LMFDB - Embedded newform 54.10.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [54,10,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, 10, 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, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	54.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,32,0,512,912] 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:	$27.8119351528$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3329}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 832 $ x^2 - x - 832
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-28.3487$ 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)

$f(q)$	$=$	\(q+16.0000 q^{2} +256.000 q^{4} +2013.83 q^{5} -4565.16 q^{7} +4096.00 q^{8} +32221.3 q^{10} +69876.3 q^{11} +45612.6 q^{13} -73042.6 q^{14} +65536.0 q^{16} -142743. q^{17} -759286. q^{19} +515541. q^{20} +1.11802e6 q^{22} +1.87376e6 q^{23} +2.10239e6 q^{25} +729802. q^{26} -1.16868e6 q^{28} +797363. q^{29} +8.93141e6 q^{31} +1.04858e6 q^{32} -2.28389e6 q^{34} -9.19347e6 q^{35} +1.97984e7 q^{37} -1.21486e7 q^{38} +8.24866e6 q^{40} -1.92076e7 q^{41} +1.38242e7 q^{43} +1.78883e7 q^{44} +2.99802e7 q^{46} +2.81809e7 q^{47} -1.95129e7 q^{49} +3.36383e7 q^{50} +1.16768e7 q^{52} -6.10387e7 q^{53} +1.40719e8 q^{55} -1.86989e7 q^{56} +1.27578e7 q^{58} -8.17810e7 q^{59} -1.42814e8 q^{61} +1.42903e8 q^{62} +1.67772e7 q^{64} +9.18562e7 q^{65} +1.68812e8 q^{67} -3.65422e7 q^{68} -1.47095e8 q^{70} +1.12191e8 q^{71} -3.08721e8 q^{73} +3.16774e8 q^{74} -1.94377e8 q^{76} -3.18996e8 q^{77} -3.23516e8 q^{79} +1.31979e8 q^{80} -3.07322e8 q^{82} -5.34107e8 q^{83} -2.87461e8 q^{85} +2.21187e8 q^{86} +2.86213e8 q^{88} -7.97823e8 q^{89} -2.08229e8 q^{91} +4.79683e8 q^{92} +4.50894e8 q^{94} -1.52907e9 q^{95} -6.43889e8 q^{97} -3.12207e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{2} + 512 q^{4} + 912 q^{5} + 6448 q^{7} + 8192 q^{8} + 14592 q^{10} + 15126 q^{11} + 28912 q^{13} + 103168 q^{14} + 131072 q^{16} + 399960 q^{17} + 8104 q^{19} + 233472 q^{20} + 242016 q^{22}+ \cdots + 982771008 q^{98}+O(q^{100}) $ 2 * q + 32 * q^2 + 512 * q^4 + 912 * q^5 + 6448 * q^7 + 8192 * q^8 + 14592 * q^10 + 15126 * q^11 + 28912 * q^13 + 103168 * q^14 + 131072 * q^16 + 399960 * q^17 + 8104 * q^19 + 233472 * q^20 + 242016 * q^22 + 2563572 * q^23 + 1363304 * q^25 + 462592 * q^26 + 1650688 * q^28 + 3059088 * q^29 + 16787920 * q^31 + 2097152 * q^32 + 6399360 * q^34 - 21328122 * q^35 + 34798684 * q^37 + 129664 * q^38 + 3735552 * q^40 - 24238968 * q^41 + 12599680 * q^43 + 3872256 * q^44 + 41017152 * q^46 - 19535772 * q^47 + 61423188 * q^49 + 21812864 * q^50 + 7401472 * q^52 + 12534912 * q^53 + 201044736 * q^55 + 26411008 * q^56 + 48945408 * q^58 - 140256744 * q^59 - 169724624 * q^61 + 268606720 * q^62 + 33554432 * q^64 + 110257512 * q^65 - 72241232 * q^67 + 102389760 * q^68 - 341249952 * q^70 - 53846280 * q^71 - 203557694 * q^73 + 556778944 * q^74 + 2074624 * q^76 - 921970176 * q^77 - 877964864 * q^79 + 59768832 * q^80 - 387823488 * q^82 - 111456714 * q^83 - 885428280 * q^85 + 201594880 * q^86 + 61956096 * q^88 - 1234260864 * q^89 - 392155912 * q^91 + 656274432 * q^92 - 312572352 * q^94 - 2374608756 * q^95 - 165265682 * q^97 + 982771008 * 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 $

$2$	16.0000	0.707107
$2$	16.0000	0.707107
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	2013.83	1.44098	0.720490	−	0.693465i	$-0.243917\pi$
$5$	2013.83	1.44098	0.720490	+	0.693465i	$0.243917\pi$
$6$	0	0
$7$	−4565.16	−0.718646	−0.359323	−	0.933213i	$-0.616992\pi$
$7$	−4565.16	−0.718646	−0.359323	+	0.933213i	$0.616992\pi$
$8$	4096.00	0.353553
$9$	0	0
$10$	32221.3	1.01893
$11$	69876.3	1.43901	0.719503	−	0.694489i	$-0.244370\pi$
$11$	69876.3	1.43901	0.719503	+	0.694489i	$0.244370\pi$
$12$	0	0
$13$	45612.6	0.442935	0.221468	−	0.975168i	$-0.428915\pi$
$13$	45612.6	0.442935	0.221468	+	0.975168i	$0.428915\pi$
$14$	−73042.6	−0.508159
$15$	0	0
$16$	65536.0	0.250000
$17$	−142743.	−0.414510	−0.207255	−	0.978287i	$-0.566453\pi$
$17$	−142743.	−0.414510	−0.207255	+	0.978287i	$0.566453\pi$
$18$	0	0
$19$	−759286.	−1.33664	−0.668319	−	0.743874i	$-0.732986\pi$
$19$	−759286.	−1.33664	−0.668319	+	0.743874i	$0.732986\pi$
$20$	515541.	0.720490
$21$	0	0
$22$	1.11802e6	1.01753
$23$	1.87376e6	1.39617	0.698086	−	0.716013i	$-0.254035\pi$
$23$	1.87376e6	1.39617	0.698086	+	0.716013i	$0.254035\pi$
$24$	0	0
$25$	2.10239e6	1.07643
$26$	729802.	0.313202
$27$	0	0
$28$	−1.16868e6	−0.359323
$29$	797363.	0.209346	0.104673	−	0.994507i	$-0.466620\pi$
$29$	797363.	0.209346	0.104673	+	0.994507i	$0.466620\pi$
$30$	0	0
$31$	8.93141e6	1.73697	0.868485	−	0.495715i	$-0.165094\pi$
$31$	8.93141e6	1.73697	0.868485	+	0.495715i	$0.165094\pi$
$32$	1.04858e6	0.176777
$33$	0	0
$34$	−2.28389e6	−0.293103
$35$	−9.19347e6	−1.03555
$36$	0	0
$37$	1.97984e7	1.73669	0.868345	−	0.495960i	$-0.165184\pi$
$37$	1.97984e7	1.73669	0.868345	+	0.495960i	$0.165184\pi$
$38$	−1.21486e7	−0.945146
$39$	0	0
$40$	8.24866e6	0.509464
$41$	−1.92076e7	−1.06156	−0.530782	−	0.847508i	$-0.678102\pi$
$41$	−1.92076e7	−1.06156	−0.530782	+	0.847508i	$0.678102\pi$
$42$	0	0
$43$	1.38242e7	0.616639	0.308320	−	0.951283i	$-0.400233\pi$
$43$	1.38242e7	0.616639	0.308320	+	0.951283i	$0.400233\pi$
$44$	1.78883e7	0.719503
$45$	0	0
$46$	2.99802e7	0.987243
$47$	2.81809e7	0.842393	0.421196	−	0.906969i	$-0.361610\pi$
$47$	2.81809e7	0.842393	0.421196	+	0.906969i	$0.361610\pi$
$48$	0	0
$49$	−1.95129e7	−0.483548
$50$	3.36383e7	0.761148
$51$	0	0
$52$	1.16768e7	0.221468
$53$	−6.10387e7	−1.06259	−0.531293	−	0.847188i	$-0.678293\pi$
$53$	−6.10387e7	−1.06259	−0.531293	+	0.847188i	$0.678293\pi$
$54$	0	0
$55$	1.40719e8	2.07358
$56$	−1.86989e7	−0.254080
$57$	0	0
$58$	1.27578e7	0.148030
$59$	−8.17810e7	−0.878654	−0.439327	−	0.898327i	$-0.644783\pi$
$59$	−8.17810e7	−0.878654	−0.439327	+	0.898327i	$0.644783\pi$
$60$	0	0
$61$	−1.42814e8	−1.32064	−0.660322	−	0.750983i	$-0.729580\pi$
$61$	−1.42814e8	−1.32064	−0.660322	+	0.750983i	$0.729580\pi$
$62$	1.42903e8	1.22822
$63$	0	0
$64$	1.67772e7	0.125000
$65$	9.18562e7	0.638261
$66$	0	0
$67$	1.68812e8	1.02345	0.511726	−	0.859149i	$-0.329006\pi$
$67$	1.68812e8	1.02345	0.511726	+	0.859149i	$0.329006\pi$
$68$	−3.65422e7	−0.207255
$69$	0	0
$70$	−1.47095e8	−0.732248
$71$	1.12191e8	0.523958	0.261979	−	0.965074i	$-0.415625\pi$
$71$	1.12191e8	0.523958	0.261979	+	0.965074i	$0.415625\pi$
$72$	0	0
$73$	−3.08721e8	−1.27237	−0.636185	−	0.771536i	$-0.719489\pi$
$73$	−3.08721e8	−1.27237	−0.636185	+	0.771536i	$0.719489\pi$
$74$	3.16774e8	1.22803
$75$	0	0
$76$	−1.94377e8	−0.668319
$77$	−3.18996e8	−1.03414
$78$	0	0
$79$	−3.23516e8	−0.934488	−0.467244	−	0.884128i	$-0.654753\pi$
$79$	−3.23516e8	−0.934488	−0.467244	+	0.884128i	$0.654753\pi$
$80$	1.31979e8	0.360245
$81$	0	0
$82$	−3.07322e8	−0.750639
$83$	−5.34107e8	−1.23531	−0.617657	−	0.786448i	$-0.711918\pi$
$83$	−5.34107e8	−1.23531	−0.617657	+	0.786448i	$0.711918\pi$
$84$	0	0
$85$	−2.87461e8	−0.597301
$86$	2.21187e8	0.436030
$87$	0	0
$88$	2.86213e8	0.508766
$89$	−7.97823e8	−1.34788	−0.673941	−	0.738785i	$-0.735400\pi$
$89$	−7.97823e8	−1.34788	−0.673941	+	0.738785i	$0.735400\pi$
$90$	0	0
$91$	−2.08229e8	−0.318313
$92$	4.79683e8	0.698086
$93$	0	0
$94$	4.50894e8	0.595662
$95$	−1.52907e9	−1.92607
$96$	0	0
$97$	−6.43889e8	−0.738479	−0.369239	−	0.929334i	$-0.620382\pi$
$97$	−6.43889e8	−0.738479	−0.369239	+	0.929334i	$0.620382\pi$
$98$	−3.12207e8	−0.341920
$99$	0	0
$100$	5.38213e8	0.538213
$101$	1.38053e9	1.32008	0.660038	−	0.751233i	$-0.270540\pi$
$101$	1.38053e9	1.32008	0.660038	+	0.751233i	$0.270540\pi$
$102$	0	0
$103$	−1.66315e9	−1.45601	−0.728003	−	0.685574i	$-0.759552\pi$
$103$	−1.66315e9	−1.45601	−0.728003	+	0.685574i	$0.759552\pi$
$104$	1.86829e8	0.156601
$105$	0	0
$106$	−9.76619e8	−0.751361
$107$	1.55743e9	1.14864	0.574318	−	0.818632i	$-0.305267\pi$
$107$	1.55743e9	1.14864	0.574318	+	0.818632i	$0.305267\pi$
$108$	0	0
$109$	1.76222e9	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$109$	1.76222e9	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$110$	2.25151e9	1.46624
$111$	0	0
$112$	−2.99182e8	−0.179661
$113$	−1.82996e9	−1.05581	−0.527907	−	0.849302i	$-0.677023\pi$
$113$	−1.82996e9	−1.05581	−0.527907	+	0.849302i	$0.677023\pi$
$114$	0	0
$115$	3.77344e9	2.01186
$116$	2.04125e8	0.104673
$117$	0	0
$118$	−1.30850e9	−0.621303
$119$	6.51645e8	0.297886
$120$	0	0
$121$	2.52475e9	1.07074
$122$	−2.28502e9	−0.933836
$123$	0	0
$124$	2.28644e9	0.868485
$125$	3.00605e8	0.110129
$126$	0	0
$127$	−1.41081e9	−0.481228	−0.240614	−	0.970621i	$-0.577349\pi$
$127$	−1.41081e9	−0.481228	−0.240614	+	0.970621i	$0.577349\pi$
$128$	2.68435e8	0.0883883
$129$	0	0
$130$	1.46970e9	0.451319
$131$	−2.87804e9	−0.853840	−0.426920	−	0.904289i	$-0.640401\pi$
$131$	−2.87804e9	−0.853840	−0.426920	+	0.904289i	$0.640401\pi$
$132$	0	0
$133$	3.46626e9	0.960570
$134$	2.70100e9	0.723689
$135$	0	0
$136$	−5.84676e8	−0.146551
$137$	4.00471e9	0.971244	0.485622	−	0.874169i	$-0.338593\pi$
$137$	4.00471e9	0.971244	0.485622	+	0.874169i	$0.338593\pi$
$138$	0	0
$139$	1.85726e9	0.421994	0.210997	−	0.977487i	$-0.432329\pi$
$139$	1.85726e9	0.421994	0.210997	+	0.977487i	$0.432329\pi$
$140$	−2.35353e9	−0.517777
$141$	0	0
$142$	1.79506e9	0.370494
$143$	3.18724e9	0.637387
$144$	0	0
$145$	1.60576e9	0.301664
$146$	−4.93954e9	−0.899702
$147$	0	0
$148$	5.06839e9	0.868345
$149$	−6.94388e9	−1.15415	−0.577077	−	0.816689i	$-0.695807\pi$
$149$	−6.94388e9	−1.15415	−0.577077	+	0.816689i	$0.695807\pi$
$150$	0	0
$151$	−5.03872e9	−0.788722	−0.394361	−	0.918956i	$-0.629034\pi$
$151$	−5.03872e9	−0.788722	−0.394361	+	0.918956i	$0.629034\pi$
$152$	−3.11003e9	−0.472573
$153$	0	0
$154$	−5.10394e9	−0.731245
$155$	1.79864e10	2.50294
$156$	0	0
$157$	−4.86249e9	−0.638720	−0.319360	−	0.947634i	$-0.603468\pi$
$157$	−4.86249e9	−0.638720	−0.319360	+	0.947634i	$0.603468\pi$
$158$	−5.17625e9	−0.660783
$159$	0	0
$160$	2.11166e9	0.254732
$161$	−8.55403e9	−1.00335
$162$	0	0
$163$	−5.20600e9	−0.577644	−0.288822	−	0.957383i	$-0.593264\pi$
$163$	−5.20600e9	−0.577644	−0.288822	+	0.957383i	$0.593264\pi$
$164$	−4.91715e9	−0.530782
$165$	0	0
$166$	−8.54572e9	−0.873499
$167$	−1.04752e10	−1.04217	−0.521083	−	0.853506i	$-0.674472\pi$
$167$	−1.04752e10	−1.04217	−0.521083	+	0.853506i	$0.674472\pi$
$168$	0	0
$169$	−8.52399e9	−0.803808
$170$	−4.59937e9	−0.422355
$171$	0	0
$172$	3.53899e9	0.308320
$173$	5.71766e9	0.485300	0.242650	−	0.970114i	$-0.421983\pi$
$173$	5.71766e9	0.485300	0.242650	+	0.970114i	$0.421983\pi$
$174$	0	0
$175$	−9.59777e9	−0.773569
$176$	4.57941e9	0.359752
$177$	0	0
$178$	−1.27652e10	−0.953096
$179$	1.76705e10	1.28650	0.643249	−	0.765657i	$-0.277586\pi$
$179$	1.76705e10	1.28650	0.643249	+	0.765657i	$0.277586\pi$
$180$	0	0
$181$	2.45903e10	1.70298	0.851492	−	0.524368i	$-0.175698\pi$
$181$	2.45903e10	1.70298	0.851492	+	0.524368i	$0.175698\pi$
$182$	−3.33166e9	−0.225082
$183$	0	0
$184$	7.67493e9	0.493622
$185$	3.98707e10	2.50254
$186$	0	0
$187$	−9.97436e9	−0.596482
$188$	7.21431e9	0.421196
$189$	0	0
$190$	−2.44652e10	−1.36194
$191$	−2.46350e9	−0.133938	−0.0669688	−	0.997755i	$-0.521333\pi$
$191$	−2.46350e9	−0.133938	−0.0669688	+	0.997755i	$0.521333\pi$
$192$	0	0
$193$	−3.39182e10	−1.75964	−0.879821	−	0.475305i	$-0.842338\pi$
$193$	−3.39182e10	−1.75964	−0.879821	+	0.475305i	$0.842338\pi$
$194$	−1.03022e10	−0.522183
$195$	0	0
$196$	−4.99531e9	−0.241774
$197$	−4.87423e9	−0.230573	−0.115286	−	0.993332i	$-0.536779\pi$
$197$	−4.87423e9	−0.230573	−0.115286	+	0.993332i	$0.536779\pi$
$198$	0	0
$199$	3.53697e10	1.59879	0.799397	−	0.600804i	$-0.205153\pi$
$199$	3.53697e10	1.59879	0.799397	+	0.600804i	$0.205153\pi$
$200$	8.61141e9	0.380574
$201$	0	0
$202$	2.20884e10	0.933434
$203$	−3.64009e9	−0.150446
$204$	0	0
$205$	−3.86809e10	−1.52969
$206$	−2.66104e10	−1.02955
$207$	0	0
$208$	2.98927e9	0.110734
$209$	−5.30561e10	−1.92343
$210$	0	0
$211$	2.03928e9	0.0708282	0.0354141	−	0.999373i	$-0.488725\pi$
$211$	2.03928e9	0.0708282	0.0354141	+	0.999373i	$0.488725\pi$
$212$	−1.56259e10	−0.531293
$213$	0	0
$214$	2.49189e10	0.812208
$215$	2.78396e10	0.888565
$216$	0	0
$217$	−4.07733e10	−1.24827
$218$	2.81956e10	0.845526
$219$	0	0
$220$	3.60241e10	1.03679
$221$	−6.51089e9	−0.183601
$222$	0	0
$223$	6.34352e10	1.71774	0.858872	−	0.512190i	$-0.171166\pi$
$223$	6.34352e10	1.71774	0.858872	+	0.512190i	$0.171166\pi$
$224$	−4.78692e9	−0.127040
$225$	0	0
$226$	−2.92793e10	−0.746573
$227$	−3.11707e10	−0.779168	−0.389584	−	0.920991i	$-0.627381\pi$
$227$	−3.11707e10	−0.779168	−0.389584	+	0.920991i	$0.627381\pi$
$228$	0	0
$229$	6.25058e10	1.50197	0.750983	−	0.660321i	$-0.229580\pi$
$229$	6.25058e10	1.50197	0.750983	+	0.660321i	$0.229580\pi$
$230$	6.03751e10	1.42260
$231$	0	0
$232$	3.26600e9	0.0740150
$233$	−6.62666e10	−1.47297	−0.736483	−	0.676456i	$-0.763515\pi$
$233$	−6.62666e10	−1.47297	−0.736483	+	0.676456i	$0.763515\pi$
$234$	0	0
$235$	5.67516e10	1.21387
$236$	−2.09359e10	−0.439327
$237$	0	0
$238$	1.04263e10	0.210637
$239$	4.18958e10	0.830578	0.415289	−	0.909690i	$-0.363680\pi$
$239$	4.18958e10	0.830578	0.415289	+	0.909690i	$0.363680\pi$
$240$	0	0
$241$	4.82826e10	0.921963	0.460982	−	0.887410i	$-0.347497\pi$
$241$	4.82826e10	0.921963	0.460982	+	0.887410i	$0.347497\pi$
$242$	4.03960e10	0.757127
$243$	0	0
$244$	−3.65603e10	−0.660322
$245$	−3.92957e10	−0.696784
$246$	0	0
$247$	−3.46330e10	−0.592044
$248$	3.65831e10	0.614112
$249$	0	0
$250$	4.80967e9	0.0778727
$251$	3.68932e10	0.586699	0.293349	−	0.956005i	$-0.405230\pi$
$251$	3.68932e10	0.586699	0.293349	+	0.956005i	$0.405230\pi$
$252$	0	0
$253$	1.30932e11	2.00910
$254$	−2.25729e10	−0.340279
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	2.08061e10	0.297503	0.148751	−	0.988875i	$-0.452475\pi$
$257$	2.08061e10	0.297503	0.148751	+	0.988875i	$0.452475\pi$
$258$	0	0
$259$	−9.03829e10	−1.24807
$260$	2.35152e10	0.319131
$261$	0	0
$262$	−4.60487e10	−0.603756
$263$	−7.53338e10	−0.970933	−0.485467	−	0.874255i	$-0.661350\pi$
$263$	−7.53338e10	−0.970933	−0.485467	+	0.874255i	$0.661350\pi$
$264$	0	0
$265$	−1.22922e11	−1.53116
$266$	5.54602e10	0.679225
$267$	0	0
$268$	4.32159e10	0.511726
$269$	−1.50056e11	−1.74730	−0.873650	−	0.486554i	$-0.838254\pi$
$269$	−1.50056e11	−1.74730	−0.873650	+	0.486554i	$0.838254\pi$
$270$	0	0
$271$	−1.87348e10	−0.211003	−0.105501	−	0.994419i	$-0.533645\pi$
$271$	−1.87348e10	−0.211003	−0.105501	+	0.994419i	$0.533645\pi$
$272$	−9.35481e9	−0.103627
$273$	0	0
$274$	6.40753e10	0.686773
$275$	1.46908e11	1.54898
$276$	0	0
$277$	1.32857e11	1.35589	0.677945	−	0.735113i	$-0.262871\pi$
$277$	1.32857e11	1.35589	0.677945	+	0.735113i	$0.262871\pi$
$278$	2.97162e10	0.298395
$279$	0	0
$280$	−3.76564e10	−0.366124
$281$	−5.17745e7	−0.000495379 0	−0.000247689	−	1.00000i	$-0.500079\pi$
$281$	−5.17745e7	−0.000495379 0	−0.000247689		1.00000i	$0.500079\pi$
$282$	0	0
$283$	−1.22551e11	−1.13574	−0.567870	−	0.823118i	$-0.692233\pi$
$283$	−1.22551e11	−1.13574	−0.567870	+	0.823118i	$0.692233\pi$
$284$	2.87210e10	0.261979
$285$	0	0
$286$	5.09959e10	0.450700
$287$	8.76859e10	0.762889
$288$	0	0
$289$	−9.82123e10	−0.828182
$290$	2.56921e10	0.213309
$291$	0	0
$292$	−7.90326e10	−0.636185
$293$	−3.96138e10	−0.314009	−0.157005	−	0.987598i	$-0.550184\pi$
$293$	−3.96138e10	−0.314009	−0.157005	+	0.987598i	$0.550184\pi$
$294$	0	0
$295$	−1.64693e11	−1.26612
$296$	8.10943e10	0.614013
$297$	0	0
$298$	−1.11102e11	−0.816111
$299$	8.54672e10	0.618414
$300$	0	0
$301$	−6.31096e10	−0.443145
$302$	−8.06195e10	−0.557711
$303$	0	0
$304$	−4.97606e10	−0.334160
$305$	−2.87603e11	−1.90302
$306$	0	0
$307$	−3.98499e10	−0.256038	−0.128019	−	0.991772i	$-0.540862\pi$
$307$	−3.98499e10	−0.256038	−0.128019	+	0.991772i	$0.540862\pi$
$308$	−8.16631e10	−0.517068
$309$	0	0
$310$	2.87782e11	1.76985
$311$	−4.38217e10	−0.265624	−0.132812	−	0.991141i	$-0.542401\pi$
$311$	−4.38217e10	−0.265624	−0.132812	+	0.991141i	$0.542401\pi$
$312$	0	0
$313$	−9.44827e10	−0.556420	−0.278210	−	0.960520i	$-0.589741\pi$
$313$	−9.44827e10	−0.556420	−0.278210	+	0.960520i	$0.589741\pi$
$314$	−7.77998e10	−0.451643
$315$	0	0
$316$	−8.28201e10	−0.467244
$317$	−1.93959e11	−1.07881	−0.539404	−	0.842047i	$-0.681350\pi$
$317$	−1.93959e11	−1.07881	−0.539404	+	0.842047i	$0.681350\pi$
$318$	0	0
$319$	5.57168e10	0.301250
$320$	3.37865e10	0.180123
$321$	0	0
$322$	−1.36864e11	−0.709478
$323$	1.08383e11	0.554050
$324$	0	0
$325$	9.58958e10	0.476787
$326$	−8.32960e10	−0.408456
$327$	0	0
$328$	−7.86744e10	−0.375320
$329$	−1.28650e11	−0.605382
$330$	0	0
$331$	5.99617e10	0.274567	0.137283	−	0.990532i	$-0.456163\pi$
$331$	5.99617e10	0.274567	0.137283	+	0.990532i	$0.456163\pi$
$332$	−1.36732e11	−0.617657
$333$	0	0
$334$	−1.67603e11	−0.736923
$335$	3.39959e11	1.47477
$336$	0	0
$337$	−1.03965e11	−0.439091	−0.219545	−	0.975602i	$-0.570457\pi$
$337$	−1.03965e11	−0.439091	−0.219545	+	0.975602i	$0.570457\pi$
$338$	−1.36384e11	−0.568378
$339$	0	0
$340$	−7.35899e10	−0.298650
$341$	6.24094e11	2.49951
$342$	0	0
$343$	2.73300e11	1.06615
$344$	5.66238e10	0.218015
$345$	0	0
$346$	9.14825e10	0.343159
$347$	9.56959e10	0.354332	0.177166	−	0.984181i	$-0.443307\pi$
$347$	9.56959e10	0.354332	0.177166	+	0.984181i	$0.443307\pi$
$348$	0	0
$349$	3.20886e10	0.115781	0.0578903	−	0.998323i	$-0.481563\pi$
$349$	3.20886e10	0.115781	0.0578903	+	0.998323i	$0.481563\pi$
$350$	−1.53564e11	−0.546996
$351$	0	0
$352$	7.32706e10	0.254383
$353$	2.21363e11	0.758784	0.379392	−	0.925236i	$-0.376133\pi$
$353$	2.21363e11	0.758784	0.379392	+	0.925236i	$0.376133\pi$
$354$	0	0
$355$	2.25934e11	0.755014
$356$	−2.04243e11	−0.673941
$357$	0	0
$358$	2.82727e11	0.909692
$359$	2.52052e11	0.800874	0.400437	−	0.916324i	$-0.368858\pi$
$359$	2.52052e11	0.800874	0.400437	+	0.916324i	$0.368858\pi$
$360$	0	0
$361$	2.53827e11	0.786603
$362$	3.93445e11	1.20419
$363$	0	0
$364$	−5.33066e10	−0.159157
$365$	−6.21713e11	−1.83346
$366$	0	0
$367$	7.22547e10	0.207907	0.103953	−	0.994582i	$-0.466851\pi$
$367$	7.22547e10	0.207907	0.103953	+	0.994582i	$0.466851\pi$
$368$	1.22799e11	0.349043
$369$	0	0
$370$	6.37931e11	1.76956
$371$	2.78651e11	0.763622
$372$	0	0
$373$	4.78354e10	0.127956	0.0639778	−	0.997951i	$-0.479621\pi$
$373$	4.78354e10	0.127956	0.0639778	+	0.997951i	$0.479621\pi$
$374$	−1.59590e11	−0.421777
$375$	0	0
$376$	1.15429e11	0.297831
$377$	3.63698e10	0.0927268
$378$	0	0
$379$	5.03865e11	1.25440	0.627202	−	0.778857i	$-0.284200\pi$
$379$	5.03865e11	1.25440	0.627202	+	0.778857i	$0.284200\pi$
$380$	−3.91443e11	−0.963036
$381$	0	0
$382$	−3.94160e10	−0.0947082
$383$	5.78875e11	1.37464	0.687322	−	0.726353i	$-0.258786\pi$
$383$	5.78875e11	1.37464	0.687322	+	0.726353i	$0.258786\pi$
$384$	0	0
$385$	−6.42405e11	−1.49017
$386$	−5.42691e11	−1.24426
$387$	0	0
$388$	−1.64835e11	−0.369239
$389$	−4.81979e11	−1.06722	−0.533611	−	0.845730i	$-0.679165\pi$
$389$	−4.81979e11	−1.06722	−0.533611	+	0.845730i	$0.679165\pi$
$390$	0	0
$391$	−2.67467e11	−0.578727
$392$	−7.99249e10	−0.170960
$393$	0	0
$394$	−7.79876e10	−0.163039
$395$	−6.51507e11	−1.34658
$396$	0	0
$397$	5.02870e10	0.101601	0.0508006	−	0.998709i	$-0.483823\pi$
$397$	5.02870e10	0.101601	0.0508006	+	0.998709i	$0.483823\pi$
$398$	5.65915e11	1.13052
$399$	0	0
$400$	1.37783e11	0.269107
$401$	−6.10395e11	−1.17886	−0.589429	−	0.807821i	$-0.700647\pi$
$401$	−6.10395e11	−1.17886	−0.589429	+	0.807821i	$0.700647\pi$
$402$	0	0
$403$	4.07385e11	0.769365
$404$	3.53415e11	0.660038
$405$	0	0
$406$	−5.82414e10	−0.106381
$407$	1.38344e12	2.49911
$408$	0	0
$409$	7.45349e11	1.31706	0.658529	−	0.752556i	$-0.271179\pi$
$409$	7.45349e11	1.31706	0.658529	+	0.752556i	$0.271179\pi$
$410$	−6.18895e11	−1.08166
$411$	0	0
$412$	−4.25766e11	−0.728003
$413$	3.73343e11	0.631441
$414$	0	0
$415$	−1.07560e12	−1.78006
$416$	4.78283e10	0.0783006
$417$	0	0
$418$	−8.48897e11	−1.36007
$419$	−2.96284e10	−0.0469619	−0.0234809	−	0.999724i	$-0.507475\pi$
$419$	−2.96284e10	−0.0469619	−0.0234809	+	0.999724i	$0.507475\pi$
$420$	0	0
$421$	5.40091e11	0.837910	0.418955	−	0.908007i	$-0.362396\pi$
$421$	5.40091e11	0.837910	0.418955	+	0.908007i	$0.362396\pi$
$422$	3.26285e10	0.0500831
$423$	0	0
$424$	−2.50014e11	−0.375681
$425$	−3.00102e11	−0.446189
$426$	0	0
$427$	6.51967e11	0.949075
$428$	3.98703e11	0.574318
$429$	0	0
$430$	4.45433e11	0.628310
$431$	−5.69766e11	−0.795333	−0.397666	−	0.917530i	$-0.630180\pi$
$431$	−5.69766e11	−0.795333	−0.397666	+	0.917530i	$0.630180\pi$
$432$	0	0
$433$	−1.15909e12	−1.58460	−0.792301	−	0.610131i	$-0.791117\pi$
$433$	−1.15909e12	−1.58460	−0.792301	+	0.610131i	$0.791117\pi$
$434$	−6.52373e11	−0.882658
$435$	0	0
$436$	4.51129e11	0.597877
$437$	−1.42272e12	−1.86618
$438$	0	0
$439$	8.78535e10	0.112893	0.0564467	−	0.998406i	$-0.482023\pi$
$439$	8.78535e10	0.112893	0.0564467	+	0.998406i	$0.482023\pi$
$440$	5.76385e11	0.733122
$441$	0	0
$442$	−1.04174e11	−0.129826
$443$	1.97723e10	0.0243916	0.0121958	−	0.999926i	$-0.496118\pi$
$443$	1.97723e10	0.0243916	0.0121958	+	0.999926i	$0.496118\pi$
$444$	0	0
$445$	−1.60668e12	−1.94227
$446$	1.01496e12	1.21463
$447$	0	0
$448$	−7.65907e10	−0.0898307
$449$	−4.74948e11	−0.551490	−0.275745	−	0.961231i	$-0.588925\pi$
$449$	−4.74948e11	−0.551490	−0.275745	+	0.961231i	$0.588925\pi$
$450$	0	0
$451$	−1.34216e12	−1.52760
$452$	−4.68469e11	−0.527907
$453$	0	0
$454$	−4.98732e11	−0.550955
$455$	−4.19338e11	−0.458684
$456$	0	0
$457$	8.27814e11	0.887789	0.443895	−	0.896079i	$-0.353596\pi$
$457$	8.27814e11	0.887789	0.443895	+	0.896079i	$0.353596\pi$
$458$	1.00009e12	1.06205
$459$	0	0
$460$	9.66001e11	1.00593
$461$	4.26042e11	0.439337	0.219669	−	0.975575i	$-0.429502\pi$
$461$	4.26042e11	0.439337	0.219669	+	0.975575i	$0.429502\pi$
$462$	0	0
$463$	9.17345e10	0.0927723	0.0463861	−	0.998924i	$-0.485230\pi$
$463$	9.17345e10	0.0927723	0.0463861	+	0.998924i	$0.485230\pi$
$464$	5.22560e10	0.0523365
$465$	0	0
$466$	−1.06027e12	−1.04154
$467$	1.26314e12	1.22892	0.614462	−	0.788947i	$-0.289373\pi$
$467$	1.26314e12	1.22892	0.614462	+	0.788947i	$0.289373\pi$
$468$	0	0
$469$	−7.70655e11	−0.735499
$470$	9.08026e11	0.858337
$471$	0	0
$472$	−3.34975e11	−0.310651
$473$	9.65982e11	0.887347
$474$	0	0
$475$	−1.59632e12	−1.43879
$476$	1.66821e11	0.148943
$477$	0	0
$478$	6.70333e11	0.587307
$479$	−1.23760e12	−1.07416	−0.537082	−	0.843530i	$-0.680473\pi$
$479$	−1.23760e12	−1.07416	−0.537082	+	0.843530i	$0.680473\pi$
$480$	0	0
$481$	9.03058e11	0.769241
$482$	7.72521e11	0.651926
$483$	0	0
$484$	6.46335e11	0.535370
$485$	−1.29668e12	−1.06413
$486$	0	0
$487$	4.32311e11	0.348270	0.174135	−	0.984722i	$-0.444287\pi$
$487$	4.32311e11	0.348270	0.174135	+	0.984722i	$0.444287\pi$
$488$	−5.84965e11	−0.466918
$489$	0	0
$490$	−6.28732e11	−0.492701
$491$	1.63142e12	1.26678	0.633388	−	0.773835i	$-0.281664\pi$
$491$	1.63142e12	1.26678	0.633388	+	0.773835i	$0.281664\pi$
$492$	0	0
$493$	−1.13818e11	−0.0867761
$494$	−5.54128e11	−0.418639
$495$	0	0
$496$	5.85329e11	0.434243
$497$	−5.12171e11	−0.376540
$498$	0	0
$499$	−7.18532e11	−0.518793	−0.259396	−	0.965771i	$-0.583524\pi$
$499$	−7.18532e11	−0.518793	−0.259396	+	0.965771i	$0.583524\pi$
$500$	7.69548e10	0.0550643
$501$	0	0
$502$	5.90292e11	0.414859
$503$	−5.17308e11	−0.360324	−0.180162	−	0.983637i	$-0.557662\pi$
$503$	−5.17308e11	−0.360324	−0.180162	+	0.983637i	$0.557662\pi$
$504$	0	0
$505$	2.78015e12	1.90220
$506$	2.09490e12	1.42065
$507$	0	0
$508$	−3.61166e11	−0.240614
$509$	7.97889e11	0.526881	0.263441	−	0.964676i	$-0.415143\pi$
$509$	7.97889e11	0.526881	0.263441	+	0.964676i	$0.415143\pi$
$510$	0	0
$511$	1.40936e12	0.914384
$512$	6.87195e10	0.0441942
$513$	0	0
$514$	3.32897e11	0.210366
$515$	−3.34930e12	−2.09808
$516$	0	0
$517$	1.96918e12	1.21221
$518$	−1.44613e12	−0.882515
$519$	0	0
$520$	3.76243e11	0.225659
$521$	−8.24635e10	−0.0490334	−0.0245167	−	0.999699i	$-0.507805\pi$
$521$	−8.24635e10	−0.0490334	−0.0245167	+	0.999699i	$0.507805\pi$
$522$	0	0
$523$	−1.54815e12	−0.904805	−0.452402	−	0.891814i	$-0.649433\pi$
$523$	−1.54815e12	−0.904805	−0.452402	+	0.891814i	$0.649433\pi$
$524$	−7.36779e11	−0.426920
$525$	0	0
$526$	−1.20534e12	−0.686553
$527$	−1.27490e12	−0.719992
$528$	0	0
$529$	1.70983e12	0.949299
$530$	−1.96675e12	−1.08270
$531$	0	0
$532$	8.87363e11	0.480285
$533$	−8.76110e11	−0.470204
$534$	0	0
$535$	3.13641e12	1.65516
$536$	6.91455e11	0.361845
$537$	0	0
$538$	−2.40089e12	−1.23553
$539$	−1.36349e12	−0.695829
$540$	0	0
$541$	−6.21988e11	−0.312172	−0.156086	−	0.987743i	$-0.549888\pi$
$541$	−6.21988e11	−0.312172	−0.156086	+	0.987743i	$0.549888\pi$
$542$	−2.99757e11	−0.149201
$543$	0	0
$544$	−1.49677e11	−0.0732757
$545$	3.54882e12	1.72306
$546$	0	0
$547$	4.17951e11	0.199610	0.0998050	−	0.995007i	$-0.468178\pi$
$547$	4.17951e11	0.199610	0.0998050	+	0.995007i	$0.468178\pi$
$548$	1.02521e12	0.485622
$549$	0	0
$550$	2.35052e12	1.09530
$551$	−6.05426e11	−0.279820
$552$	0	0
$553$	1.47690e12	0.671566
$554$	2.12571e12	0.958758
$555$	0	0
$556$	4.75459e11	0.210997
$557$	1.53061e12	0.673776	0.336888	−	0.941545i	$-0.390626\pi$
$557$	1.53061e12	0.673776	0.336888	+	0.941545i	$0.390626\pi$
$558$	0	0
$559$	6.30557e11	0.273131
$560$	−6.02503e11	−0.258889
$561$	0	0
$562$	−8.28392e8	−0.000350286 0
$563$	3.69013e12	1.54794	0.773969	−	0.633223i	$-0.218269\pi$
$563$	3.69013e12	1.54794	0.773969	+	0.633223i	$0.218269\pi$
$564$	0	0
$565$	−3.68522e12	−1.52141
$566$	−1.96082e12	−0.803090
$567$	0	0
$568$	4.59535e11	0.185247
$569$	−7.45427e11	−0.298126	−0.149063	−	0.988828i	$-0.547626\pi$
$569$	−7.45427e11	−0.298126	−0.149063	+	0.988828i	$0.547626\pi$
$570$	0	0
$571$	3.49122e12	1.37440	0.687202	−	0.726467i	$-0.258839\pi$
$571$	3.49122e12	1.37440	0.687202	+	0.726467i	$0.258839\pi$
$572$	8.15934e11	0.318693
$573$	0	0
$574$	1.40297e12	0.539444
$575$	3.93939e12	1.50288
$576$	0	0
$577$	−4.10159e12	−1.54050	−0.770249	−	0.637743i	$-0.779868\pi$
$577$	−4.10159e12	−1.54050	−0.770249	+	0.637743i	$0.779868\pi$
$578$	−1.57140e12	−0.585613
$579$	0	0
$580$	4.11073e11	0.150832
$581$	2.43829e12	0.887753
$582$	0	0
$583$	−4.26516e12	−1.52907
$584$	−1.26452e12	−0.449851
$585$	0	0
$586$	−6.33821e11	−0.222038
$587$	−4.48627e11	−0.155960	−0.0779801	−	0.996955i	$-0.524847\pi$
$587$	−4.48627e11	−0.155960	−0.0779801	+	0.996955i	$0.524847\pi$
$588$	0	0
$589$	−6.78149e12	−2.32170
$590$	−2.63509e12	−0.895285
$591$	0	0
$592$	1.29751e12	0.434173
$593$	−8.10869e11	−0.269280	−0.134640	−	0.990895i	$-0.542988\pi$
$593$	−8.10869e11	−0.269280	−0.134640	+	0.990895i	$0.542988\pi$
$594$	0	0
$595$	1.31230e12	0.429248
$596$	−1.77763e12	−0.577077
$597$	0	0
$598$	1.36748e12	0.437285
$599$	−4.83382e12	−1.53416	−0.767078	−	0.641553i	$-0.778290\pi$
$599$	−4.83382e12	−1.53416	−0.767078	+	0.641553i	$0.778290\pi$
$600$	0	0
$601$	3.86625e12	1.20880	0.604401	−	0.796680i	$-0.293413\pi$
$601$	3.86625e12	1.20880	0.604401	+	0.796680i	$0.293413\pi$
$602$	−1.00975e12	−0.313351
$603$	0	0
$604$	−1.28991e12	−0.394361
$605$	5.08442e12	1.54292
$606$	0	0
$607$	2.90770e10	0.00869362	0.00434681	−	0.999991i	$-0.498616\pi$
$607$	2.90770e10	0.00869362	0.00434681	+	0.999991i	$0.498616\pi$
$608$	−7.96169e11	−0.236287
$609$	0	0
$610$	−4.60164e12	−1.34564
$611$	1.28541e12	0.373125
$612$	0	0
$613$	8.01432e11	0.229242	0.114621	−	0.993409i	$-0.463435\pi$
$613$	8.01432e11	0.229242	0.114621	+	0.993409i	$0.463435\pi$
$614$	−6.37599e11	−0.181046
$615$	0	0
$616$	−1.30661e12	−0.365622
$617$	1.95977e12	0.544404	0.272202	−	0.962240i	$-0.412248\pi$
$617$	1.95977e12	0.544404	0.272202	+	0.962240i	$0.412248\pi$
$618$	0	0
$619$	1.92323e12	0.526531	0.263266	−	0.964723i	$-0.415200\pi$
$619$	1.92323e12	0.526531	0.263266	+	0.964723i	$0.415200\pi$
$620$	4.60451e12	1.25147
$621$	0	0
$622$	−7.01148e11	−0.187825
$623$	3.64219e12	0.968649
$624$	0	0
$625$	−3.50087e12	−0.917733
$626$	−1.51172e12	−0.393448
$627$	0	0
$628$	−1.24480e12	−0.319360
$629$	−2.82608e12	−0.719875
$630$	0	0
$631$	2.04866e12	0.514444	0.257222	−	0.966352i	$-0.417193\pi$
$631$	2.04866e12	0.514444	0.257222	+	0.966352i	$0.417193\pi$
$632$	−1.32512e12	−0.330391
$633$	0	0
$634$	−3.10335e12	−0.762832
$635$	−2.84113e12	−0.693440
$636$	0	0
$637$	−8.90036e11	−0.214181
$638$	8.91468e11	0.213016
$639$	0	0
$640$	5.40584e11	0.127366
$641$	3.30653e12	0.773592	0.386796	−	0.922165i	$-0.373582\pi$
$641$	3.30653e12	0.773592	0.386796	+	0.922165i	$0.373582\pi$
$642$	0	0
$643$	−3.81079e12	−0.879155	−0.439577	−	0.898205i	$-0.644872\pi$
$643$	−3.81079e12	−0.879155	−0.439577	+	0.898205i	$0.644872\pi$
$644$	−2.18983e12	−0.501677
$645$	0	0
$646$	1.73412e12	0.391772
$647$	−5.14649e12	−1.15463	−0.577314	−	0.816522i	$-0.695899\pi$
$647$	−5.14649e12	−1.15463	−0.577314	+	0.816522i	$0.695899\pi$
$648$	0	0
$649$	−5.71455e12	−1.26439
$650$	1.53433e12	0.337139
$651$	0	0
$652$	−1.33274e12	−0.288822
$653$	3.24645e12	0.698715	0.349357	−	0.936990i	$-0.386400\pi$
$653$	3.24645e12	0.698715	0.349357	+	0.936990i	$0.386400\pi$
$654$	0	0
$655$	−5.79589e12	−1.23037
$656$	−1.25879e12	−0.265391
$657$	0	0
$658$	−2.05841e12	−0.428070
$659$	−3.28645e12	−0.678801	−0.339400	−	0.940642i	$-0.610224\pi$
$659$	−3.28645e12	−0.678801	−0.339400	+	0.940642i	$0.610224\pi$
$660$	0	0
$661$	−6.21697e11	−0.126670	−0.0633348	−	0.997992i	$-0.520174\pi$
$661$	−6.21697e11	−0.126670	−0.0633348	+	0.997992i	$0.520174\pi$
$662$	9.59387e11	0.194148
$663$	0	0
$664$	−2.18770e12	−0.436749
$665$	6.98047e12	1.38416
$666$	0	0
$667$	1.49407e12	0.292283
$668$	−2.68165e12	−0.521083
$669$	0	0
$670$	5.43935e12	1.04282
$671$	−9.97929e12	−1.90041
$672$	0	0
$673$	9.46448e12	1.77840	0.889199	−	0.457520i	$-0.151262\pi$
$673$	9.46448e12	1.77840	0.889199	+	0.457520i	$0.151262\pi$
$674$	−1.66345e12	−0.310484
$675$	0	0
$676$	−2.18214e12	−0.401904
$677$	−3.73730e12	−0.683768	−0.341884	−	0.939742i	$-0.611065\pi$
$677$	−3.73730e12	−0.683768	−0.341884	+	0.939742i	$0.611065\pi$
$678$	0	0
$679$	2.93945e12	0.530705
$680$	−1.17744e12	−0.211178
$681$	0	0
$682$	9.98550e12	1.76742
$683$	−1.24743e12	−0.219343	−0.109671	−	0.993968i	$-0.534980\pi$
$683$	−1.24743e12	−0.219343	−0.109671	+	0.993968i	$0.534980\pi$
$684$	0	0
$685$	8.06481e12	1.39954
$686$	4.37280e12	0.753879
$687$	0	0
$688$	9.05981e11	0.154160
$689$	−2.78414e12	−0.470656
$690$	0	0
$691$	−2.65796e12	−0.443504	−0.221752	−	0.975103i	$-0.571178\pi$
$691$	−2.65796e12	−0.443504	−0.221752	+	0.975103i	$0.571178\pi$
$692$	1.46372e12	0.242650
$693$	0	0
$694$	1.53113e12	0.250551
$695$	3.74022e12	0.608086
$696$	0	0
$697$	2.74175e12	0.440029
$698$	5.13417e11	0.0818693
$699$	0	0
$700$	−2.45703e12	−0.386785
$701$	−3.12836e12	−0.489311	−0.244656	−	0.969610i	$-0.578675\pi$
$701$	−3.12836e12	−0.489311	−0.244656	+	0.969610i	$0.578675\pi$
$702$	0	0
$703$	−1.50326e13	−2.32133
$704$	1.17233e12	0.179876
$705$	0	0
$706$	3.54180e12	0.536541
$707$	−6.30233e12	−0.948666
$708$	0	0
$709$	1.08160e13	1.60753	0.803763	−	0.594950i	$-0.202828\pi$
$709$	1.08160e13	1.60753	0.803763	+	0.594950i	$0.202828\pi$
$710$	3.61495e12	0.533875
$711$	0	0
$712$	−3.26788e12	−0.476548
$713$	1.67353e13	2.42511
$714$	0	0
$715$	6.41857e12	0.918462
$716$	4.52364e12	0.643249
$717$	0	0
$718$	4.03282e12	0.566304
$719$	−5.06139e10	−0.00706300	−0.00353150	−	0.999994i	$-0.501124\pi$
$719$	−5.06139e10	−0.00706300	−0.00353150	+	0.999994i	$0.501124\pi$
$720$	0	0
$721$	7.59254e12	1.04635
$722$	4.06123e12	0.556212
$723$	0	0
$724$	6.29512e12	0.851492
$725$	1.67637e12	0.225346
$726$	0	0
$727$	3.34338e12	0.443895	0.221948	−	0.975059i	$-0.428759\pi$
$727$	3.34338e12	0.443895	0.221948	+	0.975059i	$0.428759\pi$
$728$	−8.52906e11	−0.112541
$729$	0	0
$730$	−9.94741e12	−1.29645
$731$	−1.97330e12	−0.255603
$732$	0	0
$733$	−4.16161e12	−0.532468	−0.266234	−	0.963908i	$-0.585779\pi$
$733$	−4.16161e12	−0.532468	−0.266234	+	0.963908i	$0.585779\pi$
$734$	1.15607e12	0.147012
$735$	0	0
$736$	1.96478e12	0.246811
$737$	1.17960e13	1.47275
$738$	0	0
$739$	9.32557e12	1.15020	0.575102	−	0.818081i	$-0.304962\pi$
$739$	9.32557e12	1.15020	0.575102	+	0.818081i	$0.304962\pi$
$740$	1.02069e13	1.25127
$741$	0	0
$742$	4.45842e12	0.539963
$743$	−8.93818e12	−1.07597	−0.537984	−	0.842955i	$-0.680814\pi$
$743$	−8.93818e12	−1.07597	−0.537984	+	0.842955i	$0.680814\pi$
$744$	0	0
$745$	−1.39838e13	−1.66311
$746$	7.65366e11	0.0904783
$747$	0	0
$748$	−2.55344e12	−0.298241
$749$	−7.10993e12	−0.825462
$750$	0	0
$751$	3.61027e12	0.414152	0.207076	−	0.978325i	$-0.433605\pi$
$751$	3.61027e12	0.414152	0.207076	+	0.978325i	$0.433605\pi$
$752$	1.84686e12	0.210598
$753$	0	0
$754$	5.81917e11	0.0655677
$755$	−1.01471e13	−1.13653
$756$	0	0
$757$	1.48103e13	1.63920	0.819599	−	0.572938i	$-0.194196\pi$
$757$	1.48103e13	1.63920	0.819599	+	0.572938i	$0.194196\pi$
$758$	8.06184e12	0.886998
$759$	0	0
$760$	−6.26309e12	−0.680969
$761$	−1.70198e13	−1.83960	−0.919798	−	0.392391i	$-0.871648\pi$
$761$	−1.70198e13	−1.83960	−0.919798	+	0.392391i	$0.871648\pi$
$762$	0	0
$763$	−8.04483e12	−0.859323
$764$	−6.30656e11	−0.0669688
$765$	0	0
$766$	9.26200e12	0.972020
$767$	−3.73025e12	−0.389187
$768$	0	0
$769$	−1.19791e13	−1.23526	−0.617629	−	0.786470i	$-0.711906\pi$
$769$	−1.19791e13	−1.23526	−0.617629	+	0.786470i	$0.711906\pi$
$770$	−1.02785e13	−1.05371
$771$	0	0
$772$	−8.68305e12	−0.879821
$773$	−7.00938e12	−0.706110	−0.353055	−	0.935603i	$-0.614857\pi$
$773$	−7.00938e12	−0.706110	−0.353055	+	0.935603i	$0.614857\pi$
$774$	0	0
$775$	1.87774e13	1.86972
$776$	−2.63737e12	−0.261092
$777$	0	0
$778$	−7.71167e12	−0.754640
$779$	1.45841e13	1.41893
$780$	0	0
$781$	7.83951e12	0.753979
$782$	−4.27947e12	−0.409222
$783$	0	0
$784$	−1.27880e12	−0.120887
$785$	−9.79224e12	−0.920383
$786$	0	0
$787$	−7.91010e12	−0.735014	−0.367507	−	0.930021i	$-0.619789\pi$
$787$	−7.91010e12	−0.735014	−0.367507	+	0.930021i	$0.619789\pi$
$788$	−1.24780e12	−0.115286
$789$	0	0
$790$	−1.04241e13	−0.952175
$791$	8.35404e12	0.758756
$792$	0	0
$793$	−6.51411e12	−0.584959
$794$	8.04593e11	0.0718429
$795$	0	0
$796$	9.05464e12	0.799397
$797$	7.16965e12	0.629412	0.314706	−	0.949189i	$-0.398094\pi$
$797$	7.16965e12	0.629412	0.314706	+	0.949189i	$0.398094\pi$
$798$	0	0
$799$	−4.02263e12	−0.349180
$800$	2.20452e12	0.190287
$801$	0	0
$802$	−9.76632e12	−0.833578
$803$	−2.15723e13	−1.83095
$804$	0	0
$805$	−1.72264e13	−1.44581
$806$	6.51816e12	0.544023
$807$	0	0
$808$	5.65464e12	0.466717
$809$	1.58279e13	1.29914	0.649568	−	0.760304i	$-0.274950\pi$
$809$	1.58279e13	1.29914	0.649568	+	0.760304i	$0.274950\pi$
$810$	0	0
$811$	−9.43072e11	−0.0765510	−0.0382755	−	0.999267i	$-0.512186\pi$
$811$	−9.43072e11	−0.0765510	−0.0382755	+	0.999267i	$0.512186\pi$
$812$	−9.31863e11	−0.0752229
$813$	0	0
$814$	2.21350e13	1.76714
$815$	−1.04840e13	−0.832374
$816$	0	0
$817$	−1.04965e13	−0.824224
$818$	1.19256e13	0.931300
$819$	0	0
$820$	−9.90232e12	−0.764847
$821$	1.91719e13	1.47273	0.736363	−	0.676587i	$-0.236542\pi$
$821$	1.91719e13	1.47273	0.736363	+	0.676587i	$0.236542\pi$
$822$	0	0
$823$	−9.33830e12	−0.709527	−0.354763	−	0.934956i	$-0.615439\pi$
$823$	−9.33830e12	−0.709527	−0.354763	+	0.934956i	$0.615439\pi$
$824$	−6.81225e12	−0.514776
$825$	0	0
$826$	5.97349e12	0.446496
$827$	−1.08669e13	−0.807854	−0.403927	−	0.914791i	$-0.632355\pi$
$827$	−1.08669e13	−0.807854	−0.403927	+	0.914791i	$0.632355\pi$
$828$	0	0
$829$	−3.98687e12	−0.293181	−0.146591	−	0.989197i	$-0.546830\pi$
$829$	−3.98687e12	−0.293181	−0.146591	+	0.989197i	$0.546830\pi$
$830$	−1.72096e13	−1.25869
$831$	0	0
$832$	7.65253e11	0.0553669
$833$	2.78533e12	0.200436
$834$	0	0
$835$	−2.10953e13	−1.50174
$836$	−1.35824e13	−0.961716
$837$	0	0
$838$	−4.74055e11	−0.0332071
$839$	1.87689e13	1.30770	0.653852	−	0.756622i	$-0.273152\pi$
$839$	1.87689e13	1.30770	0.653852	+	0.756622i	$0.273152\pi$
$840$	0	0
$841$	−1.38714e13	−0.956174
$842$	8.64146e12	0.592492
$843$	0	0
$844$	5.22056e11	0.0354141
$845$	−1.71659e13	−1.15827
$846$	0	0
$847$	−1.15259e13	−0.769482
$848$	−4.00023e12	−0.265646
$849$	0	0
$850$	−4.80164e12	−0.315503
$851$	3.70975e13	2.42472
$852$	0	0
$853$	−1.34814e12	−0.0871893	−0.0435947	−	0.999049i	$-0.513881\pi$
$853$	−1.34814e12	−0.0871893	−0.0435947	+	0.999049i	$0.513881\pi$
$854$	1.04315e13	0.671097
$855$	0	0
$856$	6.37924e12	0.406104
$857$	5.07330e11	0.0321275	0.0160637	−	0.999871i	$-0.494887\pi$
$857$	5.07330e11	0.0321275	0.0160637	+	0.999871i	$0.494887\pi$
$858$	0	0
$859$	2.87459e13	1.80139	0.900693	−	0.434456i	$-0.143059\pi$
$859$	2.87459e13	1.80139	0.900693	+	0.434456i	$0.143059\pi$
$860$	7.12693e12	0.444283
$861$	0	0
$862$	−9.11626e12	−0.562385
$863$	−3.00660e11	−0.0184513	−0.00922566	−	0.999957i	$-0.502937\pi$
$863$	−3.00660e11	−0.0184513	−0.00922566	+	0.999957i	$0.502937\pi$
$864$	0	0
$865$	1.15144e13	0.699309
$866$	−1.85454e13	−1.12048
$867$	0	0
$868$	−1.04380e13	−0.624133
$869$	−2.26061e13	−1.34473
$870$	0	0
$871$	7.69997e12	0.453323
$872$	7.21807e12	0.422763
$873$	0	0
$874$	−2.27635e13	−1.31959
$875$	−1.37231e12	−0.0791435
$876$	0	0
$877$	9.08872e12	0.518806	0.259403	−	0.965769i	$-0.416474\pi$
$877$	9.08872e12	0.518806	0.259403	+	0.965769i	$0.416474\pi$
$878$	1.40566e12	0.0798277
$879$	0	0
$880$	9.22217e12	0.518395
$881$	−1.67414e12	−0.0936266	−0.0468133	−	0.998904i	$-0.514907\pi$
$881$	−1.67414e12	−0.0936266	−0.0468133	+	0.998904i	$0.514907\pi$
$882$	0	0
$883$	−2.69670e13	−1.49283	−0.746413	−	0.665483i	$-0.768226\pi$
$883$	−2.69670e13	−1.49283	−0.746413	+	0.665483i	$0.768226\pi$
$884$	−1.66679e12	−0.0918005
$885$	0	0
$886$	3.16357e11	0.0172475
$887$	3.31395e13	1.79759	0.898793	−	0.438373i	$-0.144445\pi$
$887$	3.31395e13	1.79759	0.898793	+	0.438373i	$0.144445\pi$
$888$	0	0
$889$	6.44056e12	0.345832
$890$	−2.57069e13	−1.37339
$891$	0	0
$892$	1.62394e13	0.858872
$893$	−2.13974e13	−1.12597
$894$	0	0
$895$	3.55853e13	1.85382
$896$	−1.22545e12	−0.0635199
$897$	0	0
$898$	−7.59917e12	−0.389963
$899$	7.12158e12	0.363628
$900$	0	0
$901$	8.71285e12	0.440452
$902$	−2.14745e13	−1.08017
$903$	0	0
$904$	−7.49550e12	−0.373287
$905$	4.95207e13	2.45397
$906$	0	0
$907$	2.80197e13	1.37477	0.687386	−	0.726292i	$-0.258758\pi$
$907$	2.80197e13	1.37477	0.687386	+	0.726292i	$0.258758\pi$
$908$	−7.97971e12	−0.389584
$909$	0	0
$910$	−6.70941e12	−0.324338
$911$	2.45209e13	1.17952	0.589758	−	0.807580i	$-0.299223\pi$
$911$	2.45209e13	1.17952	0.589758	+	0.807580i	$0.299223\pi$
$912$	0	0
$913$	−3.73214e13	−1.77762
$914$	1.32450e13	0.627762
$915$	0	0
$916$	1.60015e13	0.750983
$917$	1.31387e13	0.613608
$918$	0	0
$919$	2.79427e13	1.29226	0.646128	−	0.763229i	$-0.276387\pi$
$919$	2.79427e13	1.29226	0.646128	+	0.763229i	$0.276387\pi$
$920$	1.54560e13	0.711299
$921$	0	0
$922$	6.81667e12	0.310658
$923$	5.11734e12	0.232079
$924$	0	0
$925$	4.16241e13	1.86942
$926$	1.46775e12	0.0655999
$927$	0	0
$928$	8.36096e11	0.0370075
$929$	−1.17529e12	−0.0517697	−0.0258848	−	0.999665i	$-0.508240\pi$
$929$	−1.17529e12	−0.0517697	−0.0258848	+	0.999665i	$0.508240\pi$
$930$	0	0
$931$	1.48159e13	0.646329
$932$	−1.69642e13	−0.736483
$933$	0	0
$934$	2.02102e13	0.868980
$935$	−2.00867e13	−0.859520
$936$	0	0
$937$	−3.99627e12	−0.169366	−0.0846830	−	0.996408i	$-0.526988\pi$
$937$	−3.99627e12	−0.169366	−0.0846830	+	0.996408i	$0.526988\pi$
$938$	−1.23305e13	−0.520076
$939$	0	0
$940$	1.45284e13	0.606936
$941$	−7.08647e12	−0.294630	−0.147315	−	0.989090i	$-0.547063\pi$
$941$	−7.08647e12	−0.294630	−0.147315	+	0.989090i	$0.547063\pi$
$942$	0	0
$943$	−3.59905e13	−1.48213
$944$	−5.35960e12	−0.219664
$945$	0	0
$946$	1.54557e13	0.627449
$947$	−4.28949e13	−1.73313	−0.866564	−	0.499066i	$-0.833676\pi$
$947$	−4.28949e13	−1.73313	−0.866564	+	0.499066i	$0.833676\pi$
$948$	0	0
$949$	−1.40816e13	−0.563578
$950$	−2.55411e13	−1.01738
$951$	0	0
$952$	2.66914e12	0.105319
$953$	1.57454e13	0.618350	0.309175	−	0.951005i	$-0.399947\pi$
$953$	1.57454e13	0.618350	0.309175	+	0.951005i	$0.399947\pi$
$954$	0	0
$955$	−4.96108e12	−0.193002
$956$	1.07253e13	0.415289
$957$	0	0
$958$	−1.98016e13	−0.759548
$959$	−1.82821e13	−0.697980
$960$	0	0
$961$	5.33305e13	2.01707
$962$	1.44489e13	0.543936
$963$	0	0
$964$	1.23603e13	0.460982
$965$	−6.83055e13	−2.53561
$966$	0	0
$967$	9.01621e12	0.331593	0.165796	−	0.986160i	$-0.446981\pi$
$967$	9.01621e12	0.331593	0.165796	+	0.986160i	$0.446981\pi$
$968$	1.03414e13	0.378564
$969$	0	0
$970$	−2.07469e13	−0.752456
$971$	−1.47845e13	−0.533730	−0.266865	−	0.963734i	$-0.585988\pi$
$971$	−1.47845e13	−0.533730	−0.266865	+	0.963734i	$0.585988\pi$
$972$	0	0
$973$	−8.47870e12	−0.303265
$974$	6.91698e12	0.246264
$975$	0	0
$976$	−9.35944e12	−0.330161
$977$	−2.19982e13	−0.772435	−0.386217	−	0.922408i	$-0.626219\pi$
$977$	−2.19982e13	−0.772435	−0.386217	+	0.922408i	$0.626219\pi$
$978$	0	0
$979$	−5.57489e13	−1.93961
$980$	−1.00597e13	−0.348392
$981$	0	0
$982$	2.61028e13	0.895746
$983$	3.06326e13	1.04639	0.523194	−	0.852214i	$-0.324740\pi$
$983$	3.06326e13	1.04639	0.523194	+	0.852214i	$0.324740\pi$
$984$	0	0
$985$	−9.81587e12	−0.332251
$986$	−1.82109e12	−0.0613599
$987$	0	0
$988$	−8.86606e12	−0.296022
$989$	2.59032e13	0.860935
$990$	0	0
$991$	−1.39052e13	−0.457978	−0.228989	−	0.973429i	$-0.573542\pi$
$991$	−1.39052e13	−0.457978	−0.228989	+	0.973429i	$0.573542\pi$
$992$	9.36526e12	0.307056
$993$	0	0
$994$	−8.19474e12	−0.266254
$995$	7.12286e13	2.30383
$996$	0	0
$997$	1.22601e13	0.392975	0.196488	−	0.980506i	$-0.437046\pi$
$997$	1.22601e13	0.392975	0.196488	+	0.980506i	$0.437046\pi$
$998$	−1.14965e13	−0.366842
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.10.a.g.1.2	yes	2
3.2	odd	2		54.10.a.f.1.1	✓	2
9.2	odd	6		162.10.c.o.109.2		4
9.4	even	3		162.10.c.l.55.1		4
9.5	odd	6		162.10.c.o.55.2		4
9.7	even	3		162.10.c.l.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.10.a.f.1.1	✓	2	3.2	odd	2
54.10.a.g.1.2	yes	2	1.1	even	1	trivial
162.10.c.l.55.1		4	9.4	even	3
162.10.c.l.109.1		4	9.7	even	3
162.10.c.o.55.2		4	9.5	odd	6
162.10.c.o.109.2		4	9.2	odd	6

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

\(f(q)\)	\(=\)	\(q+16.0000 q^{2} +256.000 q^{4} +2013.83 q^{5} -4565.16 q^{7} +4096.00 q^{8} +32221.3 q^{10} +69876.3 q^{11} +45612.6 q^{13} -73042.6 q^{14} +65536.0 q^{16} -142743. q^{17} -759286. q^{19} +515541. q^{20} +1.11802e6 q^{22} +1.87376e6 q^{23} +2.10239e6 q^{25} +729802. q^{26} -1.16868e6 q^{28} +797363. q^{29} +8.93141e6 q^{31} +1.04858e6 q^{32} -2.28389e6 q^{34} -9.19347e6 q^{35} +1.97984e7 q^{37} -1.21486e7 q^{38} +8.24866e6 q^{40} -1.92076e7 q^{41} +1.38242e7 q^{43} +1.78883e7 q^{44} +2.99802e7 q^{46} +2.81809e7 q^{47} -1.95129e7 q^{49} +3.36383e7 q^{50} +1.16768e7 q^{52} -6.10387e7 q^{53} +1.40719e8 q^{55} -1.86989e7 q^{56} +1.27578e7 q^{58} -8.17810e7 q^{59} -1.42814e8 q^{61} +1.42903e8 q^{62} +1.67772e7 q^{64} +9.18562e7 q^{65} +1.68812e8 q^{67} -3.65422e7 q^{68} -1.47095e8 q^{70} +1.12191e8 q^{71} -3.08721e8 q^{73} +3.16774e8 q^{74} -1.94377e8 q^{76} -3.18996e8 q^{77} -3.23516e8 q^{79} +1.31979e8 q^{80} -3.07322e8 q^{82} -5.34107e8 q^{83} -2.87461e8 q^{85} +2.21187e8 q^{86} +2.86213e8 q^{88} -7.97823e8 q^{89} -2.08229e8 q^{91} +4.79683e8 q^{92} +4.50894e8 q^{94} -1.52907e9 q^{95} -6.43889e8 q^{97} -3.12207e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{2} + 512 q^{4} + 912 q^{5} + 6448 q^{7} + 8192 q^{8} + 14592 q^{10} + 15126 q^{11} + 28912 q^{13} + 103168 q^{14} + 131072 q^{16} + 399960 q^{17} + 8104 q^{19} + 233472 q^{20} + 242016 q^{22}+ \cdots + 982771008 q^{98}+O(q^{100}) \) 2 * q + 32 * q^2 + 512 * q^4 + 912 * q^5 + 6448 * q^7 + 8192 * q^8 + 14592 * q^10 + 15126 * q^11 + 28912 * q^13 + 103168 * q^14 + 131072 * q^16 + 399960 * q^17 + 8104 * q^19 + 233472 * q^20 + 242016 * q^22 + 2563572 * q^23 + 1363304 * q^25 + 462592 * q^26 + 1650688 * q^28 + 3059088 * q^29 + 16787920 * q^31 + 2097152 * q^32 + 6399360 * q^34 - 21328122 * q^35 + 34798684 * q^37 + 129664 * q^38 + 3735552 * q^40 - 24238968 * q^41 + 12599680 * q^43 + 3872256 * q^44 + 41017152 * q^46 - 19535772 * q^47 + 61423188 * q^49 + 21812864 * q^50 + 7401472 * q^52 + 12534912 * q^53 + 201044736 * q^55 + 26411008 * q^56 + 48945408 * q^58 - 140256744 * q^59 - 169724624 * q^61 + 268606720 * q^62 + 33554432 * q^64 + 110257512 * q^65 - 72241232 * q^67 + 102389760 * q^68 - 341249952 * q^70 - 53846280 * q^71 - 203557694 * q^73 + 556778944 * q^74 + 2074624 * q^76 - 921970176 * q^77 - 877964864 * q^79 + 59768832 * q^80 - 387823488 * q^82 - 111456714 * q^83 - 885428280 * q^85 + 201594880 * q^86 + 61956096 * q^88 - 1234260864 * q^89 - 392155912 * q^91 + 656274432 * q^92 - 312572352 * q^94 - 2374608756 * q^95 - 165265682 * q^97 + 982771008 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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