LMFDB - Embedded newform 54.18.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [54,18,Mod(1,54)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 18, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("54.1"); S:= CuspForms(chi, 18); N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	54.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,512,0,131072,-304980] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$98.9399271661$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 6619360 $ x^2 - x - 6619360
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2573.31$ of defining polynomial
Character	$\chi$	$=$	54.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+256.000 q^{2} +65536.0 q^{4} -847149. q^{5} +8.01937e6 q^{7} +1.67772e7 q^{8} -2.16870e8 q^{10} -2.43679e8 q^{11} +1.61256e9 q^{13} +2.05296e9 q^{14} +4.29497e9 q^{16} +2.05294e9 q^{17} -1.00427e11 q^{19} -5.55188e10 q^{20} -6.23819e10 q^{22} +6.28911e11 q^{23} -4.52777e10 q^{25} +4.12816e11 q^{26} +5.25557e11 q^{28} +1.38080e12 q^{29} -1.33771e12 q^{31} +1.09951e12 q^{32} +5.25553e11 q^{34} -6.79360e12 q^{35} -2.37602e13 q^{37} -2.57093e13 q^{38} -1.42128e13 q^{40} +4.86041e12 q^{41} +1.82714e13 q^{43} -1.59698e13 q^{44} +1.61001e14 q^{46} -2.06760e14 q^{47} -1.68320e14 q^{49} -1.15911e13 q^{50} +1.05681e14 q^{52} +1.47985e14 q^{53} +2.06433e14 q^{55} +1.34543e14 q^{56} +3.53484e14 q^{58} -9.72521e14 q^{59} -5.30555e14 q^{61} -3.42454e14 q^{62} +2.81475e14 q^{64} -1.36608e15 q^{65} -4.08015e15 q^{67} +1.34542e14 q^{68} -1.73916e15 q^{70} -7.34515e15 q^{71} +5.69074e15 q^{73} -6.08260e15 q^{74} -6.58158e15 q^{76} -1.95416e15 q^{77} -6.13324e15 q^{79} -3.63848e15 q^{80} +1.24427e15 q^{82} +6.99885e15 q^{83} -1.73915e15 q^{85} +4.67748e15 q^{86} -4.08826e15 q^{88} +4.23661e15 q^{89} +1.29317e16 q^{91} +4.12163e16 q^{92} -5.29305e16 q^{94} +8.50767e16 q^{95} -7.36170e16 q^{97} -4.30900e16 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 512 q^{2} + 131072 q^{4} - 304980 q^{5} - 9246860 q^{7} + 33554432 q^{8} - 78074880 q^{10} - 155034030 q^{11} + 1702429720 q^{13} - 2367196160 q^{14} + 8589934592 q^{16} - 8486897472 q^{17} - 5707598648 q^{19}+ \cdots - 26\!\cdots\!76 q^{98}+O(q^{100}) $ 2 * q + 512 * q^2 + 131072 * q^4 - 304980 * q^5 - 9246860 * q^7 + 33554432 * q^8 - 78074880 * q^10 - 155034030 * q^11 + 1702429720 * q^13 - 2367196160 * q^14 + 8589934592 * q^16 - 8486897472 * q^17 - 5707598648 * q^19 - 19987169280 * q^20 - 39688711680 * q^22 + 266994431796 * q^23 - 514269781600 * q^25 + 435822008320 * q^26 - 606002216960 * q^28 + 113442022320 * q^29 - 6929179768316 * q^31 + 2199023255552 * q^32 - 2172645752832 * q^34 - 16154815903590 * q^35 - 14067852085580 * q^37 - 1461145253888 * q^38 - 5116715335680 * q^40 - 30739866769320 * q^41 + 58093870679560 * q^43 - 10160310190080 * q^44 + 68350574539776 * q^46 - 69202049560356 * q^47 - 102828191597796 * q^49 - 131653064089600 * q^50 + 111570434129920 * q^52 - 659688284112516 * q^53 + 254493710923140 * q^55 - 155136567541760 * q^56 + 29041157713920 * q^58 - 1288830277403880 * q^59 + 144079111616032 * q^61 - 1773870020688896 * q^62 + 562949953421312 * q^64 - 1317356094000000 * q^65 - 3203435114160680 * q^67 - 556197312724992 * q^68 - 4135632871319040 * q^70 - 11320825195049760 * q^71 - 4886170417034870 * q^73 - 3601370133908480 * q^74 - 374053184995328 * q^76 - 3484727819406708 * q^77 - 13226203174257080 * q^79 - 1309879125934080 * q^80 - 7869405892945920 * q^82 - 14304595055224350 * q^83 - 7453524098909520 * q^85 + 14872030893967360 * q^86 - 2601039408660480 * q^88 + 6721701300374040 * q^89 + 11380032424809440 * q^91 + 17497747082182656 * q^92 - 17715724687451136 * q^94 + 136430610599529420 * q^95 - 71442353414491730 * q^97 - 26324017049035776 * q^98

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	256.000	0.707107
$2$	256.000	0.707107
$3$	0	0
$3$	0	0
$4$	65536.0	0.500000
$5$	−847149.	−0.969873	−0.484936	−	0.874549i	$-0.661157\pi$
$5$	−847149.	−0.969873	−0.484936	+	0.874549i	$0.661157\pi$
$6$	0	0
$7$	8.01937e6	0.525783	0.262892	−	0.964825i	$-0.415324\pi$
$7$	8.01937e6	0.525783	0.262892	+	0.964825i	$0.415324\pi$
$8$	1.67772e7	0.353553
$9$	0	0
$10$	−2.16870e8	−0.685804
$11$	−2.43679e8	−0.342753	−0.171376	−	0.985206i	$-0.554821\pi$
$11$	−2.43679e8	−0.342753	−0.171376	+	0.985206i	$0.554821\pi$
$12$	0	0
$13$	1.61256e9	0.548274	0.274137	−	0.961691i	$-0.411608\pi$
$13$	1.61256e9	0.548274	0.274137	+	0.961691i	$0.411608\pi$
$14$	2.05296e9	0.371785
$15$	0	0
$16$	4.29497e9	0.250000
$17$	2.05294e9	0.0713774	0.0356887	−	0.999363i	$-0.488638\pi$
$17$	2.05294e9	0.0713774	0.0356887	+	0.999363i	$0.488638\pi$
$18$	0	0
$19$	−1.00427e11	−1.35658	−0.678289	−	0.734795i	$-0.737278\pi$
$19$	−1.00427e11	−1.35658	−0.678289	+	0.734795i	$0.737278\pi$
$20$	−5.55188e10	−0.484936
$21$	0	0
$22$	−6.23819e10	−0.242363
$23$	6.28911e11	1.67457	0.837284	−	0.546768i	$-0.184142\pi$
$23$	6.28911e11	1.67457	0.837284	+	0.546768i	$0.184142\pi$
$24$	0	0
$25$	−4.52777e10	−0.0593464
$26$	4.12816e11	0.387688
$27$	0	0
$28$	5.25557e11	0.262892
$29$	1.38080e12	0.512563	0.256282	−	0.966602i	$-0.417503\pi$
$29$	1.38080e12	0.512563	0.256282	+	0.966602i	$0.417503\pi$
$30$	0	0
$31$	−1.33771e12	−0.281700	−0.140850	−	0.990031i	$-0.544984\pi$
$31$	−1.33771e12	−0.281700	−0.140850	+	0.990031i	$0.544984\pi$
$32$	1.09951e12	0.176777
$33$	0	0
$34$	5.25553e11	0.0504715
$35$	−6.79360e12	−0.509943
$36$	0	0
$37$	−2.37602e13	−1.11208	−0.556038	−	0.831157i	$-0.687679\pi$
$37$	−2.37602e13	−1.11208	−0.556038	+	0.831157i	$0.687679\pi$
$38$	−2.57093e13	−0.959246
$39$	0	0
$40$	−1.42128e13	−0.342902
$41$	4.86041e12	0.0950628	0.0475314	−	0.998870i	$-0.484865\pi$
$41$	4.86041e12	0.0950628	0.0475314	+	0.998870i	$0.484865\pi$
$42$	0	0
$43$	1.82714e13	0.238391	0.119196	−	0.992871i	$-0.461968\pi$
$43$	1.82714e13	0.238391	0.119196	+	0.992871i	$0.461968\pi$
$44$	−1.59698e13	−0.171376
$45$	0	0
$46$	1.61001e14	1.18410
$47$	−2.06760e14	−1.26659	−0.633293	−	0.773912i	$-0.718297\pi$
$47$	−2.06760e14	−1.26659	−0.633293	+	0.773912i	$0.718297\pi$
$48$	0	0
$49$	−1.68320e14	−0.723552
$50$	−1.15911e13	−0.0419643
$51$	0	0
$52$	1.05681e14	0.274137
$53$	1.47985e14	0.326493	0.163247	−	0.986585i	$-0.447803\pi$
$53$	1.47985e14	0.326493	0.163247	+	0.986585i	$0.447803\pi$
$54$	0	0
$55$	2.06433e14	0.332427
$56$	1.34543e14	0.185892
$57$	0	0
$58$	3.53484e14	0.362437
$59$	−9.72521e14	−0.862297	−0.431149	−	0.902281i	$-0.641891\pi$
$59$	−9.72521e14	−0.862297	−0.431149	+	0.902281i	$0.641891\pi$
$60$	0	0
$61$	−5.30555e14	−0.354346	−0.177173	−	0.984180i	$-0.556695\pi$
$61$	−5.30555e14	−0.354346	−0.177173	+	0.984180i	$0.556695\pi$
$62$	−3.42454e14	−0.199192
$63$	0	0
$64$	2.81475e14	0.125000
$65$	−1.36608e15	−0.531756
$66$	0	0
$67$	−4.08015e15	−1.22755	−0.613777	−	0.789480i	$-0.710350\pi$
$67$	−4.08015e15	−1.22755	−0.613777	+	0.789480i	$0.710350\pi$
$68$	1.34542e14	0.0356887
$69$	0	0
$70$	−1.73916e15	−0.360584
$71$	−7.34515e15	−1.34991	−0.674955	−	0.737859i	$-0.735837\pi$
$71$	−7.34515e15	−1.34991	−0.674955	+	0.737859i	$0.735837\pi$
$72$	0	0
$73$	5.69074e15	0.825895	0.412947	−	0.910755i	$-0.364499\pi$
$73$	5.69074e15	0.825895	0.412947	+	0.910755i	$0.364499\pi$
$74$	−6.08260e15	−0.786357
$75$	0	0
$76$	−6.58158e15	−0.678289
$77$	−1.95416e15	−0.180214
$78$	0	0
$79$	−6.13324e15	−0.454841	−0.227421	−	0.973797i	$-0.573029\pi$
$79$	−6.13324e15	−0.454841	−0.227421	+	0.973797i	$0.573029\pi$
$80$	−3.63848e15	−0.242468
$81$	0	0
$82$	1.24427e15	0.0672195
$83$	6.99885e15	0.341085	0.170543	−	0.985350i	$-0.445448\pi$
$83$	6.99885e15	0.341085	0.170543	+	0.985350i	$0.445448\pi$
$84$	0	0
$85$	−1.73915e15	−0.0692270
$86$	4.67748e15	0.168568
$87$	0	0
$88$	−4.08826e15	−0.121181
$89$	4.23661e15	0.114078	0.0570392	−	0.998372i	$-0.481834\pi$
$89$	4.23661e15	0.114078	0.0570392	+	0.998372i	$0.481834\pi$
$90$	0	0
$91$	1.29317e16	0.288273
$92$	4.12163e16	0.837284
$93$	0	0
$94$	−5.29305e16	−0.895611
$95$	8.50767e16	1.31571
$96$	0	0
$97$	−7.36170e16	−0.953715	−0.476857	−	0.878981i	$-0.658224\pi$
$97$	−7.36170e16	−0.953715	−0.476857	+	0.878981i	$0.658224\pi$
$98$	−4.30900e16	−0.511629
$99$	0	0
$100$	−2.96732e15	−0.0296732
$101$	1.77435e16	0.163045	0.0815224	−	0.996672i	$-0.474022\pi$
$101$	1.77435e16	0.163045	0.0815224	+	0.996672i	$0.474022\pi$
$102$	0	0
$103$	−8.96458e16	−0.697291	−0.348645	−	0.937255i	$-0.613358\pi$
$103$	−8.96458e16	−0.697291	−0.348645	+	0.937255i	$0.613358\pi$
$104$	2.70543e16	0.193844
$105$	0	0
$106$	3.78843e16	0.230865
$107$	−6.31261e16	−0.355179	−0.177589	−	0.984105i	$-0.556830\pi$
$107$	−6.31261e16	−0.355179	−0.177589	+	0.984105i	$0.556830\pi$
$108$	0	0
$109$	−2.71956e17	−1.30730	−0.653648	−	0.756799i	$-0.726762\pi$
$109$	−2.71956e17	−1.30730	−0.653648	+	0.756799i	$0.726762\pi$
$110$	5.28468e16	0.235061
$111$	0	0
$112$	3.44429e16	0.131446
$113$	−2.70809e16	−0.0958290	−0.0479145	−	0.998851i	$-0.515257\pi$
$113$	−2.70809e16	−0.0958290	−0.0479145	+	0.998851i	$0.515257\pi$
$114$	0	0
$115$	−5.32782e17	−1.62412
$116$	9.04920e16	0.256282
$117$	0	0
$118$	−2.48965e17	−0.609736
$119$	1.64633e16	0.0375291
$120$	0	0
$121$	−4.46067e17	−0.882520
$122$	−1.35822e17	−0.250560
$123$	0	0
$124$	−8.76681e16	−0.140850
$125$	6.84681e17	1.02743
$126$	0	0
$127$	−1.20002e18	−1.57347	−0.786734	−	0.617292i	$-0.788230\pi$
$127$	−1.20002e18	−1.57347	−0.786734	+	0.617292i	$0.788230\pi$
$128$	7.20576e16	0.0883883
$129$	0	0
$130$	−3.49716e17	−0.376008
$131$	7.98700e17	0.804595	0.402298	−	0.915509i	$-0.368212\pi$
$131$	7.98700e17	0.804595	0.402298	+	0.915509i	$0.368212\pi$
$132$	0	0
$133$	−8.05361e17	−0.713266
$134$	−1.04452e18	−0.868011
$135$	0	0
$136$	3.44427e16	0.0252357
$137$	−2.33204e18	−1.60550	−0.802751	−	0.596314i	$-0.796631\pi$
$137$	−2.33204e18	−1.60550	−0.802751	+	0.596314i	$0.796631\pi$
$138$	0	0
$139$	1.66874e18	1.01569	0.507847	−	0.861447i	$-0.330441\pi$
$139$	1.66874e18	1.01569	0.507847	+	0.861447i	$0.330441\pi$
$140$	−4.45225e17	−0.254971
$141$	0	0
$142$	−1.88036e18	−0.954531
$143$	−3.92948e17	−0.187923
$144$	0	0
$145$	−1.16974e18	−0.497121
$146$	1.45683e18	0.583996
$147$	0	0
$148$	−1.55715e18	−0.556038
$149$	−4.14235e18	−1.39689	−0.698447	−	0.715662i	$-0.746125\pi$
$149$	−4.14235e18	−1.39689	−0.698447	+	0.715662i	$0.746125\pi$
$150$	0	0
$151$	−4.53805e18	−1.36636	−0.683180	−	0.730250i	$-0.739403\pi$
$151$	−4.53805e18	−1.36636	−0.683180	+	0.730250i	$0.739403\pi$
$152$	−1.68489e18	−0.479623
$153$	0	0
$154$	−5.00264e17	−0.127430
$155$	1.13324e18	0.273214
$156$	0	0
$157$	−8.91729e18	−1.92791	−0.963953	−	0.266071i	$-0.914274\pi$
$157$	−8.91729e18	−1.92791	−0.963953	+	0.266071i	$0.914274\pi$
$158$	−1.57011e18	−0.321621
$159$	0	0
$160$	−9.31450e17	−0.171451
$161$	5.04347e18	0.880460
$162$	0	0
$163$	1.02359e19	1.60891	0.804455	−	0.594014i	$-0.202458\pi$
$163$	1.02359e19	1.60891	0.804455	+	0.594014i	$0.202458\pi$
$164$	3.18532e17	0.0475314
$165$	0	0
$166$	1.79171e18	0.241184
$167$	2.13116e18	0.272600	0.136300	−	0.990668i	$-0.456479\pi$
$167$	2.13116e18	0.272600	0.136300	+	0.990668i	$0.456479\pi$
$168$	0	0
$169$	−6.05006e18	−0.699395
$170$	−4.45222e17	−0.0489509
$171$	0	0
$172$	1.19744e18	0.119196
$173$	5.29524e18	0.501758	0.250879	−	0.968019i	$-0.419280\pi$
$173$	5.29524e18	0.501758	0.250879	+	0.968019i	$0.419280\pi$
$174$	0	0
$175$	−3.63099e17	−0.0312034
$176$	−1.04660e18	−0.0856882
$177$	0	0
$178$	1.08457e18	0.0806657
$179$	1.83007e19	1.29783	0.648913	−	0.760863i	$-0.275224\pi$
$179$	1.83007e19	1.29783	0.648913	+	0.760863i	$0.275224\pi$
$180$	0	0
$181$	1.36998e19	0.883991	0.441995	−	0.897017i	$-0.354271\pi$
$181$	1.36998e19	0.883991	0.441995	+	0.897017i	$0.354271\pi$
$182$	3.31052e18	0.203840
$183$	0	0
$184$	1.05514e19	0.592049
$185$	2.01284e19	1.07857
$186$	0	0
$187$	−5.00260e17	−0.0244648
$188$	−1.35502e19	−0.633293
$189$	0	0
$190$	2.17796e19	0.930347
$191$	−1.30408e19	−0.532748	−0.266374	−	0.963870i	$-0.585826\pi$
$191$	−1.30408e19	−0.532748	−0.266374	+	0.963870i	$0.585826\pi$
$192$	0	0
$193$	1.61726e19	0.604702	0.302351	−	0.953197i	$-0.402228\pi$
$193$	1.61726e19	0.604702	0.302351	+	0.953197i	$0.402228\pi$
$194$	−1.88460e19	−0.674378
$195$	0	0
$196$	−1.10310e19	−0.361776
$197$	8.86779e17	0.0278518	0.0139259	−	0.999903i	$-0.495567\pi$
$197$	8.86779e17	0.0278518	0.0139259	+	0.999903i	$0.495567\pi$
$198$	0	0
$199$	4.62179e19	1.33217	0.666084	−	0.745876i	$-0.267969\pi$
$199$	4.62179e19	1.33217	0.666084	+	0.745876i	$0.267969\pi$
$200$	−7.59634e17	−0.0209821
$201$	0	0
$202$	4.54233e18	0.115290
$203$	1.10731e19	0.269497
$204$	0	0
$205$	−4.11750e18	−0.0921988
$206$	−2.29493e19	−0.493059
$207$	0	0
$208$	6.92590e18	0.137069
$209$	2.44720e19	0.464971
$210$	0	0
$211$	1.06983e20	1.87462	0.937312	−	0.348492i	$-0.113306\pi$
$211$	1.06983e20	1.87462	0.937312	+	0.348492i	$0.113306\pi$
$212$	9.69837e18	0.163247
$213$	0	0
$214$	−1.61603e19	−0.251149
$215$	−1.54786e19	−0.231209
$216$	0	0
$217$	−1.07276e19	−0.148113
$218$	−6.96208e19	−0.924398
$219$	0	0
$220$	1.35288e19	0.166213
$221$	3.31049e18	0.0391344
$222$	0	0
$223$	4.85282e19	0.531378	0.265689	−	0.964059i	$-0.414401\pi$
$223$	4.85282e19	0.531378	0.265689	+	0.964059i	$0.414401\pi$
$224$	8.81739e18	0.0929462
$225$	0	0
$226$	−6.93272e18	−0.0677613
$227$	1.76860e20	1.66498	0.832491	−	0.554039i	$-0.186914\pi$
$227$	1.76860e20	1.66498	0.832491	+	0.554039i	$0.186914\pi$
$228$	0	0
$229$	7.15770e19	0.625420	0.312710	−	0.949849i	$-0.398763\pi$
$229$	7.15770e19	0.625420	0.312710	+	0.949849i	$0.398763\pi$
$230$	−1.36392e20	−1.14843
$231$	0	0
$232$	2.31660e19	0.181218
$233$	−5.26580e19	−0.397136	−0.198568	−	0.980087i	$-0.563629\pi$
$233$	−5.26580e19	−0.397136	−0.198568	+	0.980087i	$0.563629\pi$
$234$	0	0
$235$	1.75156e20	1.22843
$236$	−6.37351e19	−0.431149
$237$	0	0
$238$	4.21460e18	0.0265370
$239$	−1.26421e20	−0.768136	−0.384068	−	0.923305i	$-0.625477\pi$
$239$	−1.26421e20	−0.768136	−0.384068	+	0.923305i	$0.625477\pi$
$240$	0	0
$241$	−1.04575e20	−0.591948	−0.295974	−	0.955196i	$-0.595644\pi$
$241$	−1.04575e20	−0.591948	−0.295974	+	0.955196i	$0.595644\pi$
$242$	−1.14193e20	−0.624036
$243$	0	0
$244$	−3.47705e19	−0.177173
$245$	1.42592e20	0.701754
$246$	0	0
$247$	−1.61945e20	−0.743777
$248$	−2.24430e19	−0.0995961
$249$	0	0
$250$	1.75278e20	0.726504
$251$	1.11461e20	0.446577	0.223288	−	0.974752i	$-0.428321\pi$
$251$	1.11461e20	0.446577	0.223288	+	0.974752i	$0.428321\pi$
$252$	0	0
$253$	−1.53253e20	−0.573963
$254$	−3.07206e20	−1.11261
$255$	0	0
$256$	1.84467e19	0.0625000
$257$	2.62335e20	0.859855	0.429928	−	0.902863i	$-0.358539\pi$
$257$	2.62335e20	0.859855	0.429928	+	0.902863i	$0.358539\pi$
$258$	0	0
$259$	−1.90541e20	−0.584711
$260$	−8.95274e19	−0.265878
$261$	0	0
$262$	2.04467e20	0.568935
$263$	3.27028e20	0.880970	0.440485	−	0.897760i	$-0.354806\pi$
$263$	3.27028e20	0.880970	0.440485	+	0.897760i	$0.354806\pi$
$264$	0	0
$265$	−1.25366e20	−0.316657
$266$	−2.06172e20	−0.504355
$267$	0	0
$268$	−2.67397e20	−0.613777
$269$	8.30681e20	1.84731	0.923655	−	0.383224i	$-0.125186\pi$
$269$	8.30681e20	1.84731	0.923655	+	0.383224i	$0.125186\pi$
$270$	0	0
$271$	−1.31233e20	−0.274035	−0.137017	−	0.990569i	$-0.543752\pi$
$271$	−1.31233e20	−0.274035	−0.137017	+	0.990569i	$0.543752\pi$
$272$	8.81732e18	0.0178444
$273$	0	0
$274$	−5.97002e20	−1.13526
$275$	1.10333e19	0.0203412
$276$	0	0
$277$	−4.79360e20	−0.830967	−0.415483	−	0.909601i	$-0.636388\pi$
$277$	−4.79360e20	−0.830967	−0.415483	+	0.909601i	$0.636388\pi$
$278$	4.27198e20	0.718205
$279$	0	0
$280$	−1.13978e20	−0.180292
$281$	9.88338e20	1.51671	0.758353	−	0.651844i	$-0.226004\pi$
$281$	9.88338e20	1.51671	0.758353	+	0.651844i	$0.226004\pi$
$282$	0	0
$283$	−1.00371e21	−1.45018	−0.725092	−	0.688652i	$-0.758203\pi$
$283$	−1.00371e21	−1.45018	−0.725092	+	0.688652i	$0.758203\pi$
$284$	−4.81372e20	−0.674955
$285$	0	0
$286$	−1.00595e20	−0.132881
$287$	3.89774e19	0.0499824
$288$	0	0
$289$	−8.23026e20	−0.994905
$290$	−2.99454e20	−0.351518
$291$	0	0
$292$	3.72949e20	0.412947
$293$	−1.57822e21	−1.69743	−0.848716	−	0.528850i	$-0.822624\pi$
$293$	−1.57822e21	−1.69743	−0.848716	+	0.528850i	$0.822624\pi$
$294$	0	0
$295$	8.23870e20	0.836319
$296$	−3.98629e20	−0.393179
$297$	0	0
$298$	−1.06044e21	−0.987753
$299$	1.01416e21	0.918123
$300$	0	0
$301$	1.46525e20	0.125342
$302$	−1.16174e21	−0.966162
$303$	0	0
$304$	−4.31331e20	−0.339145
$305$	4.49459e20	0.343670
$306$	0	0
$307$	−1.21206e21	−0.876696	−0.438348	−	0.898805i	$-0.644436\pi$
$307$	−1.21206e21	−0.876696	−0.438348	+	0.898805i	$0.644436\pi$
$308$	−1.28068e20	−0.0901069
$309$	0	0
$310$	2.90109e20	0.193191
$311$	4.53796e20	0.294034	0.147017	−	0.989134i	$-0.453033\pi$
$311$	4.53796e20	0.294034	0.147017	+	0.989134i	$0.453033\pi$
$312$	0	0
$313$	2.29387e21	1.40748	0.703739	−	0.710459i	$-0.251512\pi$
$313$	2.29387e21	1.40748	0.703739	+	0.710459i	$0.251512\pi$
$314$	−2.28283e21	−1.36324
$315$	0	0
$316$	−4.01948e20	−0.227421
$317$	−8.22006e20	−0.452763	−0.226382	−	0.974039i	$-0.572690\pi$
$317$	−8.22006e20	−0.452763	−0.226382	+	0.974039i	$0.572690\pi$
$318$	0	0
$319$	−3.36472e20	−0.175682
$320$	−2.38451e20	−0.121234
$321$	0	0
$322$	1.29113e21	0.622579
$323$	−2.06171e20	−0.0968291
$324$	0	0
$325$	−7.30131e19	−0.0325381
$326$	2.62039e21	1.13767
$327$	0	0
$328$	8.15442e19	0.0336098
$329$	−1.65808e21	−0.665949
$330$	0	0
$331$	−1.90795e21	−0.727828	−0.363914	−	0.931433i	$-0.618560\pi$
$331$	−1.90795e21	−0.727828	−0.363914	+	0.931433i	$0.618560\pi$
$332$	4.58677e20	0.170543
$333$	0	0
$334$	5.45576e20	0.192757
$335$	3.45650e21	1.19057
$336$	0	0
$337$	1.78453e20	0.0584346	0.0292173	−	0.999573i	$-0.490699\pi$
$337$	1.78453e20	0.0584346	0.0292173	+	0.999573i	$0.490699\pi$
$338$	−1.54882e21	−0.494547
$339$	0	0
$340$	−1.13977e20	−0.0346135
$341$	3.25972e20	0.0965536
$342$	0	0
$343$	−3.21537e21	−0.906215
$344$	3.06543e20	0.0842840
$345$	0	0
$346$	1.35558e21	0.354796
$347$	−5.68530e21	−1.45195	−0.725977	−	0.687719i	$-0.758612\pi$
$347$	−5.68530e21	−1.45195	−0.725977	+	0.687719i	$0.758612\pi$
$348$	0	0
$349$	6.37823e21	1.55126	0.775629	−	0.631189i	$-0.217433\pi$
$349$	6.37823e21	1.55126	0.775629	+	0.631189i	$0.217433\pi$
$350$	−9.29533e19	−0.0220641
$351$	0	0
$352$	−2.67928e20	−0.0605907
$353$	2.91442e21	0.643379	0.321690	−	0.946845i	$-0.395749\pi$
$353$	2.91442e21	0.643379	0.321690	+	0.946845i	$0.395749\pi$
$354$	0	0
$355$	6.22244e21	1.30924
$356$	2.77651e20	0.0570392
$357$	0	0
$358$	4.68497e21	0.917701
$359$	2.51628e21	0.481344	0.240672	−	0.970606i	$-0.422632\pi$
$359$	2.51628e21	0.481344	0.240672	+	0.970606i	$0.422632\pi$
$360$	0	0
$361$	4.60520e21	0.840305
$362$	3.50716e21	0.625076
$363$	0	0
$364$	8.47493e20	0.144137
$365$	−4.82091e21	−0.801013
$366$	0	0
$367$	−1.53497e21	−0.243466	−0.121733	−	0.992563i	$-0.538845\pi$
$367$	−1.53497e21	−0.243466	−0.121733	+	0.992563i	$0.538845\pi$
$368$	2.70115e21	0.418642
$369$	0	0
$370$	5.15287e21	0.762667
$371$	1.18675e21	0.171665
$372$	0	0
$373$	−5.11988e20	−0.0707514	−0.0353757	−	0.999374i	$-0.511263\pi$
$373$	−5.11988e20	−0.0707514	−0.0353757	+	0.999374i	$0.511263\pi$
$374$	−1.28067e20	−0.0172992
$375$	0	0
$376$	−3.46885e21	−0.447806
$377$	2.22662e21	0.281025
$378$	0	0
$379$	1.58862e22	1.91684	0.958421	−	0.285358i	$-0.0921124\pi$
$379$	1.58862e22	1.91684	0.958421	+	0.285358i	$0.0921124\pi$
$380$	5.57558e21	0.657854
$381$	0	0
$382$	−3.33846e21	−0.376710
$383$	−1.31732e22	−1.45379	−0.726893	−	0.686750i	$-0.759037\pi$
$383$	−1.31732e22	−1.45379	−0.726893	+	0.686750i	$0.759037\pi$
$384$	0	0
$385$	1.65546e21	0.174784
$386$	4.14017e21	0.427589
$387$	0	0
$388$	−4.82457e21	−0.476857
$389$	1.26152e22	1.21990	0.609948	−	0.792441i	$-0.291190\pi$
$389$	1.26152e22	1.21990	0.609948	+	0.792441i	$0.291190\pi$
$390$	0	0
$391$	1.29112e21	0.119526
$392$	−2.82395e21	−0.255814
$393$	0	0
$394$	2.27015e20	0.0196942
$395$	5.19577e21	0.441138
$396$	0	0
$397$	9.34039e21	0.759706	0.379853	−	0.925047i	$-0.375975\pi$
$397$	9.34039e21	0.759706	0.379853	+	0.925047i	$0.375975\pi$
$398$	1.18318e22	0.941985
$399$	0	0
$400$	−1.94466e20	−0.0148366
$401$	−1.60000e22	−1.19507	−0.597536	−	0.801842i	$-0.703853\pi$
$401$	−1.60000e22	−1.19507	−0.597536	+	0.801842i	$0.703853\pi$
$402$	0	0
$403$	−2.15714e21	−0.154449
$404$	1.16284e21	0.0815224
$405$	0	0
$406$	2.83472e21	0.190563
$407$	5.78987e21	0.381168
$408$	0	0
$409$	−6.83445e21	−0.431574	−0.215787	−	0.976440i	$-0.569232\pi$
$409$	−6.83445e21	−0.431574	−0.215787	+	0.976440i	$0.569232\pi$
$410$	−1.05408e21	−0.0651944
$411$	0	0
$412$	−5.87503e21	−0.348645
$413$	−7.79900e21	−0.453381
$414$	0	0
$415$	−5.92907e21	−0.330809
$416$	1.77303e21	0.0969221
$417$	0	0
$418$	6.26483e21	0.328784
$419$	−2.96662e22	−1.52561	−0.762803	−	0.646631i	$-0.776178\pi$
$419$	−2.96662e22	−1.52561	−0.762803	+	0.646631i	$0.776178\pi$
$420$	0	0
$421$	−2.76916e22	−1.36757	−0.683786	−	0.729683i	$-0.739668\pi$
$421$	−2.76916e22	−1.36757	−0.683786	+	0.729683i	$0.739668\pi$
$422$	2.73877e22	1.32556
$423$	0	0
$424$	2.48278e21	0.115433
$425$	−9.29526e19	−0.00423600
$426$	0	0
$427$	−4.25472e21	−0.186309
$428$	−4.13703e21	−0.177589
$429$	0	0
$430$	−3.96253e21	−0.163490
$431$	−4.17737e22	−1.68984	−0.844921	−	0.534891i	$-0.820353\pi$
$431$	−4.17737e22	−1.68984	−0.844921	+	0.534891i	$0.820353\pi$
$432$	0	0
$433$	−2.51263e22	−0.977194	−0.488597	−	0.872510i	$-0.662491\pi$
$433$	−2.51263e22	−0.977194	−0.488597	+	0.872510i	$0.662491\pi$
$434$	−2.74626e21	−0.104732
$435$	0	0
$436$	−1.78229e22	−0.653648
$437$	−6.31597e22	−2.27168
$438$	0	0
$439$	−2.88132e22	−0.996880	−0.498440	−	0.866924i	$-0.666094\pi$
$439$	−2.88132e22	−0.996880	−0.498440	+	0.866924i	$0.666094\pi$
$440$	3.46337e21	0.117531
$441$	0	0
$442$	8.47487e20	0.0276722
$443$	2.03106e22	0.650564	0.325282	−	0.945617i	$-0.394541\pi$
$443$	2.03106e22	0.650564	0.325282	+	0.945617i	$0.394541\pi$
$444$	0	0
$445$	−3.58904e21	−0.110642
$446$	1.24232e22	0.375741
$447$	0	0
$448$	2.25725e21	0.0657229
$449$	8.71590e21	0.249011	0.124505	−	0.992219i	$-0.460266\pi$
$449$	8.71590e21	0.249011	0.124505	+	0.992219i	$0.460266\pi$
$450$	0	0
$451$	−1.18438e21	−0.0325830
$452$	−1.77478e21	−0.0479145
$453$	0	0
$454$	4.52761e22	1.17732
$455$	−1.09551e22	−0.279589
$456$	0	0
$457$	−2.05034e22	−0.504126	−0.252063	−	0.967711i	$-0.581109\pi$
$457$	−2.05034e22	−0.504126	−0.252063	+	0.967711i	$0.581109\pi$
$458$	1.83237e22	0.442239
$459$	0	0
$460$	−3.49164e22	−0.812059
$461$	6.35546e22	1.45107	0.725537	−	0.688183i	$-0.241592\pi$
$461$	6.35546e22	1.45107	0.725537	+	0.688183i	$0.241592\pi$
$462$	0	0
$463$	−1.39293e22	−0.306542	−0.153271	−	0.988184i	$-0.548981\pi$
$463$	−1.39293e22	−0.306542	−0.153271	+	0.988184i	$0.548981\pi$
$464$	5.93048e21	0.128141
$465$	0	0
$466$	−1.34804e22	−0.280817
$467$	−2.46074e21	−0.0503352	−0.0251676	−	0.999683i	$-0.508012\pi$
$467$	−2.46074e21	−0.0503352	−0.0251676	+	0.999683i	$0.508012\pi$
$468$	0	0
$469$	−3.27202e22	−0.645427
$470$	4.48400e22	0.868629
$471$	0	0
$472$	−1.63162e22	−0.304868
$473$	−4.45237e21	−0.0817093
$474$	0	0
$475$	4.54711e21	0.0805081
$476$	1.07894e21	0.0187645
$477$	0	0
$478$	−3.23639e22	−0.543154
$479$	−5.77281e21	−0.0951777	−0.0475889	−	0.998867i	$-0.515154\pi$
$479$	−5.77281e21	−0.0951777	−0.0475889	+	0.998867i	$0.515154\pi$
$480$	0	0
$481$	−3.83147e22	−0.609723
$482$	−2.67713e22	−0.418571
$483$	0	0
$484$	−2.92335e22	−0.441260
$485$	6.23646e22	0.924982
$486$	0	0
$487$	1.18963e23	1.70378	0.851891	−	0.523719i	$-0.175456\pi$
$487$	1.18963e23	1.70378	0.851891	+	0.523719i	$0.175456\pi$
$488$	−8.90124e21	−0.125280
$489$	0	0
$490$	3.65036e22	0.496215
$491$	9.85583e22	1.31674	0.658371	−	0.752694i	$-0.271246\pi$
$491$	9.85583e22	1.31674	0.658371	+	0.752694i	$0.271246\pi$
$492$	0	0
$493$	2.83470e21	0.0365854
$494$	−4.14578e22	−0.525930
$495$	0	0
$496$	−5.74542e21	−0.0704251
$497$	−5.89035e22	−0.709760
$498$	0	0
$499$	−5.61088e22	−0.653396	−0.326698	−	0.945129i	$-0.605936\pi$
$499$	−5.61088e22	−0.653396	−0.326698	+	0.945129i	$0.605936\pi$
$500$	4.48712e22	0.513716
$501$	0	0
$502$	2.85340e22	0.315777
$503$	−3.69306e22	−0.401845	−0.200923	−	0.979607i	$-0.564394\pi$
$503$	−3.69306e22	−0.401845	−0.200923	+	0.979607i	$0.564394\pi$
$504$	0	0
$505$	−1.50314e22	−0.158133
$506$	−3.92327e22	−0.405853
$507$	0	0
$508$	−7.86447e22	−0.786734
$509$	−7.31575e22	−0.719710	−0.359855	−	0.933008i	$-0.617174\pi$
$509$	−7.31575e22	−0.719710	−0.359855	+	0.933008i	$0.617174\pi$
$510$	0	0
$511$	4.56362e22	0.434242
$512$	4.72237e21	0.0441942
$513$	0	0
$514$	6.71579e22	0.608009
$515$	7.59434e22	0.676283
$516$	0	0
$517$	5.03831e22	0.434126
$518$	−4.87786e22	−0.413453
$519$	0	0
$520$	−2.29190e22	−0.188004
$521$	−1.36272e22	−0.109973	−0.0549864	−	0.998487i	$-0.517512\pi$
$521$	−1.36272e22	−0.109973	−0.0549864	+	0.998487i	$0.517512\pi$
$522$	0	0
$523$	−1.74409e22	−0.136240	−0.0681199	−	0.997677i	$-0.521700\pi$
$523$	−1.74409e22	−0.136240	−0.0681199	+	0.997677i	$0.521700\pi$
$524$	5.23436e22	0.402298
$525$	0	0
$526$	8.37192e22	0.622940
$527$	−2.74624e21	−0.0201070
$528$	0	0
$529$	2.54480e23	1.80418
$530$	−3.20936e22	−0.223910
$531$	0	0
$532$	−5.27801e22	−0.356633
$533$	7.83772e21	0.0521205
$534$	0	0
$535$	5.34772e22	0.344478
$536$	−6.84536e22	−0.434006
$537$	0	0
$538$	2.12654e23	1.30625
$539$	4.10162e22	0.248000
$540$	0	0
$541$	2.16969e23	1.27122	0.635609	−	0.772011i	$-0.280749\pi$
$541$	2.16969e23	1.27122	0.635609	+	0.772011i	$0.280749\pi$
$542$	−3.35958e22	−0.193772
$543$	0	0
$544$	2.25723e21	0.0126179
$545$	2.30388e23	1.26791
$546$	0	0
$547$	6.59130e21	0.0351624	0.0175812	−	0.999845i	$-0.494403\pi$
$547$	6.59130e21	0.0351624	0.0175812	+	0.999845i	$0.494403\pi$
$548$	−1.52833e23	−0.802751
$549$	0	0
$550$	2.82451e21	0.0143834
$551$	−1.38669e23	−0.695332
$552$	0	0
$553$	−4.91847e22	−0.239148
$554$	−1.22716e23	−0.587582
$555$	0	0
$556$	1.09363e23	0.507847
$557$	2.85395e23	1.30520	0.652601	−	0.757702i	$-0.273678\pi$
$557$	2.85395e23	1.30520	0.652601	+	0.757702i	$0.273678\pi$
$558$	0	0
$559$	2.94638e22	0.130704
$560$	−2.91783e22	−0.127486
$561$	0	0
$562$	2.53014e23	1.07247
$563$	1.37063e23	0.572267	0.286134	−	0.958190i	$-0.407630\pi$
$563$	1.37063e23	0.572267	0.286134	+	0.958190i	$0.407630\pi$
$564$	0	0
$565$	2.29416e22	0.0929419
$566$	−2.56950e23	−1.02544
$567$	0	0
$568$	−1.23231e23	−0.477265
$569$	−3.43584e23	−1.31093	−0.655464	−	0.755226i	$-0.727527\pi$
$569$	−3.43584e23	−1.31093	−0.655464	+	0.755226i	$0.727527\pi$
$570$	0	0
$571$	1.15788e23	0.428803	0.214402	−	0.976746i	$-0.431220\pi$
$571$	1.15788e23	0.428803	0.214402	+	0.976746i	$0.431220\pi$
$572$	−2.57522e22	−0.0939613
$573$	0	0
$574$	9.97823e21	0.0353429
$575$	−2.84757e22	−0.0993797
$576$	0	0
$577$	−2.91327e23	−0.987156	−0.493578	−	0.869701i	$-0.664311\pi$
$577$	−2.91327e23	−0.987156	−0.493578	+	0.869701i	$0.664311\pi$
$578$	−2.10695e23	−0.703504
$579$	0	0
$580$	−7.66602e22	−0.248561
$581$	5.61264e22	0.179337
$582$	0	0
$583$	−3.60610e22	−0.111906
$584$	9.54748e22	0.291998
$585$	0	0
$586$	−4.04024e23	−1.20027
$587$	−2.28147e23	−0.668023	−0.334012	−	0.942569i	$-0.608402\pi$
$587$	−2.28147e23	−0.668023	−0.334012	+	0.942569i	$0.608402\pi$
$588$	0	0
$589$	1.34342e23	0.382149
$590$	2.10911e23	0.591367
$591$	0	0
$592$	−1.02049e23	−0.278019
$593$	2.89061e23	0.776292	0.388146	−	0.921598i	$-0.373116\pi$
$593$	2.89061e23	0.776292	0.388146	+	0.921598i	$0.373116\pi$
$594$	0	0
$595$	−1.39469e22	−0.0363984
$596$	−2.71473e23	−0.698447
$597$	0	0
$598$	2.59625e23	0.649211
$599$	6.48122e23	1.59782	0.798911	−	0.601449i	$-0.205410\pi$
$599$	6.48122e23	1.59782	0.798911	+	0.601449i	$0.205410\pi$
$600$	0	0
$601$	1.73835e23	0.416586	0.208293	−	0.978066i	$-0.433209\pi$
$601$	1.73835e23	0.416586	0.208293	+	0.978066i	$0.433209\pi$
$602$	3.75104e22	0.0886303
$603$	0	0
$604$	−2.97405e23	−0.683180
$605$	3.77886e23	0.855933
$606$	0	0
$607$	−4.65967e23	−1.02625	−0.513123	−	0.858315i	$-0.671512\pi$
$607$	−4.65967e23	−1.02625	−0.513123	+	0.858315i	$0.671512\pi$
$608$	−1.10421e23	−0.239811
$609$	0	0
$610$	1.15062e23	0.243012
$611$	−3.33413e23	−0.694436
$612$	0	0
$613$	6.52857e23	1.32252	0.661262	−	0.750155i	$-0.270021\pi$
$613$	6.52857e23	1.32252	0.661262	+	0.750155i	$0.270021\pi$
$614$	−3.10288e23	−0.619917
$615$	0	0
$616$	−3.27853e22	−0.0637152
$617$	7.54556e23	1.44633	0.723166	−	0.690675i	$-0.242686\pi$
$617$	7.54556e23	1.44633	0.723166	+	0.690675i	$0.242686\pi$
$618$	0	0
$619$	−3.63269e23	−0.677420	−0.338710	−	0.940891i	$-0.609991\pi$
$619$	−3.63269e23	−0.677420	−0.338710	+	0.940891i	$0.609991\pi$
$620$	7.42680e22	0.136607
$621$	0	0
$622$	1.16172e23	0.207913
$623$	3.39750e22	0.0599806
$624$	0	0
$625$	−5.45482e23	−0.937132
$626$	5.87230e23	0.995237
$627$	0	0
$628$	−5.84404e23	−0.963953
$629$	−4.87782e22	−0.0793772
$630$	0	0
$631$	−3.30279e23	−0.523156	−0.261578	−	0.965182i	$-0.584243\pi$
$631$	−3.30279e23	−0.523156	−0.261578	+	0.965182i	$0.584243\pi$
$632$	−1.02899e23	−0.160811
$633$	0	0
$634$	−2.10433e23	−0.320152
$635$	1.01660e24	1.52606
$636$	0	0
$637$	−2.71427e23	−0.396705
$638$	−8.61369e22	−0.124226
$639$	0	0
$640$	−6.10435e22	−0.0857255
$641$	−3.94560e23	−0.546789	−0.273395	−	0.961902i	$-0.588146\pi$
$641$	−3.94560e23	−0.546789	−0.273395	+	0.961902i	$0.588146\pi$
$642$	0	0
$643$	−6.95104e23	−0.938116	−0.469058	−	0.883167i	$-0.655407\pi$
$643$	−6.95104e23	−0.938116	−0.469058	+	0.883167i	$0.655407\pi$
$644$	3.30529e23	0.440230
$645$	0	0
$646$	−5.27797e22	−0.0684685
$647$	−1.22088e24	−1.56310	−0.781551	−	0.623841i	$-0.785571\pi$
$647$	−1.22088e24	−1.56310	−0.781551	+	0.623841i	$0.785571\pi$
$648$	0	0
$649$	2.36983e23	0.295555
$650$	−1.86914e22	−0.0230079
$651$	0	0
$652$	6.70820e23	0.804455
$653$	−4.60026e23	−0.544528	−0.272264	−	0.962223i	$-0.587772\pi$
$653$	−4.60026e23	−0.544528	−0.272264	+	0.962223i	$0.587772\pi$
$654$	0	0
$655$	−6.76618e23	−0.780355
$656$	2.08753e22	0.0237657
$657$	0	0
$658$	−4.24469e23	−0.470897
$659$	−3.70941e23	−0.406236	−0.203118	−	0.979154i	$-0.565108\pi$
$659$	−3.70941e23	−0.406236	−0.203118	+	0.979154i	$0.565108\pi$
$660$	0	0
$661$	−1.26164e24	−1.34655	−0.673274	−	0.739393i	$-0.735113\pi$
$661$	−1.26164e24	−1.34655	−0.673274	+	0.739393i	$0.735113\pi$
$662$	−4.88435e23	−0.514652
$663$	0	0
$664$	1.17421e23	0.120592
$665$	6.82261e23	0.691778
$666$	0	0
$667$	8.68400e23	0.858322
$668$	1.39667e23	0.136300
$669$	0	0
$670$	8.84863e23	0.841860
$671$	1.29285e23	0.121453
$672$	0	0
$673$	−2.14270e24	−1.96260	−0.981301	−	0.192478i	$-0.938348\pi$
$673$	−2.14270e24	−1.96260	−0.981301	+	0.192478i	$0.938348\pi$
$674$	4.56840e22	0.0413195
$675$	0	0
$676$	−3.96497e23	−0.349698
$677$	−9.73581e23	−0.847946	−0.423973	−	0.905675i	$-0.639365\pi$
$677$	−9.73581e23	−0.847946	−0.423973	+	0.905675i	$0.639365\pi$
$678$	0	0
$679$	−5.90362e23	−0.501447
$680$	−2.91781e22	−0.0244755
$681$	0	0
$682$	8.34489e22	0.0682737
$683$	−2.21286e24	−1.78804	−0.894020	−	0.448027i	$-0.852127\pi$
$683$	−2.21286e24	−1.78804	−0.894020	+	0.448027i	$0.852127\pi$
$684$	0	0
$685$	1.97559e24	1.55713
$686$	−8.23135e23	−0.640791
$687$	0	0
$688$	7.84751e22	0.0595978
$689$	2.38635e23	0.179008
$690$	0	0
$691$	7.53736e23	0.551640	0.275820	−	0.961209i	$-0.411051\pi$
$691$	7.53736e23	0.551640	0.275820	+	0.961209i	$0.411051\pi$
$692$	3.47029e23	0.250879
$693$	0	0
$694$	−1.45544e24	−1.02669
$695$	−1.41367e24	−0.985095
$696$	0	0
$697$	9.97815e21	0.00678534
$698$	1.63283e24	1.09691
$699$	0	0
$700$	−2.37960e22	−0.0156017
$701$	1.12387e24	0.727967	0.363983	−	0.931405i	$-0.381417\pi$
$701$	1.12387e24	0.727967	0.363983	+	0.931405i	$0.381417\pi$
$702$	0	0
$703$	2.38616e24	1.50862
$704$	−6.85897e22	−0.0428441
$705$	0	0
$706$	7.46091e23	0.454938
$707$	1.42291e23	0.0857263
$708$	0	0
$709$	−2.37394e24	−1.39629	−0.698145	−	0.715956i	$-0.745991\pi$
$709$	−2.37394e24	−1.39629	−0.698145	+	0.715956i	$0.745991\pi$
$710$	1.59294e24	0.925773
$711$	0	0
$712$	7.10786e22	0.0403328
$713$	−8.41301e23	−0.471727
$714$	0	0
$715$	3.32886e23	0.182261
$716$	1.19935e24	0.648913
$717$	0	0
$718$	6.44167e23	0.340362
$719$	−1.19702e24	−0.625037	−0.312519	−	0.949912i	$-0.601173\pi$
$719$	−1.19702e24	−0.625037	−0.312519	+	0.949912i	$0.601173\pi$
$720$	0	0
$721$	−7.18903e23	−0.366624
$722$	1.17893e24	0.594186
$723$	0	0
$724$	8.97833e23	0.441995
$725$	−6.25194e22	−0.0304188
$726$	0	0
$727$	3.83121e24	1.82093	0.910466	−	0.413583i	$-0.135723\pi$
$727$	3.83121e24	1.82093	0.910466	+	0.413583i	$0.135723\pi$
$728$	2.16958e23	0.101920
$729$	0	0
$730$	−1.23415e24	−0.566402
$731$	3.75102e22	0.0170158
$732$	0	0
$733$	3.77963e24	1.67519	0.837597	−	0.546289i	$-0.183960\pi$
$733$	3.77963e24	1.67519	0.837597	+	0.546289i	$0.183960\pi$
$734$	−3.92952e23	−0.172156
$735$	0	0
$736$	6.91495e23	0.296025
$737$	9.94249e23	0.420747
$738$	0	0
$739$	3.29096e23	0.136096	0.0680479	−	0.997682i	$-0.478323\pi$
$739$	3.29096e23	0.136096	0.0680479	+	0.997682i	$0.478323\pi$
$740$	1.31914e24	0.539287
$741$	0	0
$742$	3.03808e23	0.121385
$743$	−1.88066e24	−0.742857	−0.371428	−	0.928462i	$-0.621132\pi$
$743$	−1.88066e24	−0.742857	−0.371428	+	0.928462i	$0.621132\pi$
$744$	0	0
$745$	3.50919e24	1.35481
$746$	−1.31069e23	−0.0500288
$747$	0	0
$748$	−3.27850e22	−0.0122324
$749$	−5.06232e23	−0.186747
$750$	0	0
$751$	1.24542e24	0.449134	0.224567	−	0.974459i	$-0.427903\pi$
$751$	1.24542e24	0.449134	0.224567	+	0.974459i	$0.427903\pi$
$752$	−8.88027e23	−0.316646
$753$	0	0
$754$	5.70015e23	0.198715
$755$	3.84440e24	1.32519
$756$	0	0
$757$	4.13981e24	1.39529	0.697647	−	0.716442i	$-0.254230\pi$
$757$	4.13981e24	1.39529	0.697647	+	0.716442i	$0.254230\pi$
$758$	4.06686e24	1.35541
$759$	0	0
$760$	1.42735e24	0.465173
$761$	−4.57805e24	−1.47540	−0.737701	−	0.675127i	$-0.764089\pi$
$761$	−4.57805e24	−1.47540	−0.737701	+	0.675127i	$0.764089\pi$
$762$	0	0
$763$	−2.18092e24	−0.687354
$764$	−8.54645e23	−0.266374
$765$	0	0
$766$	−3.37233e24	−1.02798
$767$	−1.56825e24	−0.472775
$768$	0	0
$769$	−6.64989e24	−1.96083	−0.980417	−	0.196934i	$-0.936902\pi$
$769$	−6.64989e24	−1.96083	−0.980417	+	0.196934i	$0.936902\pi$
$770$	4.23798e23	0.123591
$771$	0	0
$772$	1.05988e24	0.302351
$773$	−1.75170e24	−0.494235	−0.247117	−	0.968986i	$-0.579483\pi$
$773$	−1.75170e24	−0.494235	−0.247117	+	0.968986i	$0.579483\pi$
$774$	0	0
$775$	6.05685e22	0.0167179
$776$	−1.23509e24	−0.337189
$777$	0	0
$778$	3.22949e24	0.862597
$779$	−4.88117e23	−0.128960
$780$	0	0
$781$	1.78986e24	0.462686
$782$	3.30526e23	0.0845179
$783$	0	0
$784$	−7.22930e23	−0.180888
$785$	7.55428e24	1.86982
$786$	0	0
$787$	−3.65074e22	−0.00884293	−0.00442146	−	0.999990i	$-0.501407\pi$
$787$	−3.65074e22	−0.00884293	−0.00442146	+	0.999990i	$0.501407\pi$
$788$	5.81160e22	0.0139259
$789$	0	0
$790$	1.33012e24	0.311932
$791$	−2.17172e23	−0.0503853
$792$	0	0
$793$	−8.55553e23	−0.194279
$794$	2.39114e24	0.537193
$795$	0	0
$796$	3.02894e24	0.666084
$797$	5.05348e24	1.09950	0.549750	−	0.835329i	$-0.314723\pi$
$797$	5.05348e24	1.09950	0.549750	+	0.835329i	$0.314723\pi$
$798$	0	0
$799$	−4.24466e23	−0.0904056
$800$	−4.97834e22	−0.0104911
$801$	0	0
$802$	−4.09601e24	−0.845043
$803$	−1.38672e24	−0.283078
$804$	0	0
$805$	−4.27257e24	−0.853934
$806$	−5.52227e23	−0.109212
$807$	0	0
$808$	2.97686e23	0.0576451
$809$	6.17340e23	0.118294	0.0591469	−	0.998249i	$-0.481162\pi$
$809$	6.17340e23	0.118294	0.0591469	+	0.998249i	$0.481162\pi$
$810$	0	0
$811$	4.34432e24	0.815163	0.407582	−	0.913169i	$-0.366372\pi$
$811$	4.34432e24	0.815163	0.407582	+	0.913169i	$0.366372\pi$
$812$	7.25689e23	0.134749
$813$	0	0
$814$	1.48221e24	0.269526
$815$	−8.67134e24	−1.56044
$816$	0	0
$817$	−1.83494e24	−0.323396
$818$	−1.74962e24	−0.305169
$819$	0	0
$820$	−2.69844e23	−0.0460994
$821$	2.16713e24	0.366411	0.183205	−	0.983075i	$-0.441353\pi$
$821$	2.16713e24	0.366411	0.183205	+	0.983075i	$0.441353\pi$
$822$	0	0
$823$	−9.91114e24	−1.64144	−0.820720	−	0.571331i	$-0.806427\pi$
$823$	−9.91114e24	−1.64144	−0.820720	+	0.571331i	$0.806427\pi$
$824$	−1.50401e24	−0.246529
$825$	0	0
$826$	−1.99654e24	−0.320589
$827$	−8.36444e24	−1.32935	−0.664676	−	0.747132i	$-0.731430\pi$
$827$	−8.36444e24	−1.32935	−0.664676	+	0.747132i	$0.731430\pi$
$828$	0	0
$829$	5.00772e24	0.779698	0.389849	−	0.920879i	$-0.372527\pi$
$829$	5.00772e24	0.779698	0.389849	+	0.920879i	$0.372527\pi$
$830$	−1.51784e24	−0.233918
$831$	0	0
$832$	4.53896e23	0.0685343
$833$	−3.45552e23	−0.0516453
$834$	0	0
$835$	−1.80541e24	−0.264387
$836$	1.60380e24	0.232486
$837$	0	0
$838$	−7.59454e24	−1.07877
$839$	4.36049e23	0.0613139	0.0306569	−	0.999530i	$-0.490240\pi$
$839$	4.36049e23	0.0613139	0.0306569	+	0.999530i	$0.490240\pi$
$840$	0	0
$841$	−5.35054e24	−0.737279
$842$	−7.08905e24	−0.967019
$843$	0	0
$844$	7.01124e24	0.937312
$845$	5.12530e24	0.678325
$846$	0	0
$847$	−3.57718e24	−0.464014
$848$	6.35592e23	0.0816233
$849$	0	0
$850$	−2.37959e22	−0.00299530
$851$	−1.49430e25	−1.86225
$852$	0	0
$853$	7.44576e24	0.909583	0.454792	−	0.890598i	$-0.349714\pi$
$853$	7.44576e24	0.909583	0.454792	+	0.890598i	$0.349714\pi$
$854$	−1.08921e24	−0.131740
$855$	0	0
$856$	−1.05908e24	−0.125575
$857$	9.17712e24	1.07738	0.538690	−	0.842504i	$-0.318919\pi$
$857$	9.17712e24	1.07738	0.538690	+	0.842504i	$0.318919\pi$
$858$	0	0
$859$	−5.83348e24	−0.671407	−0.335703	−	0.941968i	$-0.608974\pi$
$859$	−5.83348e24	−0.671407	−0.335703	+	0.941968i	$0.608974\pi$
$860$	−1.01441e24	−0.115605
$861$	0	0
$862$	−1.06941e25	−1.19490
$863$	1.13165e24	0.125205	0.0626025	−	0.998039i	$-0.480060\pi$
$863$	1.13165e24	0.125205	0.0626025	+	0.998039i	$0.480060\pi$
$864$	0	0
$865$	−4.48586e24	−0.486641
$866$	−6.43232e24	−0.690981
$867$	0	0
$868$	−7.03043e23	−0.0740567
$869$	1.49454e24	0.155898
$870$	0	0
$871$	−6.57949e24	−0.673036
$872$	−4.56267e24	−0.462199
$873$	0	0
$874$	−1.61689e25	−1.60632
$875$	5.49070e24	0.540206
$876$	0	0
$877$	6.91600e24	0.667358	0.333679	−	0.942687i	$-0.391710\pi$
$877$	6.91600e24	0.667358	0.333679	+	0.942687i	$0.391710\pi$
$878$	−7.37617e24	−0.704900
$879$	0	0
$880$	8.86622e23	0.0831067
$881$	1.70872e25	1.58626	0.793131	−	0.609051i	$-0.208449\pi$
$881$	1.70872e25	1.58626	0.793131	+	0.609051i	$0.208449\pi$
$882$	0	0
$883$	−9.18892e23	−0.0836755	−0.0418377	−	0.999124i	$-0.513321\pi$
$883$	−9.18892e23	−0.0836755	−0.0418377	+	0.999124i	$0.513321\pi$
$884$	2.16957e23	0.0195672
$885$	0	0
$886$	5.19950e24	0.460018
$887$	6.41621e24	0.562248	0.281124	−	0.959671i	$-0.409293\pi$
$887$	6.41621e24	0.562248	0.281124	+	0.959671i	$0.409293\pi$
$888$	0	0
$889$	−9.62343e24	−0.827303
$890$	−9.18795e23	−0.0782355
$891$	0	0
$892$	3.18035e24	0.265689
$893$	2.07643e25	1.71822
$894$	0	0
$895$	−1.55034e25	−1.25873
$896$	5.77856e23	0.0464731
$897$	0	0
$898$	2.23127e24	0.176077
$899$	−1.84711e24	−0.144389
$900$	0	0
$901$	3.03805e23	0.0233042
$902$	−3.03202e23	−0.0230397
$903$	0	0
$904$	−4.54343e23	−0.0338807
$905$	−1.16058e25	−0.857359
$906$	0	0
$907$	−3.18557e24	−0.230954	−0.115477	−	0.993310i	$-0.536840\pi$
$907$	−3.18557e24	−0.230954	−0.115477	+	0.993310i	$0.536840\pi$
$908$	1.15907e25	0.832491
$909$	0	0
$910$	−2.80450e24	−0.197699
$911$	2.04691e25	1.42953	0.714763	−	0.699367i	$-0.246535\pi$
$911$	2.04691e25	1.42953	0.714763	+	0.699367i	$0.246535\pi$
$912$	0	0
$913$	−1.70548e24	−0.116908
$914$	−5.24888e24	−0.356471
$915$	0	0
$916$	4.69087e24	0.312710
$917$	6.40507e24	0.423043
$918$	0	0
$919$	1.40089e25	0.908287	0.454144	−	0.890928i	$-0.349945\pi$
$919$	1.40089e25	0.908287	0.454144	+	0.890928i	$0.349945\pi$
$920$	−8.93860e24	−0.574213
$921$	0	0
$922$	1.62700e25	1.02606
$923$	−1.18445e25	−0.740121
$924$	0	0
$925$	1.07581e24	0.0659978
$926$	−3.56589e24	−0.216758
$927$	0	0
$928$	1.51820e24	0.0906092
$929$	2.19459e25	1.29784	0.648919	−	0.760857i	$-0.275221\pi$
$929$	2.19459e25	1.29784	0.648919	+	0.760857i	$0.275221\pi$
$930$	0	0
$931$	1.69039e25	0.981555
$932$	−3.45099e24	−0.198568
$933$	0	0
$934$	−6.29948e23	−0.0355923
$935$	4.23795e23	0.0237278
$936$	0	0
$937$	1.45560e25	0.800305	0.400152	−	0.916449i	$-0.368957\pi$
$937$	1.45560e25	0.800305	0.400152	+	0.916449i	$0.368957\pi$
$938$	−8.37638e24	−0.456386
$939$	0	0
$940$	1.14791e25	0.614213
$941$	2.49769e25	1.32442	0.662212	−	0.749316i	$-0.269618\pi$
$941$	2.49769e25	1.32442	0.662212	+	0.749316i	$0.269618\pi$
$942$	0	0
$943$	3.05677e24	0.159189
$944$	−4.17695e24	−0.215574
$945$	0	0
$946$	−1.13981e24	−0.0577772
$947$	−2.46572e25	−1.23871	−0.619355	−	0.785111i	$-0.712606\pi$
$947$	−2.46572e25	−1.23871	−0.619355	+	0.785111i	$0.712606\pi$
$948$	0	0
$949$	9.17667e24	0.452817
$950$	1.16406e24	0.0569278
$951$	0	0
$952$	2.76208e23	0.0132685
$953$	−1.40846e25	−0.670587	−0.335294	−	0.942114i	$-0.608836\pi$
$953$	−1.40846e25	−0.670587	−0.335294	+	0.942114i	$0.608836\pi$
$954$	0	0
$955$	1.10475e25	0.516698
$956$	−8.28515e24	−0.384068
$957$	0	0
$958$	−1.47784e24	−0.0673008
$959$	−1.87015e25	−0.844147
$960$	0	0
$961$	−2.07606e25	−0.920645
$962$	−9.80857e24	−0.431139
$963$	0	0
$964$	−6.85344e24	−0.295974
$965$	−1.37006e25	−0.586484
$966$	0	0
$967$	1.16961e25	0.491945	0.245972	−	0.969277i	$-0.420893\pi$
$967$	1.16961e25	0.491945	0.245972	+	0.969277i	$0.420893\pi$
$968$	−7.48377e24	−0.312018
$969$	0	0
$970$	1.59653e25	0.654061
$971$	1.45914e25	0.592563	0.296281	−	0.955101i	$-0.404253\pi$
$971$	1.45914e25	0.592563	0.296281	+	0.955101i	$0.404253\pi$
$972$	0	0
$973$	1.33822e25	0.534035
$974$	3.04544e25	1.20476
$975$	0	0
$976$	−2.27872e24	−0.0885864
$977$	2.16074e25	0.832721	0.416360	−	0.909200i	$-0.363305\pi$
$977$	2.16074e25	0.832721	0.416360	+	0.909200i	$0.363305\pi$
$978$	0	0
$979$	−1.03238e24	−0.0391007
$980$	9.34493e24	0.350877
$981$	0	0
$982$	2.52309e25	0.931077
$983$	−2.26102e25	−0.827180	−0.413590	−	0.910463i	$-0.635725\pi$
$983$	−2.26102e25	−0.827180	−0.413590	+	0.910463i	$0.635725\pi$
$984$	0	0
$985$	−7.51234e23	−0.0270127
$986$	7.25683e23	0.0258698
$987$	0	0
$988$	−1.06132e25	−0.371888
$989$	1.14911e25	0.399202
$990$	0	0
$991$	4.37346e25	1.49348	0.746739	−	0.665117i	$-0.231618\pi$
$991$	4.37346e25	1.49348	0.746739	+	0.665117i	$0.231618\pi$
$992$	−1.47083e24	−0.0497981
$993$	0	0
$994$	−1.50793e25	−0.501876
$995$	−3.91535e25	−1.29203
$996$	0	0
$997$	3.46382e25	1.12369	0.561845	−	0.827243i	$-0.310092\pi$
$997$	3.46382e25	1.12369	0.561845	+	0.827243i	$0.310092\pi$
$998$	−1.43639e25	−0.462021
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.18.a.b.1.1	yes	2
3.2	odd	2		54.18.a.a.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.18.a.a.1.2	✓	2	3.2	odd	2
54.18.a.b.1.1	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

\(f(q)\)	\(=\)	\(q+256.000 q^{2} +65536.0 q^{4} -847149. q^{5} +8.01937e6 q^{7} +1.67772e7 q^{8} -2.16870e8 q^{10} -2.43679e8 q^{11} +1.61256e9 q^{13} +2.05296e9 q^{14} +4.29497e9 q^{16} +2.05294e9 q^{17} -1.00427e11 q^{19} -5.55188e10 q^{20} -6.23819e10 q^{22} +6.28911e11 q^{23} -4.52777e10 q^{25} +4.12816e11 q^{26} +5.25557e11 q^{28} +1.38080e12 q^{29} -1.33771e12 q^{31} +1.09951e12 q^{32} +5.25553e11 q^{34} -6.79360e12 q^{35} -2.37602e13 q^{37} -2.57093e13 q^{38} -1.42128e13 q^{40} +4.86041e12 q^{41} +1.82714e13 q^{43} -1.59698e13 q^{44} +1.61001e14 q^{46} -2.06760e14 q^{47} -1.68320e14 q^{49} -1.15911e13 q^{50} +1.05681e14 q^{52} +1.47985e14 q^{53} +2.06433e14 q^{55} +1.34543e14 q^{56} +3.53484e14 q^{58} -9.72521e14 q^{59} -5.30555e14 q^{61} -3.42454e14 q^{62} +2.81475e14 q^{64} -1.36608e15 q^{65} -4.08015e15 q^{67} +1.34542e14 q^{68} -1.73916e15 q^{70} -7.34515e15 q^{71} +5.69074e15 q^{73} -6.08260e15 q^{74} -6.58158e15 q^{76} -1.95416e15 q^{77} -6.13324e15 q^{79} -3.63848e15 q^{80} +1.24427e15 q^{82} +6.99885e15 q^{83} -1.73915e15 q^{85} +4.67748e15 q^{86} -4.08826e15 q^{88} +4.23661e15 q^{89} +1.29317e16 q^{91} +4.12163e16 q^{92} -5.29305e16 q^{94} +8.50767e16 q^{95} -7.36170e16 q^{97} -4.30900e16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 512 q^{2} + 131072 q^{4} - 304980 q^{5} - 9246860 q^{7} + 33554432 q^{8} - 78074880 q^{10} - 155034030 q^{11} + 1702429720 q^{13} - 2367196160 q^{14} + 8589934592 q^{16} - 8486897472 q^{17} - 5707598648 q^{19}+ \cdots - 26\!\cdots\!76 q^{98}+O(q^{100}) \) 2 * q + 512 * q^2 + 131072 * q^4 - 304980 * q^5 - 9246860 * q^7 + 33554432 * q^8 - 78074880 * q^10 - 155034030 * q^11 + 1702429720 * q^13 - 2367196160 * q^14 + 8589934592 * q^16 - 8486897472 * q^17 - 5707598648 * q^19 - 19987169280 * q^20 - 39688711680 * q^22 + 266994431796 * q^23 - 514269781600 * q^25 + 435822008320 * q^26 - 606002216960 * q^28 + 113442022320 * q^29 - 6929179768316 * q^31 + 2199023255552 * q^32 - 2172645752832 * q^34 - 16154815903590 * q^35 - 14067852085580 * q^37 - 1461145253888 * q^38 - 5116715335680 * q^40 - 30739866769320 * q^41 + 58093870679560 * q^43 - 10160310190080 * q^44 + 68350574539776 * q^46 - 69202049560356 * q^47 - 102828191597796 * q^49 - 131653064089600 * q^50 + 111570434129920 * q^52 - 659688284112516 * q^53 + 254493710923140 * q^55 - 155136567541760 * q^56 + 29041157713920 * q^58 - 1288830277403880 * q^59 + 144079111616032 * q^61 - 1773870020688896 * q^62 + 562949953421312 * q^64 - 1317356094000000 * q^65 - 3203435114160680 * q^67 - 556197312724992 * q^68 - 4135632871319040 * q^70 - 11320825195049760 * q^71 - 4886170417034870 * q^73 - 3601370133908480 * q^74 - 374053184995328 * q^76 - 3484727819406708 * q^77 - 13226203174257080 * q^79 - 1309879125934080 * q^80 - 7869405892945920 * q^82 - 14304595055224350 * q^83 - 7453524098909520 * q^85 + 14872030893967360 * q^86 - 2601039408660480 * q^88 + 6721701300374040 * q^89 + 11380032424809440 * q^91 + 17497747082182656 * q^92 - 17715724687451136 * q^94 + 136430610599529420 * q^95 - 71442353414491730 * q^97 - 26324017049035776 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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