −48960.0 −3.17142
$$621$$ −8640.00 −0.558311
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −899.000 −0.0575360
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 10672.0 0.678120
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −12908.0 −0.814357 −0.407179 0.913349i $$-0.633487\pi$$
−0.407179 + 0.913349i $$0.633487\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 47232.0 2.94477
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 22644.0 1.40185
$$640$$ 0 0
$$641$$ −4590.00 −0.282830 −0.141415 0.989950i $$-0.545165\pi$$
−0.141415 + 0.989950i $$0.545165\pi$$
$$642$$ 0 0
$$643$$ −10168.0 −0.623619 −0.311809 0.950145i $$-0.600935\pi$$
−0.311809 + 0.950145i $$0.600935\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 32724.0 1.98843 0.994214 0.107416i $$-0.0342575\pi$$
0.994214 + 0.107416i $$0.0342575\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −29824.0 −1.79141
$$653$$ 32742.0 1.96216 0.981082 0.193591i $$-0.0620135\pi$$
0.981082 + 0.193591i $$0.0620135\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −23582.0 −1.38765 −0.693823 0.720146i $$-0.744075\pi$$
−0.693823 + 0.720146i $$0.744075\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 5344.00 0.308836
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 15920.0 0.907794
$$676$$ 17576.0 1.00000
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −35352.0 −1.98054 −0.990268 0.139170i $$-0.955556\pi$$
−0.990268 + 0.139170i $$0.955556\pi$$
$$684$$ 0 0
$$685$$ 21708.0 1.21083
$$686$$ 0 0
$$687$$ 26480.0 1.47056
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −30328.0 −1.66965 −0.834827 0.550512i $$-0.814433\pi$$
−0.834827 + 0.550512i $$0.814433\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −5184.00 −0.276937
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 46080.0 2.44603
$$709$$ −33554.0 −1.77736 −0.888679 0.458530i $$-0.848376\pi$$
−0.888679 + 0.458530i $$0.848376\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −36720.0 −1.92872
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 16128.0 0.841804
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 22644.0 1.17452 0.587259 0.809399i $$-0.300207\pi$$
0.587259 + 0.809399i $$0.300207\pi$$
$$720$$ 42624.0 2.20625
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 16400.0 0.841852
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −33284.0 −1.69799 −0.848993 0.528405i $$-0.822790\pi$$
−0.848993 + 0.528405i $$0.822790\pi$$
$$728$$ 0 0
$$729$$ −30563.0 −1.55276
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ −49392.0 −2.47871
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 62496.0 3.10459
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 39652.0 1.92666 0.963330 0.268319i $$-0.0864680\pi$$
0.963330 + 0.268319i $$0.0864680\pi$$
$$752$$ −2304.00 −0.111726
$$753$$ 5184.00 0.250884
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 31426.0 1.50885 0.754424 0.656388i $$-0.227916\pi$$
0.754424 + 0.656388i $$0.227916\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −41760.0 −1.97752
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 32768.0 1.53960
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ −64368.0 −3.00669
$$772$$ 0 0
$$773$$ 32238.0 1.50003 0.750013 0.661423i $$-0.230047\pi$$
0.750013 + 0.661423i $$0.230047\pi$$
$$774$$ 0 0
$$775$$ 67660.0 3.13602
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −21952.0 −1.00000
$$785$$ −24012.0 −1.09175
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −106272. −4.74098
$$796$$ −31520.0 −1.40351
$$797$$ 7146.00 0.317596 0.158798 0.987311i $$-0.449238\pi$$
0.158798 + 0.987311i $$0.449238\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 61938.0 2.73217
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 26624.0 1.16786
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 22320.0 0.973607
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 67104.0 2.88411
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ 3332.00 0.141125 0.0705627 0.997507i $$-0.477521\pi$$
0.0705627 + 0.997507i $$0.477521\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 31968.0 1.34174
$$829$$ −47734.0 −1.99984 −0.999922 0.0125057i $$-0.996019\pi$$
−0.999922 + 0.0125057i $$0.996019\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 27200.0 1.12326
$$838$$ 0 0
$$839$$ 22140.0 0.911034 0.455517 0.890227i $$-0.349454\pi$$
0.455517 + 0.890227i $$0.349454\pi$$
$$840$$ 0 0
$$841$$ −24389.0 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −39546.0 −1.60997
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −47232.0 −1.91268
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 46872.0 1.88807
$$852$$ −39168.0 −1.57497
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 50096.0 1.98982 0.994909 0.100779i $$-0.0321334\pi$$
0.994909 + 0.100779i $$0.0321334\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 46548.0 1.83605 0.918026 0.396521i $$-0.129783\pi$$
0.918026 + 0.396521i $$0.129783\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −39304.0 −1.53960
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −1258.00 −0.0487707
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 34542.0 1.32094 0.660471 0.750852i $$-0.270357\pi$$
0.660471 + 0.750852i $$0.270357\pi$$
$$882$$ 0 0
$$883$$ −27272.0 −1.03938 −0.519692 0.854354i $$-0.673953\pi$$
−0.519692 + 0.854354i $$0.673953\pi$$
$$884$$ 0 0
$$885$$ −103680. −3.93804
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −5344.00 −0.200595
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −36288.0 −1.35528
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ −58904.0 −2.18163
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −36900.0 −1.35536
$$906$$ 0 0
$$907$$ −21256.0 −0.778163 −0.389082 0.921203i $$-0.627208\pi$$
−0.389082 + 0.921203i $$0.627208\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −52020.0 −1.89188 −0.945938 0.324347i $$-0.894856\pi$$
−0.945938 + 0.324347i $$0.894856\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −26480.0 −0.955157
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −86366.0 −3.06994
$$926$$ 0 0
$$927$$ 43364.0 1.53642
$$928$$ 0 0
$$929$$ −56610.0 −1.99926 −0.999631 0.0271744i $$-0.991349\pi$$
−0.999631 + 0.0271744i $$0.991349\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 75744.0 2.65782
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$938$$ 0 0
$$939$$ −87856.0 −3.05333
$$940$$ 5184.00 0.179876
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −46080.0 −1.58875
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 23256.0 0.798013 0.399007 0.916948i $$-0.369355\pi$$
0.399007 + 0.916948i $$0.369355\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 47952.0 1.63507
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 93960.0 3.18374
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ −73728.0 −2.47871
$$961$$ 85809.0 2.88037
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 39960.0 1.32068 0.660339 0.750968i $$-0.270413\pi$$
0.660339 + 0.750968i $$0.270413\pi$$
$$972$$ 40256.0 1.32841
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 59454.0 1.94688 0.973440 0.228942i $$-0.0735266\pi$$
0.973440 + 0.228942i $$0.0735266\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 49392.0 1.60997
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −17748.0 −0.575863 −0.287931 0.957651i $$-0.592968\pi$$
−0.287931 + 0.957651i $$0.592968\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 51460.0 1.64953 0.824763 0.565478i $$-0.191308\pi$$
0.824763 + 0.565478i $$0.191308\pi$$
$$992$$ 0 0
$$993$$ 64960.0 2.07598
$$994$$ 0 0
$$995$$ 70920.0 2.25961
$$996$$ 0 0
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ 0 0
$$999$$ −34720.0 −1.09959
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 121.4.a.a.1.1 1
3.2 odd 2 1089.4.a.f.1.1 1
4.3 odd 2 1936.4.a.a.1.1 1
11.2 odd 10 121.4.c.a.81.1 4
11.3 even 5 121.4.c.a.9.1 4
11.4 even 5 121.4.c.a.27.1 4
11.5 even 5 121.4.c.a.3.1 4
11.6 odd 10 121.4.c.a.3.1 4
11.7 odd 10 121.4.c.a.27.1 4
11.8 odd 10 121.4.c.a.9.1 4
11.9 even 5 121.4.c.a.81.1 4
11.10 odd 2 CM 121.4.a.a.1.1 1
33.32 even 2 1089.4.a.f.1.1 1
44.43 even 2 1936.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.a.a.1.1 1 1.1 even 1 trivial
121.4.a.a.1.1 1 11.10 odd 2 CM
121.4.c.a.3.1 4 11.5 even 5
121.4.c.a.3.1 4 11.6 odd 10
121.4.c.a.9.1 4 11.3 even 5
121.4.c.a.9.1 4 11.8 odd 10
121.4.c.a.27.1 4 11.4 even 5
121.4.c.a.27.1 4 11.7 odd 10
121.4.c.a.81.1 4 11.2 odd 10
121.4.c.a.81.1 4 11.9 even 5
1089.4.a.f.1.1 1 3.2 odd 2
1089.4.a.f.1.1 1 33.32 even 2
1936.4.a.a.1.1 1 4.3 odd 2
1936.4.a.a.1.1 1 44.43 even 2