LMFDB - Embedded newform 7728.2.a.cg.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 7x^{3} + 31x^{2} - 17x + 2 $ x^6 - x^5 - 13x^4 + 7x^3 + 31x^2 - 17x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.82561$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.74592 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.74592 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.83141 q^{11} +2.46385 q^{13} -1.74592 q^{15} +1.28206 q^{17} +8.03516 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.95177 q^{25} -1.00000 q^{27} +9.64110 q^{29} +0.717936 q^{31} -1.83141 q^{33} +1.74592 q^{35} +9.64110 q^{37} -2.46385 q^{39} +0.887999 q^{41} -10.8386 q^{43} +1.74592 q^{45} -12.0134 q^{47} +1.00000 q^{49} -1.28206 q^{51} +7.55561 q^{53} +3.19749 q^{55} -8.03516 q^{57} +5.55561 q^{59} +7.02890 q^{61} +1.00000 q^{63} +4.30169 q^{65} +6.18117 q^{67} +1.00000 q^{69} +3.72420 q^{71} +7.20585 q^{73} +1.95177 q^{75} +1.83141 q^{77} -11.0927 q^{79} +1.00000 q^{81} -4.20977 q^{83} +2.23838 q^{85} -9.64110 q^{87} -12.0989 q^{89} +2.46385 q^{91} -0.717936 q^{93} +14.0287 q^{95} +8.22547 q^{97} +1.83141 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9 - 5 * q^11 + 2 * q^13 + 10 * q^17 - 3 * q^19 - 6 * q^21 - 6 * q^23 + 18 * q^25 - 6 * q^27 + 3 * q^29 + 2 * q^31 + 5 * q^33 + 3 * q^37 - 2 * q^39 + 4 * q^41 - 6 * q^43 + 2 * q^47 + 6 * q^49 - 10 * q^51 - 4 * q^53 + 15 * q^55 + 3 * q^57 - 16 * q^59 + 22 * q^61 + 6 * q^63 + 35 * q^65 - 9 * q^67 + 6 * q^69 - 11 * q^71 + 24 * q^73 - 18 * q^75 - 5 * q^77 - 18 * q^79 + 6 * q^81 - 2 * q^83 + 13 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 - 2 * q^93 - 5 * q^95 + 37 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.74592	0.780798	0.390399	−	0.920646i	$-0.372337\pi$
$5$	1.74592	0.780798	0.390399	+	0.920646i	$0.372337\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.83141	0.552191	0.276095	−	0.961130i	$-0.410959\pi$
$11$	1.83141	0.552191	0.276095	+	0.961130i	$0.410959\pi$
$12$	0	0
$13$	2.46385	0.683350	0.341675	−	0.939818i	$-0.389006\pi$
$13$	2.46385	0.683350	0.341675	+	0.939818i	$0.389006\pi$
$14$	0	0
$15$	−1.74592	−0.450794
$16$	0	0
$17$	1.28206	0.310946	0.155473	−	0.987840i	$-0.450310\pi$
$17$	1.28206	0.310946	0.155473	+	0.987840i	$0.450310\pi$
$18$	0	0
$19$	8.03516	1.84339	0.921696	−	0.387912i	$-0.126804\pi$
$19$	8.03516	1.84339	0.921696	+	0.387912i	$0.126804\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.95177	−0.390354
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.64110	1.79031	0.895153	−	0.445758i	$-0.147066\pi$
$29$	9.64110	1.79031	0.895153	+	0.445758i	$0.147066\pi$
$30$	0	0
$31$	0.717936	0.128945	0.0644725	−	0.997919i	$-0.479464\pi$
$31$	0.717936	0.128945	0.0644725	+	0.997919i	$0.479464\pi$
$32$	0	0
$33$	−1.83141	−0.318808
$34$	0	0
$35$	1.74592	0.295114
$36$	0	0
$37$	9.64110	1.58499	0.792493	−	0.609881i	$-0.208783\pi$
$37$	9.64110	1.58499	0.792493	+	0.609881i	$0.208783\pi$
$38$	0	0
$39$	−2.46385	−0.394532
$40$	0	0
$41$	0.887999	0.138682	0.0693411	−	0.997593i	$-0.477910\pi$
$41$	0.887999	0.138682	0.0693411	+	0.997593i	$0.477910\pi$
$42$	0	0
$43$	−10.8386	−1.65287	−0.826435	−	0.563033i	$-0.809635\pi$
$43$	−10.8386	−1.65287	−0.826435	+	0.563033i	$0.809635\pi$
$44$	0	0
$45$	1.74592	0.260266
$46$	0	0
$47$	−12.0134	−1.75234	−0.876170	−	0.482002i	$-0.839910\pi$
$47$	−12.0134	−1.75234	−0.876170	+	0.482002i	$0.839910\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.28206	−0.179525
$52$	0	0
$53$	7.55561	1.03784	0.518921	−	0.854822i	$-0.326334\pi$
$53$	7.55561	1.03784	0.518921	+	0.854822i	$0.326334\pi$
$54$	0	0
$55$	3.19749	0.431150
$56$	0	0
$57$	−8.03516	−1.06428
$58$	0	0
$59$	5.55561	0.723278	0.361639	−	0.932318i	$-0.382217\pi$
$59$	5.55561	0.723278	0.361639	+	0.932318i	$0.382217\pi$
$60$	0	0
$61$	7.02890	0.899959	0.449979	−	0.893039i	$-0.351431\pi$
$61$	7.02890	0.899959	0.449979	+	0.893039i	$0.351431\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	4.30169	0.533559
$66$	0	0
$67$	6.18117	0.755150	0.377575	−	0.925979i	$-0.376758\pi$
$67$	6.18117	0.755150	0.377575	+	0.925979i	$0.376758\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	3.72420	0.441981	0.220990	−	0.975276i	$-0.429071\pi$
$71$	3.72420	0.441981	0.220990	+	0.975276i	$0.429071\pi$
$72$	0	0
$73$	7.20585	0.843381	0.421691	−	0.906740i	$-0.361437\pi$
$73$	7.20585	0.843381	0.421691	+	0.906740i	$0.361437\pi$
$74$	0	0
$75$	1.95177	0.225371
$76$	0	0
$77$	1.83141	0.208709
$78$	0	0
$79$	−11.0927	−1.24802	−0.624012	−	0.781415i	$-0.714498\pi$
$79$	−11.0927	−1.24802	−0.624012	+	0.781415i	$0.714498\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.20977	−0.462083	−0.231041	−	0.972944i	$-0.574213\pi$
$83$	−4.20977	−0.462083	−0.231041	+	0.972944i	$0.574213\pi$
$84$	0	0
$85$	2.23838	0.242786
$86$	0	0
$87$	−9.64110	−1.03363
$88$	0	0
$89$	−12.0989	−1.28248	−0.641242	−	0.767339i	$-0.721581\pi$
$89$	−12.0989	−1.28248	−0.641242	+	0.767339i	$0.721581\pi$
$90$	0	0
$91$	2.46385	0.258282
$92$	0	0
$93$	−0.717936	−0.0744465
$94$	0	0
$95$	14.0287	1.43932
$96$	0	0
$97$	8.22547	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$97$	8.22547	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$98$	0	0
$99$	1.83141	0.184064
$100$	0	0
$101$	−15.3662	−1.52900	−0.764498	−	0.644626i	$-0.777013\pi$
$101$	−15.3662	−1.52900	−0.764498	+	0.644626i	$0.777013\pi$
$102$	0	0
$103$	1.63871	0.161466	0.0807332	−	0.996736i	$-0.474274\pi$
$103$	1.63871	0.161466	0.0807332	+	0.996736i	$0.474274\pi$
$104$	0	0
$105$	−1.74592	−0.170384
$106$	0	0
$107$	−8.23236	−0.795852	−0.397926	−	0.917417i	$-0.630270\pi$
$107$	−8.23236	−0.795852	−0.397926	+	0.917417i	$0.630270\pi$
$108$	0	0
$109$	−17.9872	−1.72286	−0.861432	−	0.507873i	$-0.830432\pi$
$109$	−17.9872	−1.72286	−0.861432	+	0.507873i	$0.830432\pi$
$110$	0	0
$111$	−9.64110	−0.915092
$112$	0	0
$113$	10.3816	0.976621	0.488310	−	0.872670i	$-0.337613\pi$
$113$	10.3816	0.976621	0.488310	+	0.872670i	$0.337613\pi$
$114$	0	0
$115$	−1.74592	−0.162808
$116$	0	0
$117$	2.46385	0.227783
$118$	0	0
$119$	1.28206	0.117527
$120$	0	0
$121$	−7.64594	−0.695085
$122$	0	0
$123$	−0.887999	−0.0800682
$124$	0	0
$125$	−12.1372	−1.08559
$126$	0	0
$127$	−2.36546	−0.209900	−0.104950	−	0.994477i	$-0.533468\pi$
$127$	−2.36546	−0.209900	−0.104950	+	0.994477i	$0.533468\pi$
$128$	0	0
$129$	10.8386	0.954285
$130$	0	0
$131$	−11.2238	−0.980631	−0.490316	−	0.871545i	$-0.663118\pi$
$131$	−11.2238	−0.980631	−0.490316	+	0.871545i	$0.663118\pi$
$132$	0	0
$133$	8.03516	0.696737
$134$	0	0
$135$	−1.74592	−0.150265
$136$	0	0
$137$	18.3942	1.57152	0.785761	−	0.618530i	$-0.212272\pi$
$137$	18.3942	1.57152	0.785761	+	0.618530i	$0.212272\pi$
$138$	0	0
$139$	−11.4087	−0.967677	−0.483838	−	0.875157i	$-0.660758\pi$
$139$	−11.4087	−0.967677	−0.483838	+	0.875157i	$0.660758\pi$
$140$	0	0
$141$	12.0134	1.01171
$142$	0	0
$143$	4.51233	0.377340
$144$	0	0
$145$	16.8326	1.39787
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	7.09028	0.580858	0.290429	−	0.956897i	$-0.406202\pi$
$149$	7.09028	0.580858	0.290429	+	0.956897i	$0.406202\pi$
$150$	0	0
$151$	12.7934	1.04111	0.520554	−	0.853828i	$-0.325725\pi$
$151$	12.7934	1.04111	0.520554	+	0.853828i	$0.325725\pi$
$152$	0	0
$153$	1.28206	0.103649
$154$	0	0
$155$	1.25346	0.100680
$156$	0	0
$157$	18.7531	1.49666	0.748330	−	0.663327i	$-0.230856\pi$
$157$	18.7531	1.49666	0.748330	+	0.663327i	$0.230856\pi$
$158$	0	0
$159$	−7.55561	−0.599199
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	13.6694	1.07067	0.535334	−	0.844640i	$-0.320186\pi$
$163$	13.6694	1.07067	0.535334	+	0.844640i	$0.320186\pi$
$164$	0	0
$165$	−3.19749	−0.248924
$166$	0	0
$167$	12.0352	0.931309	0.465654	−	0.884967i	$-0.345819\pi$
$167$	12.0352	0.931309	0.465654	+	0.884967i	$0.345819\pi$
$168$	0	0
$169$	−6.92942	−0.533033
$170$	0	0
$171$	8.03516	0.614464
$172$	0	0
$173$	−12.4367	−0.945546	−0.472773	−	0.881184i	$-0.656747\pi$
$173$	−12.4367	−0.945546	−0.472773	+	0.881184i	$0.656747\pi$
$174$	0	0
$175$	−1.95177	−0.147540
$176$	0	0
$177$	−5.55561	−0.417585
$178$	0	0
$179$	−2.48103	−0.185441	−0.0927204	−	0.995692i	$-0.529556\pi$
$179$	−2.48103	−0.185441	−0.0927204	+	0.995692i	$0.529556\pi$
$180$	0	0
$181$	4.96513	0.369055	0.184528	−	0.982827i	$-0.440924\pi$
$181$	4.96513	0.369055	0.184528	+	0.982827i	$0.440924\pi$
$182$	0	0
$183$	−7.02890	−0.519591
$184$	0	0
$185$	16.8326	1.23756
$186$	0	0
$187$	2.34798	0.171702
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	12.5249	0.906270	0.453135	−	0.891442i	$-0.350306\pi$
$191$	12.5249	0.906270	0.453135	+	0.891442i	$0.350306\pi$
$192$	0	0
$193$	2.71100	0.195142	0.0975709	−	0.995229i	$-0.468893\pi$
$193$	2.71100	0.195142	0.0975709	+	0.995229i	$0.468893\pi$
$194$	0	0
$195$	−4.30169	−0.308050
$196$	0	0
$197$	−27.6698	−1.97139	−0.985697	−	0.168527i	$-0.946099\pi$
$197$	−27.6698	−1.97139	−0.985697	+	0.168527i	$0.946099\pi$
$198$	0	0
$199$	−18.2149	−1.29122	−0.645608	−	0.763669i	$-0.723396\pi$
$199$	−18.2149	−1.29122	−0.645608	+	0.763669i	$0.723396\pi$
$200$	0	0
$201$	−6.18117	−0.435986
$202$	0	0
$203$	9.64110	0.676672
$204$	0	0
$205$	1.55037	0.108283
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	14.7157	1.01790
$210$	0	0
$211$	−1.83141	−0.126079	−0.0630397	−	0.998011i	$-0.520079\pi$
$211$	−1.83141	−0.126079	−0.0630397	+	0.998011i	$0.520079\pi$
$212$	0	0
$213$	−3.72420	−0.255178
$214$	0	0
$215$	−18.9233	−1.29056
$216$	0	0
$217$	0.717936	0.0487367
$218$	0	0
$219$	−7.20585	−0.486926
$220$	0	0
$221$	3.15882	0.212485
$222$	0	0
$223$	13.2923	0.890116	0.445058	−	0.895502i	$-0.353183\pi$
$223$	13.2923	0.890116	0.445058	+	0.895502i	$0.353183\pi$
$224$	0	0
$225$	−1.95177	−0.130118
$226$	0	0
$227$	3.81429	0.253163	0.126582	−	0.991956i	$-0.459599\pi$
$227$	3.81429	0.253163	0.126582	+	0.991956i	$0.459599\pi$
$228$	0	0
$229$	−17.2142	−1.13755	−0.568774	−	0.822494i	$-0.692582\pi$
$229$	−17.2142	−1.13755	−0.568774	+	0.822494i	$0.692582\pi$
$230$	0	0
$231$	−1.83141	−0.120498
$232$	0	0
$233$	4.87235	0.319198	0.159599	−	0.987182i	$-0.448980\pi$
$233$	4.87235	0.319198	0.159599	+	0.987182i	$0.448980\pi$
$234$	0	0
$235$	−20.9745	−1.36822
$236$	0	0
$237$	11.0927	0.720546
$238$	0	0
$239$	11.6170	0.751444	0.375722	−	0.926732i	$-0.377395\pi$
$239$	11.6170	0.751444	0.375722	+	0.926732i	$0.377395\pi$
$240$	0	0
$241$	2.50122	0.161118	0.0805590	−	0.996750i	$-0.474329\pi$
$241$	2.50122	0.161118	0.0805590	+	0.996750i	$0.474329\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.74592	0.111543
$246$	0	0
$247$	19.7975	1.25968
$248$	0	0
$249$	4.20977	0.266783
$250$	0	0
$251$	−3.54031	−0.223462	−0.111731	−	0.993738i	$-0.535640\pi$
$251$	−3.54031	−0.223462	−0.111731	+	0.993738i	$0.535640\pi$
$252$	0	0
$253$	−1.83141	−0.115140
$254$	0	0
$255$	−2.23838	−0.140173
$256$	0	0
$257$	−14.8621	−0.927071	−0.463536	−	0.886078i	$-0.653419\pi$
$257$	−14.8621	−0.927071	−0.463536	+	0.886078i	$0.653419\pi$
$258$	0	0
$259$	9.64110	0.599069
$260$	0	0
$261$	9.64110	0.596769
$262$	0	0
$263$	−26.4086	−1.62842	−0.814211	−	0.580570i	$-0.802830\pi$
$263$	−26.4086	−1.62842	−0.814211	+	0.580570i	$0.802830\pi$
$264$	0	0
$265$	13.1915	0.810346
$266$	0	0
$267$	12.0989	0.740443
$268$	0	0
$269$	13.5348	0.825232	0.412616	−	0.910905i	$-0.364615\pi$
$269$	13.5348	0.825232	0.412616	+	0.910905i	$0.364615\pi$
$270$	0	0
$271$	−2.43672	−0.148020	−0.0740101	−	0.997257i	$-0.523580\pi$
$271$	−2.43672	−0.148020	−0.0740101	+	0.997257i	$0.523580\pi$
$272$	0	0
$273$	−2.46385	−0.149119
$274$	0	0
$275$	−3.57449	−0.215550
$276$	0	0
$277$	−3.95661	−0.237730	−0.118865	−	0.992910i	$-0.537926\pi$
$277$	−3.95661	−0.237730	−0.118865	+	0.992910i	$0.537926\pi$
$278$	0	0
$279$	0.717936	0.0429817
$280$	0	0
$281$	−30.3610	−1.81118	−0.905592	−	0.424149i	$-0.860573\pi$
$281$	−30.3610	−1.81118	−0.905592	+	0.424149i	$0.860573\pi$
$282$	0	0
$283$	6.68285	0.397254	0.198627	−	0.980075i	$-0.436352\pi$
$283$	6.68285	0.397254	0.198627	+	0.980075i	$0.436352\pi$
$284$	0	0
$285$	−14.0287	−0.830991
$286$	0	0
$287$	0.887999	0.0524169
$288$	0	0
$289$	−15.3563	−0.903312
$290$	0	0
$291$	−8.22547	−0.482186
$292$	0	0
$293$	29.3960	1.71733	0.858665	−	0.512537i	$-0.171294\pi$
$293$	29.3960	1.71733	0.858665	+	0.512537i	$0.171294\pi$
$294$	0	0
$295$	9.69964	0.564735
$296$	0	0
$297$	−1.83141	−0.106269
$298$	0	0
$299$	−2.46385	−0.142488
$300$	0	0
$301$	−10.8386	−0.624726
$302$	0	0
$303$	15.3662	0.882766
$304$	0	0
$305$	12.2719	0.702686
$306$	0	0
$307$	−19.0757	−1.08871	−0.544354	−	0.838855i	$-0.683225\pi$
$307$	−19.0757	−1.08871	−0.544354	+	0.838855i	$0.683225\pi$
$308$	0	0
$309$	−1.63871	−0.0932227
$310$	0	0
$311$	29.9364	1.69754	0.848768	−	0.528766i	$-0.177345\pi$
$311$	29.9364	1.69754	0.848768	+	0.528766i	$0.177345\pi$
$312$	0	0
$313$	−17.1230	−0.967852	−0.483926	−	0.875109i	$-0.660790\pi$
$313$	−17.1230	−0.967852	−0.483926	+	0.875109i	$0.660790\pi$
$314$	0	0
$315$	1.74592	0.0983714
$316$	0	0
$317$	6.38162	0.358428	0.179214	−	0.983810i	$-0.442645\pi$
$317$	6.38162	0.358428	0.179214	+	0.983810i	$0.442645\pi$
$318$	0	0
$319$	17.6568	0.988591
$320$	0	0
$321$	8.23236	0.459486
$322$	0	0
$323$	10.3016	0.573196
$324$	0	0
$325$	−4.80887	−0.266748
$326$	0	0
$327$	17.9872	0.994696
$328$	0	0
$329$	−12.0134	−0.662322
$330$	0	0
$331$	−11.1167	−0.611028	−0.305514	−	0.952188i	$-0.598828\pi$
$331$	−11.1167	−0.611028	−0.305514	+	0.952188i	$0.598828\pi$
$332$	0	0
$333$	9.64110	0.528329
$334$	0	0
$335$	10.7918	0.589620
$336$	0	0
$337$	14.7405	0.802967	0.401484	−	0.915866i	$-0.368495\pi$
$337$	14.7405	0.802967	0.401484	+	0.915866i	$0.368495\pi$
$338$	0	0
$339$	−10.3816	−0.563852
$340$	0	0
$341$	1.31483	0.0712023
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.74592	0.0939971
$346$	0	0
$347$	11.8091	0.633948	0.316974	−	0.948434i	$-0.397333\pi$
$347$	11.8091	0.633948	0.316974	+	0.948434i	$0.397333\pi$
$348$	0	0
$349$	1.35548	0.0725572	0.0362786	−	0.999342i	$-0.488450\pi$
$349$	1.35548	0.0725572	0.0362786	+	0.999342i	$0.488450\pi$
$350$	0	0
$351$	−2.46385	−0.131511
$352$	0	0
$353$	−10.1493	−0.540191	−0.270095	−	0.962834i	$-0.587055\pi$
$353$	−10.1493	−0.540191	−0.270095	+	0.962834i	$0.587055\pi$
$354$	0	0
$355$	6.50214	0.345098
$356$	0	0
$357$	−1.28206	−0.0678540
$358$	0	0
$359$	−23.9252	−1.26273	−0.631363	−	0.775488i	$-0.717504\pi$
$359$	−23.9252	−1.26273	−0.631363	+	0.775488i	$0.717504\pi$
$360$	0	0
$361$	45.5638	2.39810
$362$	0	0
$363$	7.64594	0.401308
$364$	0	0
$365$	12.5808	0.658511
$366$	0	0
$367$	1.61782	0.0844498	0.0422249	−	0.999108i	$-0.486555\pi$
$367$	1.61782	0.0844498	0.0422249	+	0.999108i	$0.486555\pi$
$368$	0	0
$369$	0.887999	0.0462274
$370$	0	0
$371$	7.55561	0.392268
$372$	0	0
$373$	32.6983	1.69305	0.846526	−	0.532347i	$-0.178690\pi$
$373$	32.6983	1.69305	0.846526	+	0.532347i	$0.178690\pi$
$374$	0	0
$375$	12.1372	0.626763
$376$	0	0
$377$	23.7543	1.22341
$378$	0	0
$379$	−28.6373	−1.47100	−0.735500	−	0.677525i	$-0.763053\pi$
$379$	−28.6373	−1.47100	−0.735500	+	0.677525i	$0.763053\pi$
$380$	0	0
$381$	2.36546	0.121186
$382$	0	0
$383$	35.7673	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$383$	35.7673	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$384$	0	0
$385$	3.19749	0.162959
$386$	0	0
$387$	−10.8386	−0.550956
$388$	0	0
$389$	−23.7843	−1.20591	−0.602957	−	0.797774i	$-0.706011\pi$
$389$	−23.7843	−1.20591	−0.602957	+	0.797774i	$0.706011\pi$
$390$	0	0
$391$	−1.28206	−0.0648368
$392$	0	0
$393$	11.2238	0.566168
$394$	0	0
$395$	−19.3669	−0.974454
$396$	0	0
$397$	19.8094	0.994203	0.497102	−	0.867692i	$-0.334398\pi$
$397$	19.8094	0.994203	0.497102	+	0.867692i	$0.334398\pi$
$398$	0	0
$399$	−8.03516	−0.402261
$400$	0	0
$401$	−0.790522	−0.0394768	−0.0197384	−	0.999805i	$-0.506283\pi$
$401$	−0.790522	−0.0394768	−0.0197384	+	0.999805i	$0.506283\pi$
$402$	0	0
$403$	1.76889	0.0881146
$404$	0	0
$405$	1.74592	0.0867554
$406$	0	0
$407$	17.6568	0.875215
$408$	0	0
$409$	−13.2232	−0.653843	−0.326922	−	0.945051i	$-0.606011\pi$
$409$	−13.2232	−0.653843	−0.326922	+	0.945051i	$0.606011\pi$
$410$	0	0
$411$	−18.3942	−0.907319
$412$	0	0
$413$	5.55561	0.273374
$414$	0	0
$415$	−7.34992	−0.360793
$416$	0	0
$417$	11.4087	0.558688
$418$	0	0
$419$	10.8538	0.530244	0.265122	−	0.964215i	$-0.414588\pi$
$419$	10.8538	0.530244	0.265122	+	0.964215i	$0.414588\pi$
$420$	0	0
$421$	16.5014	0.804229	0.402115	−	0.915589i	$-0.368275\pi$
$421$	16.5014	0.804229	0.402115	+	0.915589i	$0.368275\pi$
$422$	0	0
$423$	−12.0134	−0.584113
$424$	0	0
$425$	−2.50229	−0.121379
$426$	0	0
$427$	7.02890	0.340152
$428$	0	0
$429$	−4.51233	−0.217857
$430$	0	0
$431$	29.5387	1.42283	0.711414	−	0.702773i	$-0.248055\pi$
$431$	29.5387	1.42283	0.711414	+	0.702773i	$0.248055\pi$
$432$	0	0
$433$	41.0795	1.97415	0.987077	−	0.160247i	$-0.0512292\pi$
$433$	41.0795	1.97415	0.987077	+	0.160247i	$0.0512292\pi$
$434$	0	0
$435$	−16.8326	−0.807060
$436$	0	0
$437$	−8.03516	−0.384374
$438$	0	0
$439$	5.81663	0.277612	0.138806	−	0.990320i	$-0.455673\pi$
$439$	5.81663	0.277612	0.138806	+	0.990320i	$0.455673\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	6.56279	0.311807	0.155904	−	0.987772i	$-0.450171\pi$
$443$	6.56279	0.311807	0.155904	+	0.987772i	$0.450171\pi$
$444$	0	0
$445$	−21.1237	−1.00136
$446$	0	0
$447$	−7.09028	−0.335359
$448$	0	0
$449$	−28.5482	−1.34727	−0.673636	−	0.739063i	$-0.735268\pi$
$449$	−28.5482	−1.34727	−0.673636	+	0.739063i	$0.735268\pi$
$450$	0	0
$451$	1.62629	0.0765790
$452$	0	0
$453$	−12.7934	−0.601085
$454$	0	0
$455$	4.30169	0.201666
$456$	0	0
$457$	40.7468	1.90606	0.953028	−	0.302884i	$-0.0979493\pi$
$457$	40.7468	1.90606	0.953028	+	0.302884i	$0.0979493\pi$
$458$	0	0
$459$	−1.28206	−0.0598416
$460$	0	0
$461$	−38.2576	−1.78184	−0.890918	−	0.454164i	$-0.849938\pi$
$461$	−38.2576	−1.78184	−0.890918	+	0.454164i	$0.849938\pi$
$462$	0	0
$463$	−33.6234	−1.56261	−0.781305	−	0.624149i	$-0.785446\pi$
$463$	−33.6234	−1.56261	−0.781305	+	0.624149i	$0.785446\pi$
$464$	0	0
$465$	−1.25346	−0.0581277
$466$	0	0
$467$	−23.7179	−1.09753	−0.548767	−	0.835975i	$-0.684902\pi$
$467$	−23.7179	−1.09753	−0.548767	+	0.835975i	$0.684902\pi$
$468$	0	0
$469$	6.18117	0.285420
$470$	0	0
$471$	−18.7531	−0.864097
$472$	0	0
$473$	−19.8499	−0.912699
$474$	0	0
$475$	−15.6828	−0.719575
$476$	0	0
$477$	7.55561	0.345947
$478$	0	0
$479$	0.736346	0.0336445	0.0168223	−	0.999858i	$-0.494645\pi$
$479$	0.736346	0.0336445	0.0168223	+	0.999858i	$0.494645\pi$
$480$	0	0
$481$	23.7543	1.08310
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	14.3610	0.652100
$486$	0	0
$487$	2.88253	0.130620	0.0653100	−	0.997865i	$-0.479196\pi$
$487$	2.88253	0.130620	0.0653100	+	0.997865i	$0.479196\pi$
$488$	0	0
$489$	−13.6694	−0.618150
$490$	0	0
$491$	−17.7386	−0.800530	−0.400265	−	0.916399i	$-0.631082\pi$
$491$	−17.7386	−0.800530	−0.400265	+	0.916399i	$0.631082\pi$
$492$	0	0
$493$	12.3605	0.556689
$494$	0	0
$495$	3.19749	0.143717
$496$	0	0
$497$	3.72420	0.167053
$498$	0	0
$499$	−3.21126	−0.143756	−0.0718779	−	0.997413i	$-0.522899\pi$
$499$	−3.21126	−0.143756	−0.0718779	+	0.997413i	$0.522899\pi$
$500$	0	0
$501$	−12.0352	−0.537691
$502$	0	0
$503$	17.4226	0.776835	0.388418	−	0.921483i	$-0.373022\pi$
$503$	17.4226	0.776835	0.388418	+	0.921483i	$0.373022\pi$
$504$	0	0
$505$	−26.8282	−1.19384
$506$	0	0
$507$	6.92942	0.307746
$508$	0	0
$509$	11.0266	0.488745	0.244372	−	0.969681i	$-0.421418\pi$
$509$	11.0266	0.488745	0.244372	+	0.969681i	$0.421418\pi$
$510$	0	0
$511$	7.20585	0.318768
$512$	0	0
$513$	−8.03516	−0.354761
$514$	0	0
$515$	2.86105	0.126073
$516$	0	0
$517$	−22.0015	−0.967626
$518$	0	0
$519$	12.4367	0.545911
$520$	0	0
$521$	−16.6281	−0.728490	−0.364245	−	0.931303i	$-0.618673\pi$
$521$	−16.6281	−0.728490	−0.364245	+	0.931303i	$0.618673\pi$
$522$	0	0
$523$	27.9530	1.22230	0.611151	−	0.791514i	$-0.290707\pi$
$523$	27.9530	1.22230	0.611151	+	0.791514i	$0.290707\pi$
$524$	0	0
$525$	1.95177	0.0851822
$526$	0	0
$527$	0.920440	0.0400950
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	5.55561	0.241093
$532$	0	0
$533$	2.18790	0.0947685
$534$	0	0
$535$	−14.3730	−0.621400
$536$	0	0
$537$	2.48103	0.107064
$538$	0	0
$539$	1.83141	0.0788844
$540$	0	0
$541$	−2.76514	−0.118883	−0.0594413	−	0.998232i	$-0.518932\pi$
$541$	−2.76514	−0.118883	−0.0594413	+	0.998232i	$0.518932\pi$
$542$	0	0
$543$	−4.96513	−0.213074
$544$	0	0
$545$	−31.4042	−1.34521
$546$	0	0
$547$	33.9077	1.44979	0.724895	−	0.688860i	$-0.241888\pi$
$547$	33.9077	1.44979	0.724895	+	0.688860i	$0.241888\pi$
$548$	0	0
$549$	7.02890	0.299986
$550$	0	0
$551$	77.4678	3.30024
$552$	0	0
$553$	−11.0927	−0.471708
$554$	0	0
$555$	−16.8326	−0.714503
$556$	0	0
$557$	8.80544	0.373098	0.186549	−	0.982446i	$-0.440270\pi$
$557$	8.80544	0.373098	0.186549	+	0.982446i	$0.440270\pi$
$558$	0	0
$559$	−26.7047	−1.12949
$560$	0	0
$561$	−2.34798	−0.0991320
$562$	0	0
$563$	24.8576	1.04762	0.523811	−	0.851834i	$-0.324510\pi$
$563$	24.8576	1.04762	0.523811	+	0.851834i	$0.324510\pi$
$564$	0	0
$565$	18.1255	0.762544
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	27.2924	1.14416	0.572079	−	0.820198i	$-0.306137\pi$
$569$	27.2924	1.14416	0.572079	+	0.820198i	$0.306137\pi$
$570$	0	0
$571$	16.9777	0.710493	0.355246	−	0.934773i	$-0.384397\pi$
$571$	16.9777	0.710493	0.355246	+	0.934773i	$0.384397\pi$
$572$	0	0
$573$	−12.5249	−0.523235
$574$	0	0
$575$	1.95177	0.0813944
$576$	0	0
$577$	16.3498	0.680651	0.340326	−	0.940308i	$-0.389463\pi$
$577$	16.3498	0.680651	0.340326	+	0.940308i	$0.389463\pi$
$578$	0	0
$579$	−2.71100	−0.112665
$580$	0	0
$581$	−4.20977	−0.174651
$582$	0	0
$583$	13.8374	0.573087
$584$	0	0
$585$	4.30169	0.177853
$586$	0	0
$587$	35.1986	1.45280	0.726402	−	0.687271i	$-0.241191\pi$
$587$	35.1986	1.45280	0.726402	+	0.687271i	$0.241191\pi$
$588$	0	0
$589$	5.76873	0.237696
$590$	0	0
$591$	27.6698	1.13818
$592$	0	0
$593$	−27.1131	−1.11340	−0.556702	−	0.830712i	$-0.687933\pi$
$593$	−27.1131	−1.11340	−0.556702	+	0.830712i	$0.687933\pi$
$594$	0	0
$595$	2.23838	0.0917646
$596$	0	0
$597$	18.2149	0.745484
$598$	0	0
$599$	4.63630	0.189434	0.0947170	−	0.995504i	$-0.469805\pi$
$599$	4.63630	0.189434	0.0947170	+	0.995504i	$0.469805\pi$
$600$	0	0
$601$	31.3810	1.28006	0.640029	−	0.768351i	$-0.278922\pi$
$601$	31.3810	1.28006	0.640029	+	0.768351i	$0.278922\pi$
$602$	0	0
$603$	6.18117	0.251717
$604$	0	0
$605$	−13.3492	−0.542722
$606$	0	0
$607$	−18.9694	−0.769945	−0.384973	−	0.922928i	$-0.625789\pi$
$607$	−18.9694	−0.769945	−0.384973	+	0.922928i	$0.625789\pi$
$608$	0	0
$609$	−9.64110	−0.390677
$610$	0	0
$611$	−29.5994	−1.19746
$612$	0	0
$613$	3.16559	0.127857	0.0639284	−	0.997954i	$-0.479637\pi$
$613$	3.16559	0.127857	0.0639284	+	0.997954i	$0.479637\pi$
$614$	0	0
$615$	−1.55037	−0.0625171
$616$	0	0
$617$	20.5868	0.828795	0.414398	−	0.910096i	$-0.363992\pi$
$617$	20.5868	0.828795	0.414398	+	0.910096i	$0.363992\pi$
$618$	0	0
$619$	27.6153	1.10995	0.554977	−	0.831866i	$-0.312727\pi$
$619$	27.6153	1.10995	0.554977	+	0.831866i	$0.312727\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−12.0989	−0.484733
$624$	0	0
$625$	−11.4318	−0.457270
$626$	0	0
$627$	−14.7157	−0.587687
$628$	0	0
$629$	12.3605	0.492846
$630$	0	0
$631$	38.3147	1.52528	0.762642	−	0.646821i	$-0.223902\pi$
$631$	38.3147	1.52528	0.762642	+	0.646821i	$0.223902\pi$
$632$	0	0
$633$	1.83141	0.0727920
$634$	0	0
$635$	−4.12990	−0.163890
$636$	0	0
$637$	2.46385	0.0976215
$638$	0	0
$639$	3.72420	0.147327
$640$	0	0
$641$	−27.7566	−1.09632	−0.548161	−	0.836373i	$-0.684672\pi$
$641$	−27.7566	−1.09632	−0.548161	+	0.836373i	$0.684672\pi$
$642$	0	0
$643$	13.6258	0.537349	0.268675	−	0.963231i	$-0.413414\pi$
$643$	13.6258	0.537349	0.268675	+	0.963231i	$0.413414\pi$
$644$	0	0
$645$	18.9233	0.745104
$646$	0	0
$647$	−12.8544	−0.505359	−0.252680	−	0.967550i	$-0.581312\pi$
$647$	−12.8544	−0.505359	−0.252680	+	0.967550i	$0.581312\pi$
$648$	0	0
$649$	10.1746	0.399388
$650$	0	0
$651$	−0.717936	−0.0281381
$652$	0	0
$653$	−46.3496	−1.81380	−0.906899	−	0.421348i	$-0.861557\pi$
$653$	−46.3496	−1.81380	−0.906899	+	0.421348i	$0.861557\pi$
$654$	0	0
$655$	−19.5959	−0.765675
$656$	0	0
$657$	7.20585	0.281127
$658$	0	0
$659$	6.60306	0.257219	0.128609	−	0.991695i	$-0.458949\pi$
$659$	6.60306	0.257219	0.128609	+	0.991695i	$0.458949\pi$
$660$	0	0
$661$	27.2798	1.06106	0.530531	−	0.847666i	$-0.321993\pi$
$661$	27.2798	1.06106	0.530531	+	0.847666i	$0.321993\pi$
$662$	0	0
$663$	−3.15882	−0.122678
$664$	0	0
$665$	14.0287	0.544011
$666$	0	0
$667$	−9.64110	−0.373305
$668$	0	0
$669$	−13.2923	−0.513908
$670$	0	0
$671$	12.8728	0.496949
$672$	0	0
$673$	12.6026	0.485794	0.242897	−	0.970052i	$-0.421902\pi$
$673$	12.6026	0.485794	0.242897	+	0.970052i	$0.421902\pi$
$674$	0	0
$675$	1.95177	0.0751236
$676$	0	0
$677$	10.8091	0.415428	0.207714	−	0.978190i	$-0.433398\pi$
$677$	10.8091	0.415428	0.207714	+	0.978190i	$0.433398\pi$
$678$	0	0
$679$	8.22547	0.315665
$680$	0	0
$681$	−3.81429	−0.146164
$682$	0	0
$683$	14.3016	0.547235	0.273618	−	0.961839i	$-0.411780\pi$
$683$	14.3016	0.547235	0.273618	+	0.961839i	$0.411780\pi$
$684$	0	0
$685$	32.1148	1.22704
$686$	0	0
$687$	17.2142	0.656764
$688$	0	0
$689$	18.6159	0.709210
$690$	0	0
$691$	14.9041	0.566980	0.283490	−	0.958975i	$-0.408508\pi$
$691$	14.9041	0.566980	0.283490	+	0.958975i	$0.408508\pi$
$692$	0	0
$693$	1.83141	0.0695695
$694$	0	0
$695$	−19.9187	−0.755560
$696$	0	0
$697$	1.13847	0.0431227
$698$	0	0
$699$	−4.87235	−0.184289
$700$	0	0
$701$	−32.8318	−1.24004	−0.620019	−	0.784586i	$-0.712875\pi$
$701$	−32.8318	−1.24004	−0.620019	+	0.784586i	$0.712875\pi$
$702$	0	0
$703$	77.4678	2.92175
$704$	0	0
$705$	20.9745	0.789945
$706$	0	0
$707$	−15.3662	−0.577906
$708$	0	0
$709$	0.331769	0.0124598	0.00622992	−	0.999981i	$-0.498017\pi$
$709$	0.331769	0.0124598	0.00622992	+	0.999981i	$0.498017\pi$
$710$	0	0
$711$	−11.0927	−0.416008
$712$	0	0
$713$	−0.717936	−0.0268869
$714$	0	0
$715$	7.87815	0.294626
$716$	0	0
$717$	−11.6170	−0.433846
$718$	0	0
$719$	−40.6033	−1.51425	−0.757123	−	0.653272i	$-0.773396\pi$
$719$	−40.6033	−1.51425	−0.757123	+	0.653272i	$0.773396\pi$
$720$	0	0
$721$	1.63871	0.0610286
$722$	0	0
$723$	−2.50122	−0.0930215
$724$	0	0
$725$	−18.8172	−0.698853
$726$	0	0
$727$	49.8972	1.85059	0.925293	−	0.379254i	$-0.123819\pi$
$727$	49.8972	1.85059	0.925293	+	0.379254i	$0.123819\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.8958	−0.513954
$732$	0	0
$733$	−44.7426	−1.65261	−0.826303	−	0.563225i	$-0.809560\pi$
$733$	−44.7426	−1.65261	−0.826303	+	0.563225i	$0.809560\pi$
$734$	0	0
$735$	−1.74592	−0.0643992
$736$	0	0
$737$	11.3202	0.416987
$738$	0	0
$739$	38.7371	1.42497	0.712483	−	0.701689i	$-0.247570\pi$
$739$	38.7371	1.42497	0.712483	+	0.701689i	$0.247570\pi$
$740$	0	0
$741$	−19.7975	−0.727278
$742$	0	0
$743$	−33.3728	−1.22433	−0.612165	−	0.790730i	$-0.709701\pi$
$743$	−33.3728	−1.22433	−0.612165	+	0.790730i	$0.709701\pi$
$744$	0	0
$745$	12.3790	0.453533
$746$	0	0
$747$	−4.20977	−0.154028
$748$	0	0
$749$	−8.23236	−0.300804
$750$	0	0
$751$	−42.9022	−1.56552	−0.782761	−	0.622322i	$-0.786189\pi$
$751$	−42.9022	−1.56552	−0.782761	+	0.622322i	$0.786189\pi$
$752$	0	0
$753$	3.54031	0.129016
$754$	0	0
$755$	22.3362	0.812896
$756$	0	0
$757$	−51.5878	−1.87499	−0.937495	−	0.347998i	$-0.886862\pi$
$757$	−51.5878	−1.87499	−0.937495	+	0.347998i	$0.886862\pi$
$758$	0	0
$759$	1.83141	0.0664760
$760$	0	0
$761$	20.0107	0.725388	0.362694	−	0.931908i	$-0.381857\pi$
$761$	20.0107	0.725388	0.362694	+	0.931908i	$0.381857\pi$
$762$	0	0
$763$	−17.9872	−0.651181
$764$	0	0
$765$	2.23838	0.0809288
$766$	0	0
$767$	13.6882	0.494252
$768$	0	0
$769$	43.5805	1.57155	0.785776	−	0.618511i	$-0.212264\pi$
$769$	43.5805	1.57155	0.785776	+	0.618511i	$0.212264\pi$
$770$	0	0
$771$	14.8621	0.535245
$772$	0	0
$773$	−28.8981	−1.03939	−0.519697	−	0.854351i	$-0.673955\pi$
$773$	−28.8981	−1.03939	−0.519697	+	0.854351i	$0.673955\pi$
$774$	0	0
$775$	−1.40124	−0.0503342
$776$	0	0
$777$	−9.64110	−0.345872
$778$	0	0
$779$	7.13522	0.255646
$780$	0	0
$781$	6.82053	0.244058
$782$	0	0
$783$	−9.64110	−0.344545
$784$	0	0
$785$	32.7414	1.16859
$786$	0	0
$787$	−2.83165	−0.100937	−0.0504687	−	0.998726i	$-0.516072\pi$
$787$	−2.83165	−0.100937	−0.0504687	+	0.998726i	$0.516072\pi$
$788$	0	0
$789$	26.4086	0.940169
$790$	0	0
$791$	10.3816	0.369128
$792$	0	0
$793$	17.3182	0.614987
$794$	0	0
$795$	−13.1915	−0.467853
$796$	0	0
$797$	18.0468	0.639250	0.319625	−	0.947544i	$-0.396443\pi$
$797$	18.0468	0.639250	0.319625	+	0.947544i	$0.396443\pi$
$798$	0	0
$799$	−15.4020	−0.544884
$800$	0	0
$801$	−12.0989	−0.427495
$802$	0	0
$803$	13.1969	0.465707
$804$	0	0
$805$	−1.74592	−0.0615355
$806$	0	0
$807$	−13.5348	−0.476448
$808$	0	0
$809$	−19.1601	−0.673632	−0.336816	−	0.941570i	$-0.609350\pi$
$809$	−19.1601	−0.673632	−0.336816	+	0.941570i	$0.609350\pi$
$810$	0	0
$811$	2.91018	0.102190	0.0510952	−	0.998694i	$-0.483729\pi$
$811$	2.91018	0.102190	0.0510952	+	0.998694i	$0.483729\pi$
$812$	0	0
$813$	2.43672	0.0854595
$814$	0	0
$815$	23.8656	0.835976
$816$	0	0
$817$	−87.0898	−3.04689
$818$	0	0
$819$	2.46385	0.0860940
$820$	0	0
$821$	−14.0870	−0.491641	−0.245821	−	0.969315i	$-0.579057\pi$
$821$	−14.0870	−0.491641	−0.245821	+	0.969315i	$0.579057\pi$
$822$	0	0
$823$	−7.70069	−0.268429	−0.134215	−	0.990952i	$-0.542851\pi$
$823$	−7.70069	−0.268429	−0.134215	+	0.990952i	$0.542851\pi$
$824$	0	0
$825$	3.57449	0.124448
$826$	0	0
$827$	−19.5791	−0.680831	−0.340415	−	0.940275i	$-0.610568\pi$
$827$	−19.5791	−0.680831	−0.340415	+	0.940275i	$0.610568\pi$
$828$	0	0
$829$	49.7813	1.72898	0.864488	−	0.502654i	$-0.167643\pi$
$829$	49.7813	1.72898	0.864488	+	0.502654i	$0.167643\pi$
$830$	0	0
$831$	3.95661	0.137253
$832$	0	0
$833$	1.28206	0.0444209
$834$	0	0
$835$	21.0124	0.727164
$836$	0	0
$837$	−0.717936	−0.0248155
$838$	0	0
$839$	12.1281	0.418710	0.209355	−	0.977840i	$-0.432864\pi$
$839$	12.1281	0.418710	0.209355	+	0.977840i	$0.432864\pi$
$840$	0	0
$841$	63.9508	2.20520
$842$	0	0
$843$	30.3610	1.04569
$844$	0	0
$845$	−12.0982	−0.416191
$846$	0	0
$847$	−7.64594	−0.262718
$848$	0	0
$849$	−6.68285	−0.229355
$850$	0	0
$851$	−9.64110	−0.330493
$852$	0	0
$853$	−13.5523	−0.464024	−0.232012	−	0.972713i	$-0.574531\pi$
$853$	−13.5523	−0.464024	−0.232012	+	0.972713i	$0.574531\pi$
$854$	0	0
$855$	14.0287	0.479773
$856$	0	0
$857$	−37.2754	−1.27330	−0.636651	−	0.771152i	$-0.719681\pi$
$857$	−37.2754	−1.27330	−0.636651	+	0.771152i	$0.719681\pi$
$858$	0	0
$859$	−3.73449	−0.127419	−0.0637095	−	0.997968i	$-0.520293\pi$
$859$	−3.73449	−0.127419	−0.0637095	+	0.997968i	$0.520293\pi$
$860$	0	0
$861$	−0.887999	−0.0302629
$862$	0	0
$863$	−57.2597	−1.94914	−0.974572	−	0.224077i	$-0.928063\pi$
$863$	−57.2597	−1.94914	−0.974572	+	0.224077i	$0.928063\pi$
$864$	0	0
$865$	−21.7135	−0.738281
$866$	0	0
$867$	15.3563	0.521528
$868$	0	0
$869$	−20.3152	−0.689147
$870$	0	0
$871$	15.2295	0.516032
$872$	0	0
$873$	8.22547	0.278390
$874$	0	0
$875$	−12.1372	−0.410313
$876$	0	0
$877$	−5.44955	−0.184018	−0.0920091	−	0.995758i	$-0.529329\pi$
$877$	−5.44955	−0.184018	−0.0920091	+	0.995758i	$0.529329\pi$
$878$	0	0
$879$	−29.3960	−0.991501
$880$	0	0
$881$	−17.7267	−0.597227	−0.298613	−	0.954374i	$-0.596524\pi$
$881$	−17.7267	−0.597227	−0.298613	+	0.954374i	$0.596524\pi$
$882$	0	0
$883$	20.3877	0.686102	0.343051	−	0.939317i	$-0.388540\pi$
$883$	20.3877	0.686102	0.343051	+	0.939317i	$0.388540\pi$
$884$	0	0
$885$	−9.69964	−0.326050
$886$	0	0
$887$	−16.2271	−0.544851	−0.272426	−	0.962177i	$-0.587826\pi$
$887$	−16.2271	−0.544851	−0.272426	+	0.962177i	$0.587826\pi$
$888$	0	0
$889$	−2.36546	−0.0793349
$890$	0	0
$891$	1.83141	0.0613545
$892$	0	0
$893$	−96.5299	−3.23025
$894$	0	0
$895$	−4.33168	−0.144792
$896$	0	0
$897$	2.46385	0.0822657
$898$	0	0
$899$	6.92169	0.230851
$900$	0	0
$901$	9.68677	0.322713
$902$	0	0
$903$	10.8386	0.360686
$904$	0	0
$905$	8.66872	0.288158
$906$	0	0
$907$	33.3761	1.10823	0.554117	−	0.832439i	$-0.313056\pi$
$907$	33.3761	1.10823	0.554117	+	0.832439i	$0.313056\pi$
$908$	0	0
$909$	−15.3662	−0.509665
$910$	0	0
$911$	32.5819	1.07949	0.539744	−	0.841829i	$-0.318521\pi$
$911$	32.5819	1.07949	0.539744	+	0.841829i	$0.318521\pi$
$912$	0	0
$913$	−7.70982	−0.255158
$914$	0	0
$915$	−12.2719	−0.405696
$916$	0	0
$917$	−11.2238	−0.370644
$918$	0	0
$919$	−34.6410	−1.14270	−0.571350	−	0.820707i	$-0.693580\pi$
$919$	−34.6410	−1.14270	−0.571350	+	0.820707i	$0.693580\pi$
$920$	0	0
$921$	19.0757	0.628566
$922$	0	0
$923$	9.17588	0.302028
$924$	0	0
$925$	−18.8172	−0.618706
$926$	0	0
$927$	1.63871	0.0538221
$928$	0	0
$929$	33.8884	1.11184	0.555921	−	0.831235i	$-0.312366\pi$
$929$	33.8884	1.11184	0.555921	+	0.831235i	$0.312366\pi$
$930$	0	0
$931$	8.03516	0.263342
$932$	0	0
$933$	−29.9364	−0.980072
$934$	0	0
$935$	4.09939	0.134064
$936$	0	0
$937$	28.0353	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$937$	28.0353	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$938$	0	0
$939$	17.1230	0.558790
$940$	0	0
$941$	−11.1806	−0.364476	−0.182238	−	0.983254i	$-0.558334\pi$
$941$	−11.1806	−0.364476	−0.182238	+	0.983254i	$0.558334\pi$
$942$	0	0
$943$	−0.887999	−0.0289172
$944$	0	0
$945$	−1.74592	−0.0567947
$946$	0	0
$947$	−26.4133	−0.858317	−0.429159	−	0.903229i	$-0.641190\pi$
$947$	−26.4133	−0.858317	−0.429159	+	0.903229i	$0.641190\pi$
$948$	0	0
$949$	17.7542	0.576325
$950$	0	0
$951$	−6.38162	−0.206938
$952$	0	0
$953$	56.9189	1.84379	0.921893	−	0.387446i	$-0.126643\pi$
$953$	56.9189	1.84379	0.921893	+	0.387446i	$0.126643\pi$
$954$	0	0
$955$	21.8675	0.707614
$956$	0	0
$957$	−17.6568	−0.570763
$958$	0	0
$959$	18.3942	0.593980
$960$	0	0
$961$	−30.4846	−0.983373
$962$	0	0
$963$	−8.23236	−0.265284
$964$	0	0
$965$	4.73318	0.152366
$966$	0	0
$967$	−26.4117	−0.849343	−0.424672	−	0.905348i	$-0.639610\pi$
$967$	−26.4117	−0.849343	−0.424672	+	0.905348i	$0.639610\pi$
$968$	0	0
$969$	−10.3016	−0.330935
$970$	0	0
$971$	15.1284	0.485492	0.242746	−	0.970090i	$-0.421952\pi$
$971$	15.1284	0.485492	0.242746	+	0.970090i	$0.421952\pi$
$972$	0	0
$973$	−11.4087	−0.365747
$974$	0	0
$975$	4.80887	0.154007
$976$	0	0
$977$	−30.5610	−0.977733	−0.488867	−	0.872359i	$-0.662589\pi$
$977$	−30.5610	−0.977733	−0.488867	+	0.872359i	$0.662589\pi$
$978$	0	0
$979$	−22.1581	−0.708176
$980$	0	0
$981$	−17.9872	−0.574288
$982$	0	0
$983$	−17.2810	−0.551177	−0.275589	−	0.961276i	$-0.588873\pi$
$983$	−17.2810	−0.551177	−0.275589	+	0.961276i	$0.588873\pi$
$984$	0	0
$985$	−48.3093	−1.53926
$986$	0	0
$987$	12.0134	0.382392
$988$	0	0
$989$	10.8386	0.344647
$990$	0	0
$991$	−11.0986	−0.352560	−0.176280	−	0.984340i	$-0.556406\pi$
$991$	−11.0986	−0.352560	−0.176280	+	0.984340i	$0.556406\pi$
$992$	0	0
$993$	11.1167	0.352777
$994$	0	0
$995$	−31.8016	−1.00818
$996$	0	0
$997$	44.1450	1.39809	0.699043	−	0.715080i	$-0.253610\pi$
$997$	44.1450	1.39809	0.699043	+	0.715080i	$0.253610\pi$
$998$	0	0
$999$	−9.64110	−0.305031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cg.1.5		6
4.3	odd	2		3864.2.a.x.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.x.1.5	✓	6	4.3	odd	2
7728.2.a.cg.1.5		6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.74592 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.74592 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.83141 q^{11} +2.46385 q^{13} -1.74592 q^{15} +1.28206 q^{17} +8.03516 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.95177 q^{25} -1.00000 q^{27} +9.64110 q^{29} +0.717936 q^{31} -1.83141 q^{33} +1.74592 q^{35} +9.64110 q^{37} -2.46385 q^{39} +0.887999 q^{41} -10.8386 q^{43} +1.74592 q^{45} -12.0134 q^{47} +1.00000 q^{49} -1.28206 q^{51} +7.55561 q^{53} +3.19749 q^{55} -8.03516 q^{57} +5.55561 q^{59} +7.02890 q^{61} +1.00000 q^{63} +4.30169 q^{65} +6.18117 q^{67} +1.00000 q^{69} +3.72420 q^{71} +7.20585 q^{73} +1.95177 q^{75} +1.83141 q^{77} -11.0927 q^{79} +1.00000 q^{81} -4.20977 q^{83} +2.23838 q^{85} -9.64110 q^{87} -12.0989 q^{89} +2.46385 q^{91} -0.717936 q^{93} +14.0287 q^{95} +8.22547 q^{97} +1.83141 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9 - 5 * q^11 + 2 * q^13 + 10 * q^17 - 3 * q^19 - 6 * q^21 - 6 * q^23 + 18 * q^25 - 6 * q^27 + 3 * q^29 + 2 * q^31 + 5 * q^33 + 3 * q^37 - 2 * q^39 + 4 * q^41 - 6 * q^43 + 2 * q^47 + 6 * q^49 - 10 * q^51 - 4 * q^53 + 15 * q^55 + 3 * q^57 - 16 * q^59 + 22 * q^61 + 6 * q^63 + 35 * q^65 - 9 * q^67 + 6 * q^69 - 11 * q^71 + 24 * q^73 - 18 * q^75 - 5 * q^77 - 18 * q^79 + 6 * q^81 - 2 * q^83 + 13 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 - 2 * q^93 - 5 * q^95 + 37 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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