LMFDB - Embedded newform 54.16.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [54,16,Mod(1,54)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("54.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	54.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,256,0,32768,-71472] 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:	$77.0544927540$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5146}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 5146 $ x^2 - 5146
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$71.7356$ 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)

$f(q)$	$=$	\(q+128.000 q^{2} +16384.0 q^{4} +142455. q^{5} -882046. q^{7} +2.09715e6 q^{8} +1.82343e7 q^{10} -1.32259e7 q^{11} -3.16589e8 q^{13} -1.12902e8 q^{14} +2.68435e8 q^{16} +8.98145e8 q^{17} -3.58030e9 q^{19} +2.33399e9 q^{20} -1.69292e9 q^{22} -2.48401e10 q^{23} -1.02241e10 q^{25} -4.05234e10 q^{26} -1.44514e10 q^{28} +8.44737e10 q^{29} -9.50671e8 q^{31} +3.43597e10 q^{32} +1.14963e11 q^{34} -1.25652e11 q^{35} -1.42782e11 q^{37} -4.58278e11 q^{38} +2.98750e11 q^{40} +1.47061e12 q^{41} -1.58998e12 q^{43} -2.16693e11 q^{44} -3.17954e12 q^{46} +2.03678e11 q^{47} -3.96956e12 q^{49} -1.30868e12 q^{50} -5.18699e12 q^{52} -2.22358e12 q^{53} -1.88410e12 q^{55} -1.84978e12 q^{56} +1.08126e13 q^{58} -6.14211e12 q^{59} -2.73906e13 q^{61} -1.21686e11 q^{62} +4.39805e12 q^{64} -4.50997e13 q^{65} +4.97262e13 q^{67} +1.47152e13 q^{68} -1.60835e13 q^{70} -7.41880e13 q^{71} -4.47897e13 q^{73} -1.82761e13 q^{74} -5.86596e13 q^{76} +1.16659e13 q^{77} +5.33619e13 q^{79} +3.82401e13 q^{80} +1.88238e14 q^{82} -2.22672e14 q^{83} +1.27945e14 q^{85} -2.03518e14 q^{86} -2.77367e13 q^{88} -3.77610e14 q^{89} +2.79246e14 q^{91} -4.06981e14 q^{92} +2.60708e13 q^{94} -5.10033e14 q^{95} +1.25521e15 q^{97} -5.08103e14 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 256 q^{2} + 32768 q^{4} - 71472 q^{5} - 896378 q^{7} + 4194304 q^{8} - 9148416 q^{10} + 21628848 q^{11} - 151348262 q^{13} - 114736384 q^{14} + 536870912 q^{16} - 71108496 q^{17} + 1418861578 q^{19}+ \cdots - 11\!\cdots\!28 q^{98}+O(q^{100}) $ 2 * q + 256 * q^2 + 32768 * q^4 - 71472 * q^5 - 896378 * q^7 + 4194304 * q^8 - 9148416 * q^10 + 21628848 * q^11 - 151348262 * q^13 - 114736384 * q^14 + 536870912 * q^16 - 71108496 * q^17 + 1418861578 * q^19 - 1170997248 * q^20 + 2768492544 * q^22 - 22216171152 * q^23 + 5023241894 * q^25 - 19372577536 * q^26 - 14686257152 * q^28 - 69636203520 * q^29 - 187701744632 * q^31 + 68719476736 * q^32 - 9101887488 * q^34 - 122586139536 * q^35 - 721738522490 * q^37 + 181614281984 * q^38 - 149887647744 * q^40 + 1134127875840 * q^41 - 735115869968 * q^43 + 354367045632 * q^44 - 2843669907456 * q^46 - 1143668814288 * q^47 - 8716912355676 * q^49 + 642974962432 * q^50 - 2479689924608 * q^52 - 7010634953376 * q^53 - 9340483492800 * q^55 - 1879840915456 * q^56 - 8913434050560 * q^58 - 4464695524944 * q^59 - 25346208518402 * q^61 - 24025823312896 * q^62 + 8796093022208 * q^64 - 80449197973872 * q^65 - 36471044633486 * q^67 - 1165041598464 * q^68 - 15691025860608 * q^70 - 32664609209376 * q^71 + 57989177090770 * q^73 - 92382530878720 * q^74 + 23246628093952 * q^76 + 11166322109712 * q^77 + 278861646180442 * q^79 - 19185618911232 * q^80 + 145168368107520 * q^82 - 493886855787936 * q^83 + 335295249747264 * q^85 - 94094831355904 * q^86 + 45358981840896 * q^88 - 1075161917501136 * q^89 + 276877654459406 * q^91 - 363989748154368 * q^92 - 146389608228864 * q^94 - 1579489781398512 * q^95 + 287158074477898 * q^97 - 1115764781526528 * 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 $

$2$	128.000	0.707107
$2$	128.000	0.707107
$3$	0	0
$3$	0	0
$4$	16384.0	0.500000
$5$	142455.	0.815462	0.407731	−	0.913102i	$-0.366320\pi$
$5$	142455.	0.815462	0.407731	+	0.913102i	$0.366320\pi$
$6$	0	0
$7$	−882046.	−0.404814	−0.202407	−	0.979301i	$-0.564876\pi$
$7$	−882046.	−0.404814	−0.202407	+	0.979301i	$0.564876\pi$
$8$	2.09715e6	0.353553
$9$	0	0
$10$	1.82343e7	0.576618
$11$	−1.32259e7	−0.204635	−0.102318	−	0.994752i	$-0.532626\pi$
$11$	−1.32259e7	−0.204635	−0.102318	+	0.994752i	$0.532626\pi$
$12$	0	0
$13$	−3.16589e8	−1.39933	−0.699665	−	0.714471i	$-0.746668\pi$
$13$	−3.16589e8	−1.39933	−0.699665	+	0.714471i	$0.746668\pi$
$14$	−1.12902e8	−0.286247
$15$	0	0
$16$	2.68435e8	0.250000
$17$	8.98145e8	0.530859	0.265430	−	0.964130i	$-0.414486\pi$
$17$	8.98145e8	0.530859	0.265430	+	0.964130i	$0.414486\pi$
$18$	0	0
$19$	−3.58030e9	−0.918898	−0.459449	−	0.888204i	$-0.651953\pi$
$19$	−3.58030e9	−0.918898	−0.459449	+	0.888204i	$0.651953\pi$
$20$	2.33399e9	0.407731
$21$	0	0
$22$	−1.69292e9	−0.144699
$23$	−2.48401e10	−1.52123	−0.760615	−	0.649203i	$-0.775102\pi$
$23$	−2.48401e10	−1.52123	−0.760615	+	0.649203i	$0.775102\pi$
$24$	0	0
$25$	−1.02241e10	−0.335022
$26$	−4.05234e10	−0.989476
$27$	0	0
$28$	−1.44514e10	−0.202407
$29$	8.44737e10	0.909362	0.454681	−	0.890654i	$-0.349753\pi$
$29$	8.44737e10	0.909362	0.454681	+	0.890654i	$0.349753\pi$
$30$	0	0
$31$	−9.50671e8	−0.00620608	−0.00310304	−	0.999995i	$-0.500988\pi$
$31$	−9.50671e8	−0.00620608	−0.00310304	+	0.999995i	$0.500988\pi$
$32$	3.43597e10	0.176777
$33$	0	0
$34$	1.14963e11	0.375374
$35$	−1.25652e11	−0.330111
$36$	0	0
$37$	−1.42782e11	−0.247263	−0.123632	−	0.992328i	$-0.539454\pi$
$37$	−1.42782e11	−0.247263	−0.123632	+	0.992328i	$0.539454\pi$
$38$	−4.58278e11	−0.649759
$39$	0	0
$40$	2.98750e11	0.288309
$41$	1.47061e12	1.17929	0.589643	−	0.807664i	$-0.299268\pi$
$41$	1.47061e12	1.17929	0.589643	+	0.807664i	$0.299268\pi$
$42$	0	0
$43$	−1.58998e12	−0.892028	−0.446014	−	0.895026i	$-0.647157\pi$
$43$	−1.58998e12	−0.892028	−0.446014	+	0.895026i	$0.647157\pi$
$44$	−2.16693e11	−0.102318
$45$	0	0
$46$	−3.17954e12	−1.07567
$47$	2.03678e11	0.0586421	0.0293211	−	0.999570i	$-0.490665\pi$
$47$	2.03678e11	0.0586421	0.0293211	+	0.999570i	$0.490665\pi$
$48$	0	0
$49$	−3.96956e12	−0.836125
$50$	−1.30868e12	−0.236896
$51$	0	0
$52$	−5.18699e12	−0.699665
$53$	−2.22358e12	−0.260007	−0.130003	−	0.991514i	$-0.541499\pi$
$53$	−2.22358e12	−0.260007	−0.130003	+	0.991514i	$0.541499\pi$
$54$	0	0
$55$	−1.88410e12	−0.166872
$56$	−1.84978e12	−0.143124
$57$	0	0
$58$	1.08126e13	0.643016
$59$	−6.14211e12	−0.321312	−0.160656	−	0.987010i	$-0.551361\pi$
$59$	−6.14211e12	−0.321312	−0.160656	+	0.987010i	$0.551361\pi$
$60$	0	0
$61$	−2.73906e13	−1.11591	−0.557953	−	0.829873i	$-0.688413\pi$
$61$	−2.73906e13	−1.11591	−0.557953	+	0.829873i	$0.688413\pi$
$62$	−1.21686e11	−0.00438836
$63$	0	0
$64$	4.39805e12	0.125000
$65$	−4.50997e13	−1.14110
$66$	0	0
$67$	4.97262e13	1.00236	0.501181	−	0.865343i	$-0.332899\pi$
$67$	4.97262e13	1.00236	0.501181	+	0.865343i	$0.332899\pi$
$68$	1.47152e13	0.265430
$69$	0	0
$70$	−1.60835e13	−0.233423
$71$	−7.41880e13	−0.968046	−0.484023	−	0.875055i	$-0.660825\pi$
$71$	−7.41880e13	−0.968046	−0.484023	+	0.875055i	$0.660825\pi$
$72$	0	0
$73$	−4.47897e13	−0.474522	−0.237261	−	0.971446i	$-0.576250\pi$
$73$	−4.47897e13	−0.474522	−0.237261	+	0.971446i	$0.576250\pi$
$74$	−1.82761e13	−0.174842
$75$	0	0
$76$	−5.86596e13	−0.459449
$77$	1.16659e13	0.0828393
$78$	0	0
$79$	5.33619e13	0.312628	0.156314	−	0.987707i	$-0.450039\pi$
$79$	5.33619e13	0.312628	0.156314	+	0.987707i	$0.450039\pi$
$80$	3.82401e13	0.203865
$81$	0	0
$82$	1.88238e14	0.833881
$83$	−2.22672e14	−0.900698	−0.450349	−	0.892853i	$-0.648700\pi$
$83$	−2.22672e14	−0.900698	−0.450349	+	0.892853i	$0.648700\pi$
$84$	0	0
$85$	1.27945e14	0.432895
$86$	−2.03518e14	−0.630759
$87$	0	0
$88$	−2.77367e13	−0.0723495
$89$	−3.77610e14	−0.904936	−0.452468	−	0.891781i	$-0.649456\pi$
$89$	−3.77610e14	−0.904936	−0.452468	+	0.891781i	$0.649456\pi$
$90$	0	0
$91$	2.79246e14	0.566469
$92$	−4.06981e14	−0.760615
$93$	0	0
$94$	2.60708e13	0.0414663
$95$	−5.10033e14	−0.749326
$96$	0	0
$97$	1.25521e15	1.57736	0.788678	−	0.614807i	$-0.210766\pi$
$97$	1.25521e15	1.57736	0.788678	+	0.614807i	$0.210766\pi$
$98$	−5.08103e14	−0.591230
$99$	0	0
$100$	−1.67511e14	−0.167511
$101$	−1.28364e15	−1.19133	−0.595664	−	0.803234i	$-0.703111\pi$
$101$	−1.28364e15	−1.19133	−0.595664	+	0.803234i	$0.703111\pi$
$102$	0	0
$103$	−1.70326e15	−1.36459	−0.682295	−	0.731077i	$-0.739018\pi$
$103$	−1.70326e15	−1.36459	−0.682295	+	0.731077i	$0.739018\pi$
$104$	−6.63935e14	−0.494738
$105$	0	0
$106$	−2.84619e14	−0.183852
$107$	−2.61477e15	−1.57418	−0.787091	−	0.616837i	$-0.788414\pi$
$107$	−2.61477e15	−1.57418	−0.787091	+	0.616837i	$0.788414\pi$
$108$	0	0
$109$	1.08119e15	0.566504	0.283252	−	0.959046i	$-0.408587\pi$
$109$	1.08119e15	0.566504	0.283252	+	0.959046i	$0.408587\pi$
$110$	−2.41165e14	−0.117996
$111$	0	0
$112$	−2.36772e14	−0.101204
$113$	−7.13821e12	−0.00285431	−0.00142716	−	0.999999i	$-0.500454\pi$
$113$	−7.13821e12	−0.00285431	−0.00142716	+	0.999999i	$0.500454\pi$
$114$	0	0
$115$	−3.53861e15	−1.24050
$116$	1.38402e15	0.454681
$117$	0	0
$118$	−7.86190e14	−0.227202
$119$	−7.92205e14	−0.214900
$120$	0	0
$121$	−4.00232e15	−0.958124
$122$	−3.50599e15	−0.789064
$123$	0	0
$124$	−1.55758e13	−0.00310304
$125$	−5.80386e15	−1.08866
$126$	0	0
$127$	4.55166e15	0.757952	0.378976	−	0.925407i	$-0.376276\pi$
$127$	4.55166e15	0.757952	0.378976	+	0.925407i	$0.376276\pi$
$128$	5.62950e14	0.0883883
$129$	0	0
$130$	−5.77277e15	−0.806880
$131$	−1.09991e16	−1.45151	−0.725756	−	0.687952i	$-0.758510\pi$
$131$	−1.09991e16	−1.45151	−0.725756	+	0.687952i	$0.758510\pi$
$132$	0	0
$133$	3.15799e15	0.371983
$134$	6.36496e15	0.708777
$135$	0	0
$136$	1.88355e15	0.187687
$137$	7.16733e15	0.676010	0.338005	−	0.941144i	$-0.390248\pi$
$137$	7.16733e15	0.676010	0.338005	+	0.941144i	$0.390248\pi$
$138$	0	0
$139$	1.29993e16	1.09979	0.549894	−	0.835234i	$-0.314668\pi$
$139$	1.29993e16	1.09979	0.549894	+	0.835234i	$0.314668\pi$
$140$	−2.05868e15	−0.165055
$141$	0	0
$142$	−9.49607e15	−0.684512
$143$	4.18717e15	0.286352
$144$	0	0
$145$	1.20337e16	0.741550
$146$	−5.73308e15	−0.335538
$147$	0	0
$148$	−2.33934e15	−0.123632
$149$	2.15681e16	1.08372	0.541858	−	0.840470i	$-0.317721\pi$
$149$	2.15681e16	1.08372	0.541858	+	0.840470i	$0.317721\pi$
$150$	0	0
$151$	−7.86334e15	−0.357503	−0.178752	−	0.983894i	$-0.557206\pi$
$151$	−7.86334e15	−0.357503	−0.178752	+	0.983894i	$0.557206\pi$
$152$	−7.50843e15	−0.324880
$153$	0	0
$154$	1.49323e15	0.0585762
$155$	−1.35428e14	−0.00506082
$156$	0	0
$157$	−1.23936e16	−0.420680	−0.210340	−	0.977628i	$-0.567457\pi$
$157$	−1.23936e16	−0.420680	−0.210340	+	0.977628i	$0.567457\pi$
$158$	6.83032e15	0.221061
$159$	0	0
$160$	4.89473e15	0.144155
$161$	2.19101e16	0.615816
$162$	0	0
$163$	6.01364e16	1.54074	0.770371	−	0.637595i	$-0.220071\pi$
$163$	6.01364e16	1.54074	0.770371	+	0.637595i	$0.220071\pi$
$164$	2.40945e16	0.589643
$165$	0	0
$166$	−2.85020e16	−0.636889
$167$	7.07413e16	1.51112	0.755561	−	0.655078i	$-0.227364\pi$
$167$	7.07413e16	1.51112	0.755561	+	0.655078i	$0.227364\pi$
$168$	0	0
$169$	4.90426e16	0.958126
$170$	1.63770e16	0.306103
$171$	0	0
$172$	−2.60503e16	−0.446014
$173$	1.17889e17	1.93253	0.966265	−	0.257551i	$-0.0829155\pi$
$173$	1.17889e17	1.93253	0.966265	+	0.257551i	$0.0829155\pi$
$174$	0	0
$175$	9.01810e15	0.135622
$176$	−3.55030e15	−0.0511588
$177$	0	0
$178$	−4.83340e16	−0.639886
$179$	−2.95480e16	−0.375086	−0.187543	−	0.982256i	$-0.560052\pi$
$179$	−2.95480e16	−0.375086	−0.187543	+	0.982256i	$0.560052\pi$
$180$	0	0
$181$	1.32418e17	1.54653	0.773266	−	0.634082i	$-0.218622\pi$
$181$	1.32418e17	1.54653	0.773266	+	0.634082i	$0.218622\pi$
$182$	3.57435e16	0.400554
$183$	0	0
$184$	−5.20935e16	−0.537836
$185$	−2.03400e16	−0.201634
$186$	0	0
$187$	−1.18788e16	−0.108633
$188$	3.33706e15	0.0293211
$189$	0	0
$190$	−6.52842e16	−0.529854
$191$	−7.58644e16	−0.591954	−0.295977	−	0.955195i	$-0.595645\pi$
$191$	−7.58644e16	−0.591954	−0.295977	+	0.955195i	$0.595645\pi$
$192$	0	0
$193$	−2.00487e17	−1.44680	−0.723398	−	0.690432i	$-0.757421\pi$
$193$	−2.00487e17	−1.44680	−0.723398	+	0.690432i	$0.757421\pi$
$194$	1.60667e17	1.11536
$195$	0	0
$196$	−6.50372e16	−0.418063
$197$	1.60938e17	0.995777	0.497889	−	0.867241i	$-0.334109\pi$
$197$	1.60938e17	0.995777	0.497889	+	0.867241i	$0.334109\pi$
$198$	0	0
$199$	−5.77124e16	−0.331033	−0.165517	−	0.986207i	$-0.552929\pi$
$199$	−5.77124e16	−0.331033	−0.165517	+	0.986207i	$0.552929\pi$
$200$	−2.14414e16	−0.118448
$201$	0	0
$202$	−1.64305e17	−0.842396
$203$	−7.45097e16	−0.368123
$204$	0	0
$205$	2.09497e17	0.961663
$206$	−2.18018e17	−0.964911
$207$	0	0
$208$	−8.49836e16	−0.349833
$209$	4.73527e16	0.188039
$210$	0	0
$211$	3.64057e17	1.34602	0.673009	−	0.739635i	$-0.265002\pi$
$211$	3.64057e17	1.34602	0.673009	+	0.739635i	$0.265002\pi$
$212$	−3.64312e16	−0.130003
$213$	0	0
$214$	−3.34690e17	−1.11311
$215$	−2.26501e17	−0.727415
$216$	0	0
$217$	8.38535e14	0.00251231
$218$	1.38392e17	0.400579
$219$	0	0
$220$	−3.08691e16	−0.0834361
$221$	−2.84343e17	−0.742848
$222$	0	0
$223$	3.09914e17	0.756753	0.378376	−	0.925652i	$-0.376483\pi$
$223$	3.09914e17	0.756753	0.378376	+	0.925652i	$0.376483\pi$
$224$	−3.03069e16	−0.0715618
$225$	0	0
$226$	−9.13691e14	−0.00201830
$227$	8.59847e17	1.83750	0.918750	−	0.394839i	$-0.129200\pi$
$227$	8.59847e17	1.83750	0.918750	+	0.394839i	$0.129200\pi$
$228$	0	0
$229$	6.97827e17	1.39631	0.698155	−	0.715947i	$-0.254005\pi$
$229$	6.97827e17	1.39631	0.698155	+	0.715947i	$0.254005\pi$
$230$	−4.52942e17	−0.877169
$231$	0	0
$232$	1.77154e17	0.321508
$233$	−8.97443e17	−1.57702	−0.788511	−	0.615021i	$-0.789147\pi$
$233$	−8.97443e17	−1.57702	−0.788511	+	0.615021i	$0.789147\pi$
$234$	0	0
$235$	2.90150e16	0.0478204
$236$	−1.00632e17	−0.160656
$237$	0	0
$238$	−1.01402e17	−0.151957
$239$	−9.46030e17	−1.37379	−0.686895	−	0.726756i	$-0.741027\pi$
$239$	−9.46030e17	−1.37379	−0.686895	+	0.726756i	$0.741027\pi$
$240$	0	0
$241$	−1.79649e17	−0.245075	−0.122537	−	0.992464i	$-0.539103\pi$
$241$	−1.79649e17	−0.245075	−0.122537	+	0.992464i	$0.539103\pi$
$242$	−5.12297e17	−0.677496
$243$	0	0
$244$	−4.48767e17	−0.557953
$245$	−5.65484e17	−0.681828
$246$	0	0
$247$	1.13348e18	1.28584
$248$	−1.99370e15	−0.00219418
$249$	0	0
$250$	−7.42894e17	−0.769799
$251$	−1.10491e18	−1.11115	−0.555576	−	0.831466i	$-0.687502\pi$
$251$	−1.10491e18	−1.11115	−0.555576	+	0.831466i	$0.687502\pi$
$252$	0	0
$253$	3.28533e17	0.311297
$254$	5.82612e17	0.535953
$255$	0	0
$256$	7.20576e16	0.0625000
$257$	−5.92377e16	−0.0498999	−0.0249500	−	0.999689i	$-0.507943\pi$
$257$	−5.92377e16	−0.0498999	−0.0249500	+	0.999689i	$0.507943\pi$
$258$	0	0
$259$	1.25940e17	0.100096
$260$	−7.38914e17	−0.570550
$261$	0	0
$262$	−1.40788e18	−1.02637
$263$	8.12014e17	0.575301	0.287651	−	0.957735i	$-0.407126\pi$
$263$	8.12014e17	0.575301	0.287651	+	0.957735i	$0.407126\pi$
$264$	0	0
$265$	−3.16761e17	−0.212025
$266$	4.04223e17	0.263032
$267$	0	0
$268$	8.14714e17	0.501181
$269$	−1.93296e18	−1.15632	−0.578162	−	0.815922i	$-0.696230\pi$
$269$	−1.93296e18	−1.15632	−0.578162	+	0.815922i	$0.696230\pi$
$270$	0	0
$271$	−9.19659e17	−0.520424	−0.260212	−	0.965552i	$-0.583792\pi$
$271$	−9.19659e17	−0.520424	−0.260212	+	0.965552i	$0.583792\pi$
$272$	2.41094e17	0.132715
$273$	0	0
$274$	9.17418e17	0.478011
$275$	1.35223e17	0.0685573
$276$	0	0
$277$	1.27571e18	0.612566	0.306283	−	0.951941i	$-0.400915\pi$
$277$	1.27571e18	0.612566	0.306283	+	0.951941i	$0.400915\pi$
$278$	1.66391e18	0.777667
$279$	0	0
$280$	−2.63512e17	−0.116712
$281$	3.50755e18	1.51254	0.756269	−	0.654261i	$-0.227020\pi$
$281$	3.50755e18	1.51254	0.756269	+	0.654261i	$0.227020\pi$
$282$	0	0
$283$	−2.80748e18	−1.14794	−0.573969	−	0.818877i	$-0.694597\pi$
$283$	−2.80748e18	−1.14794	−0.573969	+	0.818877i	$0.694597\pi$
$284$	−1.21550e18	−0.484023
$285$	0	0
$286$	5.35958e17	0.202482
$287$	−1.29715e18	−0.477392
$288$	0	0
$289$	−2.05576e18	−0.718188
$290$	1.54032e18	0.524355
$291$	0	0
$292$	−7.33834e17	−0.237261
$293$	6.99683e17	0.220493	0.110246	−	0.993904i	$-0.464836\pi$
$293$	6.99683e17	0.220493	0.110246	+	0.993904i	$0.464836\pi$
$294$	0	0
$295$	−8.74976e17	−0.262018
$296$	−2.99435e17	−0.0874208
$297$	0	0
$298$	2.76072e18	0.766303
$299$	7.86410e18	2.12870
$300$	0	0
$301$	1.40244e18	0.361106
$302$	−1.00651e18	−0.252793
$303$	0	0
$304$	−9.61079e17	−0.229725
$305$	−3.90193e18	−0.909978
$306$	0	0
$307$	−3.12249e18	−0.693368	−0.346684	−	0.937982i	$-0.612692\pi$
$307$	−3.12249e18	−0.693368	−0.346684	+	0.937982i	$0.612692\pi$
$308$	1.91133e17	0.0414196
$309$	0	0
$310$	−1.73348e16	−0.00357854
$311$	8.18278e18	1.64891	0.824457	−	0.565925i	$-0.191481\pi$
$311$	8.18278e18	1.64891	0.824457	+	0.565925i	$0.191481\pi$
$312$	0	0
$313$	−2.49917e18	−0.479969	−0.239985	−	0.970777i	$-0.577142\pi$
$313$	−2.49917e18	−0.479969	−0.239985	+	0.970777i	$0.577142\pi$
$314$	−1.58639e18	−0.297466
$315$	0	0
$316$	8.74281e17	0.156314
$317$	6.11655e18	1.06798	0.533989	−	0.845492i	$-0.320692\pi$
$317$	6.11655e18	1.06798	0.533989	+	0.845492i	$0.320692\pi$
$318$	0	0
$319$	−1.11724e18	−0.186087
$320$	6.26525e17	0.101933
$321$	0	0
$322$	2.80450e18	0.435448
$323$	−3.21563e18	−0.487806
$324$	0	0
$325$	3.23682e18	0.468807
$326$	7.69746e18	1.08947
$327$	0	0
$328$	3.08410e18	0.416941
$329$	−1.79653e17	−0.0237392
$330$	0	0
$331$	3.13929e18	0.396388	0.198194	−	0.980163i	$-0.436492\pi$
$331$	3.13929e18	0.396388	0.198194	+	0.980163i	$0.436492\pi$
$332$	−3.64825e18	−0.450349
$333$	0	0
$334$	9.05489e18	1.06852
$335$	7.08376e18	0.817388
$336$	0	0
$337$	−2.61048e18	−0.288069	−0.144034	−	0.989573i	$-0.546008\pi$
$337$	−2.61048e18	−0.288069	−0.144034	+	0.989573i	$0.546008\pi$
$338$	6.27745e18	0.677498
$339$	0	0
$340$	2.09626e18	0.216448
$341$	1.25735e16	0.00126998
$342$	0	0
$343$	7.68890e18	0.743290
$344$	−3.33443e18	−0.315380
$345$	0	0
$346$	1.50897e19	1.36650
$347$	−1.72861e19	−1.53189	−0.765943	−	0.642908i	$-0.777728\pi$
$347$	−1.72861e19	−1.53189	−0.765943	+	0.642908i	$0.777728\pi$
$348$	0	0
$349$	−1.33509e19	−1.13323	−0.566616	−	0.823982i	$-0.691748\pi$
$349$	−1.33509e19	−1.13323	−0.566616	+	0.823982i	$0.691748\pi$
$350$	1.15432e18	0.0958991
$351$	0	0
$352$	−4.54439e17	−0.0361747
$353$	2.16624e19	1.68809	0.844047	−	0.536269i	$-0.180167\pi$
$353$	2.16624e19	1.68809	0.844047	+	0.536269i	$0.180167\pi$
$354$	0	0
$355$	−1.05685e19	−0.789405
$356$	−6.18675e18	−0.452468
$357$	0	0
$358$	−3.78215e18	−0.265226
$359$	−1.24130e19	−0.852446	−0.426223	−	0.904618i	$-0.640156\pi$
$359$	−1.24130e19	−0.852446	−0.426223	+	0.904618i	$0.640156\pi$
$360$	0	0
$361$	−2.36258e18	−0.155626
$362$	1.69495e19	1.09356
$363$	0	0
$364$	4.57516e18	0.283235
$365$	−6.38052e18	−0.386954
$366$	0	0
$367$	−3.39227e17	−0.0197467	−0.00987336	−	0.999951i	$-0.503143\pi$
$367$	−3.39227e17	−0.0197467	−0.00987336	+	0.999951i	$0.503143\pi$
$368$	−6.66797e18	−0.380308
$369$	0	0
$370$	−2.60352e18	−0.142577
$371$	1.96130e18	0.105254
$372$	0	0
$373$	−3.52841e19	−1.81871	−0.909353	−	0.416026i	$-0.863422\pi$
$373$	−3.52841e19	−1.81871	−0.909353	+	0.416026i	$0.863422\pi$
$374$	−1.52048e18	−0.0768148
$375$	0	0
$376$	4.27143e17	0.0207331
$377$	−2.67434e19	−1.27250
$378$	0	0
$379$	−8.02300e18	−0.366896	−0.183448	−	0.983029i	$-0.558726\pi$
$379$	−8.02300e18	−0.366896	−0.183448	+	0.983029i	$0.558726\pi$
$380$	−8.35638e18	−0.374663
$381$	0	0
$382$	−9.71065e18	−0.418575
$383$	−3.22865e19	−1.36468	−0.682338	−	0.731037i	$-0.739037\pi$
$383$	−3.22865e19	−1.36468	−0.682338	+	0.731037i	$0.739037\pi$
$384$	0	0
$385$	1.66186e18	0.0675522
$386$	−2.56624e19	−1.02304
$387$	0	0
$388$	2.05654e19	0.788678
$389$	−3.25042e19	−1.22269	−0.611346	−	0.791364i	$-0.709371\pi$
$389$	−3.25042e19	−1.22269	−0.611346	+	0.791364i	$0.709371\pi$
$390$	0	0
$391$	−2.23100e19	−0.807559
$392$	−8.32476e18	−0.295615
$393$	0	0
$394$	2.06001e19	0.704121
$395$	7.60168e18	0.254936
$396$	0	0
$397$	2.08495e19	0.673235	0.336618	−	0.941641i	$-0.390717\pi$
$397$	2.08495e19	0.673235	0.336618	+	0.941641i	$0.390717\pi$
$398$	−7.38719e18	−0.234076
$399$	0	0
$400$	−2.74450e18	−0.0837556
$401$	2.30643e19	0.690807	0.345404	−	0.938454i	$-0.387742\pi$
$401$	2.30643e19	0.690807	0.345404	+	0.938454i	$0.387742\pi$
$402$	0	0
$403$	3.00972e17	0.00868436
$404$	−2.10311e19	−0.595664
$405$	0	0
$406$	−9.53725e18	−0.260302
$407$	1.88842e18	0.0505988
$408$	0	0
$409$	2.27668e19	0.588000	0.294000	−	0.955805i	$-0.405014\pi$
$409$	2.27668e19	0.588000	0.294000	+	0.955805i	$0.405014\pi$
$410$	2.68156e19	0.679998
$411$	0	0
$412$	−2.79062e19	−0.682295
$413$	5.41762e18	0.130072
$414$	0	0
$415$	−3.17207e19	−0.734484
$416$	−1.08779e19	−0.247369
$417$	0	0
$418$	6.06115e18	0.132964
$419$	−3.85389e19	−0.830412	−0.415206	−	0.909727i	$-0.636291\pi$
$419$	−3.85389e19	−0.830412	−0.415206	+	0.909727i	$0.636291\pi$
$420$	0	0
$421$	4.40422e19	0.915699	0.457850	−	0.889030i	$-0.348620\pi$
$421$	4.40422e19	0.915699	0.457850	+	0.889030i	$0.348620\pi$
$422$	4.65993e19	0.951778
$423$	0	0
$424$	−4.66319e18	−0.0919262
$425$	−9.18269e18	−0.177850
$426$	0	0
$427$	2.41598e19	0.451735
$428$	−4.28404e19	−0.787091
$429$	0	0
$430$	−2.89922e19	−0.514360
$431$	−4.02778e19	−0.702241	−0.351120	−	0.936330i	$-0.614199\pi$
$431$	−4.02778e19	−0.702241	−0.351120	+	0.936330i	$0.614199\pi$
$432$	0	0
$433$	6.06056e19	1.02060	0.510298	−	0.859998i	$-0.329535\pi$
$433$	6.06056e19	1.02060	0.510298	+	0.859998i	$0.329535\pi$
$434$	1.07333e17	0.00177647
$435$	0	0
$436$	1.77142e19	0.283252
$437$	8.89351e19	1.39786
$438$	0	0
$439$	−5.39590e18	−0.0819559	−0.0409779	−	0.999160i	$-0.513047\pi$
$439$	−5.39590e18	−0.0819559	−0.0409779	+	0.999160i	$0.513047\pi$
$440$	−3.95124e18	−0.0589982
$441$	0	0
$442$	−3.63959e19	−0.525273
$443$	3.03203e19	0.430235	0.215118	−	0.976588i	$-0.430987\pi$
$443$	3.03203e19	0.430235	0.215118	+	0.976588i	$0.430987\pi$
$444$	0	0
$445$	−5.37925e19	−0.737941
$446$	3.96689e19	0.535105
$447$	0	0
$448$	−3.87928e18	−0.0506018
$449$	2.57015e19	0.329694	0.164847	−	0.986319i	$-0.447287\pi$
$449$	2.57015e19	0.329694	0.164847	+	0.986319i	$0.447287\pi$
$450$	0	0
$451$	−1.94502e19	−0.241323
$452$	−1.16952e17	−0.00142716
$453$	0	0
$454$	1.10060e20	1.29931
$455$	3.97801e19	0.461934
$456$	0	0
$457$	1.12808e20	1.26756	0.633780	−	0.773514i	$-0.281503\pi$
$457$	1.12808e20	1.26756	0.633780	+	0.773514i	$0.281503\pi$
$458$	8.93218e19	0.987340
$459$	0	0
$460$	−5.79765e19	−0.620252
$461$	−1.61480e20	−1.69966	−0.849829	−	0.527059i	$-0.823295\pi$
$461$	−1.61480e20	−1.69966	−0.849829	+	0.527059i	$0.823295\pi$
$462$	0	0
$463$	1.61498e20	1.64555	0.822773	−	0.568370i	$-0.192426\pi$
$463$	1.61498e20	1.64555	0.822773	+	0.568370i	$0.192426\pi$
$464$	2.26757e19	0.227340
$465$	0	0
$466$	−1.14873e20	−1.11512
$467$	1.26466e20	1.20809	0.604043	−	0.796952i	$-0.293556\pi$
$467$	1.26466e20	1.20809	0.604043	+	0.796952i	$0.293556\pi$
$468$	0	0
$469$	−4.38608e19	−0.405770
$470$	3.71392e18	0.0338141
$471$	0	0
$472$	−1.28809e19	−0.113601
$473$	2.10289e19	0.182540
$474$	0	0
$475$	3.66052e19	0.307851
$476$	−1.29795e19	−0.107450
$477$	0	0
$478$	−1.21092e20	−0.971417
$479$	7.68671e19	0.607050	0.303525	−	0.952824i	$-0.401836\pi$
$479$	7.68671e19	0.607050	0.303525	+	0.952824i	$0.401836\pi$
$480$	0	0
$481$	4.52031e19	0.346003
$482$	−2.29951e19	−0.173294
$483$	0	0
$484$	−6.55741e19	−0.479062
$485$	1.78812e20	1.28627
$486$	0	0
$487$	9.62520e19	0.671340	0.335670	−	0.941980i	$-0.391037\pi$
$487$	9.62520e19	0.671340	0.335670	+	0.941980i	$0.391037\pi$
$488$	−5.74422e19	−0.394532
$489$	0	0
$490$	−7.23820e19	−0.482125
$491$	1.39899e20	0.917703	0.458852	−	0.888513i	$-0.348261\pi$
$491$	1.39899e20	0.917703	0.458852	+	0.888513i	$0.348261\pi$
$492$	0	0
$493$	7.58697e19	0.482743
$494$	1.45086e20	0.909228
$495$	0	0
$496$	−2.55194e17	−0.00155152
$497$	6.54373e19	0.391879
$498$	0	0
$499$	−1.96076e20	−1.13938	−0.569692	−	0.821858i	$-0.692937\pi$
$499$	−1.96076e20	−1.13938	−0.569692	+	0.821858i	$0.692937\pi$
$500$	−9.50905e19	−0.544330
$501$	0	0
$502$	−1.41428e20	−0.785703
$503$	2.29665e20	1.25700	0.628500	−	0.777810i	$-0.283669\pi$
$503$	2.29665e20	1.25700	0.628500	+	0.777810i	$0.283669\pi$
$504$	0	0
$505$	−1.82861e20	−0.971482
$506$	4.20522e19	0.220120
$507$	0	0
$508$	7.45743e19	0.378976
$509$	−6.90568e19	−0.345798	−0.172899	−	0.984940i	$-0.555313\pi$
$509$	−6.90568e19	−0.345798	−0.172899	+	0.984940i	$0.555313\pi$
$510$	0	0
$511$	3.95065e19	0.192093
$512$	9.22337e18	0.0441942
$513$	0	0
$514$	−7.58243e18	−0.0352846
$515$	−2.42639e20	−1.11277
$516$	0	0
$517$	−2.69382e18	−0.0120002
$518$	1.61203e19	0.0707784
$519$	0	0
$520$	−9.45810e19	−0.403440
$521$	3.69117e19	0.155196	0.0775980	−	0.996985i	$-0.475275\pi$
$521$	3.69117e19	0.155196	0.0775980	+	0.996985i	$0.475275\pi$
$522$	0	0
$523$	−1.78641e20	−0.729823	−0.364912	−	0.931042i	$-0.618901\pi$
$523$	−1.78641e20	−0.729823	−0.364912	+	0.931042i	$0.618901\pi$
$524$	−1.80209e20	−0.725756
$525$	0	0
$526$	1.03938e20	0.406800
$527$	−8.53840e17	−0.00329456
$528$	0	0
$529$	3.50396e20	1.31414
$530$	−4.05454e19	−0.149925
$531$	0	0
$532$	5.17405e19	0.185992
$533$	−4.65579e20	−1.65021
$534$	0	0
$535$	−3.72488e20	−1.28368
$536$	1.04283e20	0.354388
$537$	0	0
$538$	−2.47418e20	−0.817645
$539$	5.25010e19	0.171101
$540$	0	0
$541$	6.12863e20	1.94260	0.971302	−	0.237850i	$-0.0764428\pi$
$541$	6.12863e20	1.94260	0.971302	+	0.237850i	$0.0764428\pi$
$542$	−1.17716e20	−0.367995
$543$	0	0
$544$	3.08600e19	0.0938436
$545$	1.54021e20	0.461962
$546$	0	0
$547$	2.38524e20	0.696030	0.348015	−	0.937489i	$-0.386856\pi$
$547$	2.38524e20	0.696030	0.348015	+	0.937489i	$0.386856\pi$
$548$	1.17430e20	0.338005
$549$	0	0
$550$	1.73085e19	0.0484774
$551$	−3.02441e20	−0.835611
$552$	0	0
$553$	−4.70676e19	−0.126556
$554$	1.63291e20	0.433150
$555$	0	0
$556$	2.12981e20	0.549894
$557$	−3.10723e20	−0.791516	−0.395758	−	0.918355i	$-0.629518\pi$
$557$	−3.10723e20	−0.791516	−0.395758	+	0.918355i	$0.629518\pi$
$558$	0	0
$559$	5.03370e20	1.24824
$560$	−3.37295e19	−0.0825277
$561$	0	0
$562$	4.48966e20	1.06953
$563$	−7.74946e20	−1.82162	−0.910812	−	0.412822i	$-0.864543\pi$
$563$	−7.74946e20	−1.82162	−0.910812	+	0.412822i	$0.864543\pi$
$564$	0	0
$565$	−1.01688e18	−0.00232758
$566$	−3.59357e20	−0.811714
$567$	0	0
$568$	−1.55584e20	−0.342256
$569$	−5.95598e20	−1.29304	−0.646519	−	0.762898i	$-0.723776\pi$
$569$	−5.95598e20	−1.29304	−0.646519	+	0.762898i	$0.723776\pi$
$570$	0	0
$571$	−7.01047e20	−1.48244	−0.741219	−	0.671263i	$-0.765752\pi$
$571$	−7.01047e20	−1.48244	−0.741219	+	0.671263i	$0.765752\pi$
$572$	6.86026e19	0.143176
$573$	0	0
$574$	−1.66035e20	−0.337567
$575$	2.53967e20	0.509646
$576$	0	0
$577$	−1.40957e20	−0.275592	−0.137796	−	0.990461i	$-0.544002\pi$
$577$	−1.40957e20	−0.275592	−0.137796	+	0.990461i	$0.544002\pi$
$578$	−2.63137e20	−0.507836
$579$	0	0
$580$	1.97161e20	0.370775
$581$	1.96407e20	0.364615
$582$	0	0
$583$	2.94089e19	0.0532065
$584$	−9.39307e19	−0.167769
$585$	0	0
$586$	8.95594e19	0.155912
$587$	−2.73754e19	−0.0470516	−0.0235258	−	0.999723i	$-0.507489\pi$
$587$	−2.73754e19	−0.0470516	−0.0235258	+	0.999723i	$0.507489\pi$
$588$	0	0
$589$	3.40369e18	0.00570276
$590$	−1.11997e20	−0.185275
$591$	0	0
$592$	−3.83277e19	−0.0618159
$593$	6.58066e20	1.04800	0.523998	−	0.851720i	$-0.324440\pi$
$593$	6.58066e20	1.04800	0.523998	+	0.851720i	$0.324440\pi$
$594$	0	0
$595$	−1.12854e20	−0.175242
$596$	3.53372e20	0.541858
$597$	0	0
$598$	1.00661e21	1.50522
$599$	−4.40604e20	−0.650650	−0.325325	−	0.945602i	$-0.605474\pi$
$599$	−4.40604e20	−0.650650	−0.325325	+	0.945602i	$0.605474\pi$
$600$	0	0
$601$	−8.46415e20	−1.21906	−0.609530	−	0.792763i	$-0.708642\pi$
$601$	−8.46415e20	−1.21906	−0.609530	+	0.792763i	$0.708642\pi$
$602$	1.79512e20	0.255340
$603$	0	0
$604$	−1.28833e20	−0.178752
$605$	−5.70152e20	−0.781314
$606$	0	0
$607$	−1.23203e21	−1.64704	−0.823522	−	0.567284i	$-0.807994\pi$
$607$	−1.23203e21	−1.64704	−0.823522	+	0.567284i	$0.807994\pi$
$608$	−1.23018e20	−0.162440
$609$	0	0
$610$	−4.99447e20	−0.643452
$611$	−6.44821e19	−0.0820598
$612$	0	0
$613$	2.00993e20	0.249590	0.124795	−	0.992183i	$-0.460173\pi$
$613$	2.00993e20	0.249590	0.124795	+	0.992183i	$0.460173\pi$
$614$	−3.99679e20	−0.490285
$615$	0	0
$616$	2.44651e19	0.0292881
$617$	9.83192e20	1.16279	0.581393	−	0.813623i	$-0.302508\pi$
$617$	9.83192e20	1.16279	0.581393	+	0.813623i	$0.302508\pi$
$618$	0	0
$619$	3.04341e20	0.351302	0.175651	−	0.984453i	$-0.443797\pi$
$619$	3.04341e20	0.351302	0.175651	+	0.984453i	$0.443797\pi$
$620$	−2.21885e18	−0.00253041
$621$	0	0
$622$	1.04740e21	1.16596
$623$	3.33069e20	0.366331
$624$	0	0
$625$	−5.14777e20	−0.552738
$626$	−3.19894e20	−0.339389
$627$	0	0
$628$	−2.03058e20	−0.210340
$629$	−1.28239e20	−0.131262
$630$	0	0
$631$	−5.73711e20	−0.573420	−0.286710	−	0.958017i	$-0.592562\pi$
$631$	−5.73711e20	−0.573420	−0.286710	+	0.958017i	$0.592562\pi$
$632$	1.11908e20	0.110531
$633$	0	0
$634$	7.82919e20	0.755174
$635$	6.48407e20	0.618081
$636$	0	0
$637$	1.25672e21	1.17002
$638$	−1.43007e20	−0.131584
$639$	0	0
$640$	8.01952e19	0.0720773
$641$	1.90026e21	1.68802	0.844011	−	0.536326i	$-0.180188\pi$
$641$	1.90026e21	1.68802	0.844011	+	0.536326i	$0.180188\pi$
$642$	0	0
$643$	−1.45547e21	−1.26305	−0.631526	−	0.775354i	$-0.717571\pi$
$643$	−1.45547e21	−1.26305	−0.631526	+	0.775354i	$0.717571\pi$
$644$	3.58976e20	0.307908
$645$	0	0
$646$	−4.11600e20	−0.344931
$647$	−1.36017e21	−1.12670	−0.563352	−	0.826217i	$-0.690489\pi$
$647$	−1.36017e21	−1.12670	−0.563352	+	0.826217i	$0.690489\pi$
$648$	0	0
$649$	8.12349e19	0.0657518
$650$	4.14314e20	0.331497
$651$	0	0
$652$	9.85274e20	0.770371
$653$	−1.13573e21	−0.877863	−0.438932	−	0.898520i	$-0.644643\pi$
$653$	−1.13573e21	−0.877863	−0.438932	+	0.898520i	$0.644643\pi$
$654$	0	0
$655$	−1.56687e21	−1.18365
$656$	3.94764e20	0.294822
$657$	0	0
$658$	−2.29956e19	−0.0167861
$659$	1.43598e21	1.03635	0.518175	−	0.855274i	$-0.326611\pi$
$659$	1.43598e21	1.03635	0.518175	+	0.855274i	$0.326611\pi$
$660$	0	0
$661$	−6.23697e20	−0.440010	−0.220005	−	0.975499i	$-0.570607\pi$
$661$	−6.23697e20	−0.440010	−0.220005	+	0.975499i	$0.570607\pi$
$662$	4.01828e20	0.280289
$663$	0	0
$664$	−4.66976e20	−0.318445
$665$	4.49872e20	0.303338
$666$	0	0
$667$	−2.09834e21	−1.38335
$668$	1.15903e21	0.755561
$669$	0	0
$670$	9.06722e20	0.577980
$671$	3.62265e20	0.228354
$672$	0	0
$673$	2.77612e21	1.71130	0.855648	−	0.517559i	$-0.173159\pi$
$673$	2.77612e21	1.71130	0.855648	+	0.517559i	$0.173159\pi$
$674$	−3.34141e20	−0.203695
$675$	0	0
$676$	8.03513e20	0.479063
$677$	2.56725e21	1.51374	0.756872	−	0.653563i	$-0.226727\pi$
$677$	2.56725e21	1.51374	0.756872	+	0.653563i	$0.226727\pi$
$678$	0	0
$679$	−1.10716e21	−0.638536
$680$	2.68321e20	0.153052
$681$	0	0
$682$	1.60941e18	0.000898013 0
$683$	6.84777e20	0.377915	0.188957	−	0.981985i	$-0.439489\pi$
$683$	6.84777e20	0.377915	0.188957	+	0.981985i	$0.439489\pi$
$684$	0	0
$685$	1.02102e21	0.551260
$686$	9.84179e20	0.525585
$687$	0	0
$688$	−4.26807e20	−0.223007
$689$	7.03961e20	0.363835
$690$	0	0
$691$	1.87100e21	0.946214	0.473107	−	0.881005i	$-0.343132\pi$
$691$	1.87100e21	0.946214	0.473107	+	0.881005i	$0.343132\pi$
$692$	1.93149e21	0.966265
$693$	0	0
$694$	−2.21262e21	−1.08321
$695$	1.85182e21	0.896835
$696$	0	0
$697$	1.32082e21	0.626035
$698$	−1.70891e21	−0.801316
$699$	0	0
$700$	1.47753e20	0.0678109
$701$	−3.19721e21	−1.45173	−0.725864	−	0.687838i	$-0.758560\pi$
$701$	−3.19721e21	−1.45173	−0.725864	+	0.687838i	$0.758560\pi$
$702$	0	0
$703$	5.11202e20	0.227210
$704$	−5.81681e19	−0.0255794
$705$	0	0
$706$	2.77279e21	1.19366
$707$	1.13223e21	0.482267
$708$	0	0
$709$	3.11862e21	1.30052	0.650258	−	0.759714i	$-0.274661\pi$
$709$	3.11862e21	1.30052	0.650258	+	0.759714i	$0.274661\pi$
$710$	−1.35276e21	−0.558193
$711$	0	0
$712$	−7.91905e20	−0.319943
$713$	2.36148e19	0.00944087
$714$	0	0
$715$	5.96485e20	0.233509
$716$	−4.84115e20	−0.187543
$717$	0	0
$718$	−1.58886e21	−0.602771
$719$	−2.86542e21	−1.07577	−0.537887	−	0.843017i	$-0.680777\pi$
$719$	−2.86542e21	−1.07577	−0.537887	+	0.843017i	$0.680777\pi$
$720$	0	0
$721$	1.50236e21	0.552406
$722$	−3.02410e20	−0.110044
$723$	0	0
$724$	2.16954e21	0.773266
$725$	−8.63665e20	−0.304656
$726$	0	0
$727$	−4.40337e21	−1.52152	−0.760759	−	0.649035i	$-0.775173\pi$
$727$	−4.40337e21	−1.52152	−0.760759	+	0.649035i	$0.775173\pi$
$728$	5.85621e20	0.200277
$729$	0	0
$730$	−8.16707e20	−0.273618
$731$	−1.42803e21	−0.473542
$732$	0	0
$733$	−1.38060e21	−0.448527	−0.224264	−	0.974529i	$-0.571998\pi$
$733$	−1.38060e21	−0.448527	−0.224264	+	0.974529i	$0.571998\pi$
$734$	−4.34211e19	−0.0139630
$735$	0	0
$736$	−8.53500e20	−0.268918
$737$	−6.57674e20	−0.205118
$738$	0	0
$739$	−1.81922e21	−0.555970	−0.277985	−	0.960585i	$-0.589667\pi$
$739$	−1.81922e21	−0.555970	−0.277985	+	0.960585i	$0.589667\pi$
$740$	−3.33251e20	−0.100817
$741$	0	0
$742$	2.51047e20	0.0744261
$743$	−5.98318e19	−0.0175597	−0.00877983	−	0.999961i	$-0.502795\pi$
$743$	−5.98318e19	−0.0175597	−0.00877983	+	0.999961i	$0.502795\pi$
$744$	0	0
$745$	3.07250e21	0.883729
$746$	−4.51636e21	−1.28602
$747$	0	0
$748$	−1.94622e20	−0.0543163
$749$	2.30635e21	0.637251
$750$	0	0
$751$	1.33359e21	0.361178	0.180589	−	0.983559i	$-0.442200\pi$
$751$	1.33359e21	0.361178	0.180589	+	0.983559i	$0.442200\pi$
$752$	5.46744e19	0.0146605
$753$	0	0
$754$	−3.42316e21	−0.899792
$755$	−1.12017e21	−0.291530
$756$	0	0
$757$	−1.97282e21	−0.503348	−0.251674	−	0.967812i	$-0.580981\pi$
$757$	−1.97282e21	−0.503348	−0.251674	+	0.967812i	$0.580981\pi$
$758$	−1.02694e21	−0.259434
$759$	0	0
$760$	−1.06962e21	−0.264927
$761$	−3.75846e21	−0.921775	−0.460888	−	0.887458i	$-0.652469\pi$
$761$	−3.75846e21	−0.921775	−0.460888	+	0.887458i	$0.652469\pi$
$762$	0	0
$763$	−9.53659e20	−0.229329
$764$	−1.24296e21	−0.295977
$765$	0	0
$766$	−4.13267e21	−0.964972
$767$	1.94452e21	0.449622
$768$	0	0
$769$	2.87302e21	0.651465	0.325733	−	0.945462i	$-0.394389\pi$
$769$	2.87302e21	0.651465	0.325733	+	0.945462i	$0.394389\pi$
$770$	2.12718e20	0.0477667
$771$	0	0
$772$	−3.28478e21	−0.723398
$773$	−6.42966e21	−1.40230	−0.701151	−	0.713012i	$-0.747330\pi$
$773$	−6.42966e21	−1.40230	−0.701151	+	0.713012i	$0.747330\pi$
$774$	0	0
$775$	9.71972e18	0.00207917
$776$	2.63237e21	0.557679
$777$	0	0
$778$	−4.16053e21	−0.864573
$779$	−5.26523e21	−1.08364
$780$	0	0
$781$	9.81204e20	0.198096
$782$	−2.85568e21	−0.571031
$783$	0	0
$784$	−1.06557e21	−0.209031
$785$	−1.76554e21	−0.343048
$786$	0	0
$787$	−1.15144e21	−0.219498	−0.109749	−	0.993959i	$-0.535005\pi$
$787$	−1.15144e21	−0.219498	−0.109749	+	0.993959i	$0.535005\pi$
$788$	2.63681e21	0.497889
$789$	0	0
$790$	9.73015e20	0.180267
$791$	6.29623e18	0.00115547
$792$	0	0
$793$	8.67155e21	1.56152
$794$	2.66874e21	0.476049
$795$	0	0
$796$	−9.45561e20	−0.165517
$797$	−3.96643e21	−0.687801	−0.343900	−	0.939006i	$-0.611748\pi$
$797$	−3.96643e21	−0.687801	−0.343900	+	0.939006i	$0.611748\pi$
$798$	0	0
$799$	1.82932e20	0.0311307
$800$	−3.51296e20	−0.0592241
$801$	0	0
$802$	2.95223e21	0.488474
$803$	5.92384e20	0.0971039
$804$	0	0
$805$	3.12121e21	0.502174
$806$	3.85244e19	0.00614077
$807$	0	0
$808$	−2.69198e21	−0.421198
$809$	1.77767e21	0.275573	0.137786	−	0.990462i	$-0.456001\pi$
$809$	1.77767e21	0.275573	0.137786	+	0.990462i	$0.456001\pi$
$810$	0	0
$811$	−6.01666e21	−0.915587	−0.457793	−	0.889059i	$-0.651360\pi$
$811$	−6.01666e21	−0.915587	−0.457793	+	0.889059i	$0.651360\pi$
$812$	−1.22077e21	−0.184061
$813$	0	0
$814$	2.41718e20	0.0357787
$815$	8.56674e21	1.25642
$816$	0	0
$817$	5.69261e21	0.819683
$818$	2.91415e21	0.415778
$819$	0	0
$820$	3.43239e21	0.480831
$821$	−2.90625e21	−0.403422	−0.201711	−	0.979445i	$-0.564650\pi$
$821$	−2.90625e21	−0.403422	−0.201711	+	0.979445i	$0.564650\pi$
$822$	0	0
$823$	−7.21810e21	−0.983839	−0.491920	−	0.870641i	$-0.663705\pi$
$823$	−7.21810e21	−0.983839	−0.491920	+	0.870641i	$0.663705\pi$
$824$	−3.57200e21	−0.482455
$825$	0	0
$826$	6.93456e20	0.0919747
$827$	−1.65170e21	−0.217090	−0.108545	−	0.994092i	$-0.534619\pi$
$827$	−1.65170e21	−0.217090	−0.108545	+	0.994092i	$0.534619\pi$
$828$	0	0
$829$	1.06355e22	1.37277	0.686384	−	0.727239i	$-0.259197\pi$
$829$	1.06355e22	1.37277	0.686384	+	0.727239i	$0.259197\pi$
$830$	−4.06025e21	−0.519359
$831$	0	0
$832$	−1.39237e21	−0.174916
$833$	−3.56524e21	−0.443865
$834$	0	0
$835$	1.00775e22	1.23226
$836$	7.75827e20	0.0940194
$837$	0	0
$838$	−4.93298e21	−0.587190
$839$	−8.51705e21	−1.00479	−0.502394	−	0.864639i	$-0.667547\pi$
$839$	−8.51705e21	−1.00479	−0.502394	+	0.864639i	$0.667547\pi$
$840$	0	0
$841$	−1.49337e21	−0.173061
$842$	5.63740e21	0.647497
$843$	0	0
$844$	5.96470e21	0.673009
$845$	6.98637e21	0.781315
$846$	0	0
$847$	3.53023e21	0.387863
$848$	−5.96889e20	−0.0650016
$849$	0	0
$850$	−1.17538e21	−0.125759
$851$	3.54672e21	0.376145
$852$	0	0
$853$	8.59532e20	0.0895662	0.0447831	−	0.998997i	$-0.485740\pi$
$853$	8.59532e20	0.0895662	0.0447831	+	0.998997i	$0.485740\pi$
$854$	3.09245e21	0.319425
$855$	0	0
$856$	−5.48357e21	−0.556557
$857$	1.58497e22	1.59465	0.797325	−	0.603551i	$-0.206248\pi$
$857$	1.58497e22	1.59465	0.797325	+	0.603551i	$0.206248\pi$
$858$	0	0
$859$	−1.85041e22	−1.82945	−0.914725	−	0.404078i	$-0.867593\pi$
$859$	−1.85041e22	−1.82945	−0.914725	+	0.404078i	$0.867593\pi$
$860$	−3.71100e21	−0.363707
$861$	0	0
$862$	−5.15555e21	−0.496559
$863$	1.11276e22	1.06248	0.531240	−	0.847221i	$-0.321726\pi$
$863$	1.11276e22	1.06248	0.531240	+	0.847221i	$0.321726\pi$
$864$	0	0
$865$	1.67939e22	1.57590
$866$	7.75752e21	0.721670
$867$	0	0
$868$	1.37386e19	0.00125616
$869$	−7.05759e20	−0.0639747
$870$	0	0
$871$	−1.57428e22	−1.40264
$872$	2.26742e21	0.200289
$873$	0	0
$874$	1.13837e22	0.988433
$875$	5.11927e21	0.440705
$876$	0	0
$877$	1.92958e22	1.63293	0.816463	−	0.577398i	$-0.195932\pi$
$877$	1.92958e22	1.63293	0.816463	+	0.577398i	$0.195932\pi$
$878$	−6.90675e20	−0.0579516
$879$	0	0
$880$	−5.05759e20	−0.0417180
$881$	2.00911e22	1.64318	0.821589	−	0.570081i	$-0.193088\pi$
$881$	2.00911e22	1.64318	0.821589	+	0.570081i	$0.193088\pi$
$882$	0	0
$883$	−1.28025e22	−1.02941	−0.514705	−	0.857367i	$-0.672099\pi$
$883$	−1.28025e22	−1.02941	−0.514705	+	0.857367i	$0.672099\pi$
$884$	−4.65867e21	−0.371424
$885$	0	0
$886$	3.88100e21	0.304222
$887$	−6.34248e20	−0.0492983	−0.0246492	−	0.999696i	$-0.507847\pi$
$887$	−6.34248e20	−0.0492983	−0.0246492	+	0.999696i	$0.507847\pi$
$888$	0	0
$889$	−4.01477e21	−0.306830
$890$	−6.88544e21	−0.521803
$891$	0	0
$892$	5.07762e21	0.378376
$893$	−7.29228e20	−0.0538862
$894$	0	0
$895$	−4.20928e21	−0.305868
$896$	−4.96548e20	−0.0357809
$897$	0	0
$898$	3.28979e21	0.233129
$899$	−8.03067e19	−0.00564357
$900$	0	0
$901$	−1.99710e21	−0.138027
$902$	−2.48962e21	−0.170641
$903$	0	0
$904$	−1.49699e19	−0.00100915
$905$	1.88637e22	1.26114
$906$	0	0
$907$	−1.10108e22	−0.724043	−0.362021	−	0.932170i	$-0.617913\pi$
$907$	−1.10108e22	−0.724043	−0.362021	+	0.932170i	$0.617913\pi$
$908$	1.40877e22	0.918750
$909$	0	0
$910$	5.09185e21	0.326637
$911$	−2.11226e22	−1.34388	−0.671939	−	0.740606i	$-0.734538\pi$
$911$	−2.11226e22	−1.34388	−0.671939	+	0.740606i	$0.734538\pi$
$912$	0	0
$913$	2.94503e21	0.184314
$914$	1.44394e22	0.896300
$915$	0	0
$916$	1.14332e22	0.698155
$917$	9.70167e21	0.587593
$918$	0	0
$919$	−1.49884e22	−0.893076	−0.446538	−	0.894765i	$-0.647343\pi$
$919$	−1.49884e22	−0.893076	−0.446538	+	0.894765i	$0.647343\pi$
$920$	−7.42100e21	−0.438585
$921$	0	0
$922$	−2.06694e22	−1.20184
$923$	2.34871e22	1.35462
$924$	0	0
$925$	1.45981e21	0.0828387
$926$	2.06718e22	1.16358
$927$	0	0
$928$	2.90250e21	0.160754
$929$	−1.69245e22	−0.929818	−0.464909	−	0.885359i	$-0.653913\pi$
$929$	−1.69245e22	−0.929818	−0.464909	+	0.885359i	$0.653913\pi$
$930$	0	0
$931$	1.42122e22	0.768314
$932$	−1.47037e22	−0.788511
$933$	0	0
$934$	1.61877e22	0.854246
$935$	−1.69220e21	−0.0885856
$936$	0	0
$937$	−2.63983e22	−1.35997	−0.679984	−	0.733227i	$-0.738013\pi$
$937$	−2.63983e22	−1.35997	−0.679984	+	0.733227i	$0.738013\pi$
$938$	−5.61419e21	−0.286923
$939$	0	0
$940$	4.75382e20	0.0239102
$941$	1.16595e22	0.581780	0.290890	−	0.956757i	$-0.406049\pi$
$941$	1.16595e22	0.581780	0.290890	+	0.956757i	$0.406049\pi$
$942$	0	0
$943$	−3.65302e22	−1.79397
$944$	−1.64876e21	−0.0803281
$945$	0	0
$946$	2.69170e21	0.129076
$947$	−3.30603e22	−1.57283	−0.786415	−	0.617699i	$-0.788065\pi$
$947$	−3.30603e22	−1.57283	−0.786415	+	0.617699i	$0.788065\pi$
$948$	0	0
$949$	1.41799e22	0.664013
$950$	4.68547e21	0.217684
$951$	0	0
$952$	−1.66137e21	−0.0759785
$953$	−2.65079e22	−1.20276	−0.601379	−	0.798964i	$-0.705382\pi$
$953$	−2.65079e22	−1.20276	−0.601379	+	0.798964i	$0.705382\pi$
$954$	0	0
$955$	−1.08073e22	−0.482716
$956$	−1.54998e22	−0.686895
$957$	0	0
$958$	9.83899e21	0.429249
$959$	−6.32191e21	−0.273659
$960$	0	0
$961$	−2.34644e22	−0.999961
$962$	5.78600e21	0.244661
$963$	0	0
$964$	−2.94338e21	−0.122537
$965$	−2.85605e22	−1.17981
$966$	0	0
$967$	−9.03923e21	−0.367649	−0.183824	−	0.982959i	$-0.558848\pi$
$967$	−9.03923e21	−0.367649	−0.183824	+	0.982959i	$0.558848\pi$
$968$	−8.39348e21	−0.338748
$969$	0	0
$970$	2.28879e22	0.909532
$971$	7.82937e21	0.308733	0.154366	−	0.988014i	$-0.450666\pi$
$971$	7.82937e21	0.308733	0.154366	+	0.988014i	$0.450666\pi$
$972$	0	0
$973$	−1.14660e22	−0.445210
$974$	1.23202e22	0.474709
$975$	0	0
$976$	−7.35260e21	−0.278976
$977$	−5.07858e22	−1.91220	−0.956100	−	0.293041i	$-0.905333\pi$
$977$	−5.07858e22	−1.91220	−0.956100	+	0.293041i	$0.905333\pi$
$978$	0	0
$979$	4.99423e21	0.185182
$980$	−9.26489e21	−0.340914
$981$	0	0
$982$	1.79070e22	0.648914
$983$	2.33198e21	0.0838637	0.0419318	−	0.999120i	$-0.486649\pi$
$983$	2.33198e21	0.0838637	0.0419318	+	0.999120i	$0.486649\pi$
$984$	0	0
$985$	2.29265e22	0.812018
$986$	9.71132e21	0.341351
$987$	0	0
$988$	1.85710e22	0.642921
$989$	3.94953e22	1.35698
$990$	0	0
$991$	1.16356e22	0.393766	0.196883	−	0.980427i	$-0.436918\pi$
$991$	1.16356e22	0.393766	0.196883	+	0.980427i	$0.436918\pi$
$992$	−3.26648e19	−0.00109709
$993$	0	0
$994$	8.37597e21	0.277100
$995$	−8.22144e21	−0.269945
$996$	0	0
$997$	3.35334e22	1.08458	0.542292	−	0.840190i	$-0.317557\pi$
$997$	3.35334e22	1.08458	0.542292	+	0.840190i	$0.317557\pi$
$998$	−2.50977e22	−0.805666
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.16.a.c.1.2	yes	2
3.2	odd	2		54.16.a.b.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.16.a.b.1.1	✓	2	3.2	odd	2
54.16.a.c.1.2	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:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

\(f(q)\)	\(=\)	\(q+128.000 q^{2} +16384.0 q^{4} +142455. q^{5} -882046. q^{7} +2.09715e6 q^{8} +1.82343e7 q^{10} -1.32259e7 q^{11} -3.16589e8 q^{13} -1.12902e8 q^{14} +2.68435e8 q^{16} +8.98145e8 q^{17} -3.58030e9 q^{19} +2.33399e9 q^{20} -1.69292e9 q^{22} -2.48401e10 q^{23} -1.02241e10 q^{25} -4.05234e10 q^{26} -1.44514e10 q^{28} +8.44737e10 q^{29} -9.50671e8 q^{31} +3.43597e10 q^{32} +1.14963e11 q^{34} -1.25652e11 q^{35} -1.42782e11 q^{37} -4.58278e11 q^{38} +2.98750e11 q^{40} +1.47061e12 q^{41} -1.58998e12 q^{43} -2.16693e11 q^{44} -3.17954e12 q^{46} +2.03678e11 q^{47} -3.96956e12 q^{49} -1.30868e12 q^{50} -5.18699e12 q^{52} -2.22358e12 q^{53} -1.88410e12 q^{55} -1.84978e12 q^{56} +1.08126e13 q^{58} -6.14211e12 q^{59} -2.73906e13 q^{61} -1.21686e11 q^{62} +4.39805e12 q^{64} -4.50997e13 q^{65} +4.97262e13 q^{67} +1.47152e13 q^{68} -1.60835e13 q^{70} -7.41880e13 q^{71} -4.47897e13 q^{73} -1.82761e13 q^{74} -5.86596e13 q^{76} +1.16659e13 q^{77} +5.33619e13 q^{79} +3.82401e13 q^{80} +1.88238e14 q^{82} -2.22672e14 q^{83} +1.27945e14 q^{85} -2.03518e14 q^{86} -2.77367e13 q^{88} -3.77610e14 q^{89} +2.79246e14 q^{91} -4.06981e14 q^{92} +2.60708e13 q^{94} -5.10033e14 q^{95} +1.25521e15 q^{97} -5.08103e14 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 256 q^{2} + 32768 q^{4} - 71472 q^{5} - 896378 q^{7} + 4194304 q^{8} - 9148416 q^{10} + 21628848 q^{11} - 151348262 q^{13} - 114736384 q^{14} + 536870912 q^{16} - 71108496 q^{17} + 1418861578 q^{19}+ \cdots - 11\!\cdots\!28 q^{98}+O(q^{100}) \) 2 * q + 256 * q^2 + 32768 * q^4 - 71472 * q^5 - 896378 * q^7 + 4194304 * q^8 - 9148416 * q^10 + 21628848 * q^11 - 151348262 * q^13 - 114736384 * q^14 + 536870912 * q^16 - 71108496 * q^17 + 1418861578 * q^19 - 1170997248 * q^20 + 2768492544 * q^22 - 22216171152 * q^23 + 5023241894 * q^25 - 19372577536 * q^26 - 14686257152 * q^28 - 69636203520 * q^29 - 187701744632 * q^31 + 68719476736 * q^32 - 9101887488 * q^34 - 122586139536 * q^35 - 721738522490 * q^37 + 181614281984 * q^38 - 149887647744 * q^40 + 1134127875840 * q^41 - 735115869968 * q^43 + 354367045632 * q^44 - 2843669907456 * q^46 - 1143668814288 * q^47 - 8716912355676 * q^49 + 642974962432 * q^50 - 2479689924608 * q^52 - 7010634953376 * q^53 - 9340483492800 * q^55 - 1879840915456 * q^56 - 8913434050560 * q^58 - 4464695524944 * q^59 - 25346208518402 * q^61 - 24025823312896 * q^62 + 8796093022208 * q^64 - 80449197973872 * q^65 - 36471044633486 * q^67 - 1165041598464 * q^68 - 15691025860608 * q^70 - 32664609209376 * q^71 + 57989177090770 * q^73 - 92382530878720 * q^74 + 23246628093952 * q^76 + 11166322109712 * q^77 + 278861646180442 * q^79 - 19185618911232 * q^80 + 145168368107520 * q^82 - 493886855787936 * q^83 + 335295249747264 * q^85 - 94094831355904 * q^86 + 45358981840896 * q^88 - 1075161917501136 * q^89 + 276877654459406 * q^91 - 363989748154368 * q^92 - 146389608228864 * q^94 - 1579489781398512 * q^95 + 287158074477898 * q^97 - 1115764781526528 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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