LMFDB - Embedded newform 5376.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5376,2,Mod(1,5376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5376, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5376.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5376 = 2^{8} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5376.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:	$42.9275761266$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.19664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 2x + 2 $ x^4 - 2x^3 - 5x^2 + 2*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.28139$ of defining polynomial
Character	$\chi$	$=$	5376.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.467138 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.467138 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.87666 q^{11} +4.56279 q^{13} -0.467138 q^{15} +6.09565 q^{17} -1.34379 q^{19} -1.00000 q^{21} +4.09565 q^{23} -4.78178 q^{25} +1.00000 q^{27} -7.78178 q^{29} +4.40952 q^{31} +4.87666 q^{33} +0.467138 q^{35} -4.40952 q^{37} +4.56279 q^{39} +6.09565 q^{41} +4.15327 q^{43} -0.467138 q^{45} -6.68759 q^{47} +1.00000 q^{49} +6.09565 q^{51} -1.34379 q^{53} -2.27807 q^{55} -1.34379 q^{57} +4.00000 q^{59} +5.49706 q^{61} -1.00000 q^{63} -2.13145 q^{65} -5.90658 q^{67} +4.09565 q^{69} +4.72339 q^{71} +12.0599 q^{73} -4.78178 q^{75} -4.87666 q^{77} -16.1913 q^{79} +1.00000 q^{81} +13.7533 q^{83} -2.84751 q^{85} -7.78178 q^{87} -7.96420 q^{89} -4.56279 q^{91} +4.40952 q^{93} +0.627737 q^{95} -12.8789 q^{97} +4.87666 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 6 q^{11} - 4 q^{13} - 2 q^{15} + 2 q^{17} + 8 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} + 4 q^{27} + 4 q^{31} + 6 q^{33} + 2 q^{35} - 4 q^{37} - 4 q^{39} + 2 q^{41} + 8 q^{43} - 2 q^{45} + 4 q^{49} + 2 q^{51} + 8 q^{53} + 4 q^{55} + 8 q^{57} + 16 q^{59} - 4 q^{63} - 8 q^{65} + 12 q^{67} - 6 q^{69} + 14 q^{71} + 4 q^{73} + 12 q^{75} - 6 q^{77} - 20 q^{79} + 4 q^{81} + 28 q^{83} + 20 q^{85} - 10 q^{89} + 4 q^{91} + 4 q^{93} + 20 q^{95} + 20 q^{97} + 6 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9 + 6 * q^11 - 4 * q^13 - 2 * q^15 + 2 * q^17 + 8 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 + 4 * q^27 + 4 * q^31 + 6 * q^33 + 2 * q^35 - 4 * q^37 - 4 * q^39 + 2 * q^41 + 8 * q^43 - 2 * q^45 + 4 * q^49 + 2 * q^51 + 8 * q^53 + 4 * q^55 + 8 * q^57 + 16 * q^59 - 4 * q^63 - 8 * q^65 + 12 * q^67 - 6 * q^69 + 14 * q^71 + 4 * q^73 + 12 * q^75 - 6 * q^77 - 20 * q^79 + 4 * q^81 + 28 * q^83 + 20 * q^85 - 10 * q^89 + 4 * q^91 + 4 * q^93 + 20 * q^95 + 20 * q^97 + 6 * 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
$5$	−0.467138	−0.208910	−0.104455	−	0.994530i	$-0.533310\pi$
$5$	−0.467138	−0.208910	−0.104455	+	0.994530i	$0.533310\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.87666	1.47037	0.735184	−	0.677868i	$-0.237096\pi$
$11$	4.87666	1.47037	0.735184	+	0.677868i	$0.237096\pi$
$12$	0	0
$13$	4.56279	1.26549	0.632745	−	0.774360i	$-0.281928\pi$
$13$	4.56279	1.26549	0.632745	+	0.774360i	$0.281928\pi$
$14$	0	0
$15$	−0.467138	−0.120614
$16$	0	0
$17$	6.09565	1.47841	0.739206	−	0.673479i	$-0.235201\pi$
$17$	6.09565	1.47841	0.739206	+	0.673479i	$0.235201\pi$
$18$	0	0
$19$	−1.34379	−0.308288	−0.154144	−	0.988048i	$-0.549262\pi$
$19$	−1.34379	−0.308288	−0.154144	+	0.988048i	$0.549262\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.09565	0.854002	0.427001	−	0.904251i	$-0.359570\pi$
$23$	4.09565	0.854002	0.427001	+	0.904251i	$0.359570\pi$
$24$	0	0
$25$	−4.78178	−0.956357
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.78178	−1.44504	−0.722520	−	0.691350i	$-0.757016\pi$
$29$	−7.78178	−1.44504	−0.722520	+	0.691350i	$0.757016\pi$
$30$	0	0
$31$	4.40952	0.791973	0.395987	−	0.918256i	$-0.370403\pi$
$31$	4.40952	0.791973	0.395987	+	0.918256i	$0.370403\pi$
$32$	0	0
$33$	4.87666	0.848917
$34$	0	0
$35$	0.467138	0.0789607
$36$	0	0
$37$	−4.40952	−0.724921	−0.362460	−	0.931999i	$-0.618063\pi$
$37$	−4.40952	−0.724921	−0.362460	+	0.931999i	$0.618063\pi$
$38$	0	0
$39$	4.56279	0.730631
$40$	0	0
$41$	6.09565	0.951981	0.475990	−	0.879450i	$-0.342090\pi$
$41$	6.09565	0.951981	0.475990	+	0.879450i	$0.342090\pi$
$42$	0	0
$43$	4.15327	0.633368	0.316684	−	0.948531i	$-0.397431\pi$
$43$	4.15327	0.633368	0.316684	+	0.948531i	$0.397431\pi$
$44$	0	0
$45$	−0.467138	−0.0696368
$46$	0	0
$47$	−6.68759	−0.975485	−0.487743	−	0.872988i	$-0.662180\pi$
$47$	−6.68759	−0.975485	−0.487743	+	0.872988i	$0.662180\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.09565	0.853562
$52$	0	0
$53$	−1.34379	−0.184584	−0.0922922	−	0.995732i	$-0.529419\pi$
$53$	−1.34379	−0.184584	−0.0922922	+	0.995732i	$0.529419\pi$
$54$	0	0
$55$	−2.27807	−0.307175
$56$	0	0
$57$	−1.34379	−0.177990
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	5.49706	0.703827	0.351913	−	0.936033i	$-0.385531\pi$
$61$	5.49706	0.703827	0.351913	+	0.936033i	$0.385531\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.13145	−0.264374
$66$	0	0
$67$	−5.90658	−0.721604	−0.360802	−	0.932642i	$-0.617497\pi$
$67$	−5.90658	−0.721604	−0.360802	+	0.932642i	$0.617497\pi$
$68$	0	0
$69$	4.09565	0.493058
$70$	0	0
$71$	4.72339	0.560563	0.280281	−	0.959918i	$-0.409572\pi$
$71$	4.72339	0.560563	0.280281	+	0.959918i	$0.409572\pi$
$72$	0	0
$73$	12.0599	1.41150	0.705749	−	0.708462i	$-0.250610\pi$
$73$	12.0599	1.41150	0.705749	+	0.708462i	$0.250610\pi$
$74$	0	0
$75$	−4.78178	−0.552153
$76$	0	0
$77$	−4.87666	−0.555747
$78$	0	0
$79$	−16.1913	−1.82166	−0.910832	−	0.412778i	$-0.864559\pi$
$79$	−16.1913	−1.82166	−0.910832	+	0.412778i	$0.864559\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.7533	1.50962	0.754811	−	0.655942i	$-0.227728\pi$
$83$	13.7533	1.50962	0.754811	+	0.655942i	$0.227728\pi$
$84$	0	0
$85$	−2.84751	−0.308856
$86$	0	0
$87$	−7.78178	−0.834295
$88$	0	0
$89$	−7.96420	−0.844204	−0.422102	−	0.906548i	$-0.638708\pi$
$89$	−7.96420	−0.844204	−0.422102	+	0.906548i	$0.638708\pi$
$90$	0	0
$91$	−4.56279	−0.478310
$92$	0	0
$93$	4.40952	0.457246
$94$	0	0
$95$	0.627737	0.0644044
$96$	0	0
$97$	−12.8789	−1.30765	−0.653827	−	0.756644i	$-0.726837\pi$
$97$	−12.8789	−1.30765	−0.653827	+	0.756644i	$0.726837\pi$
$98$	0	0
$99$	4.87666	0.490122
$100$	0	0
$101$	2.22045	0.220943	0.110472	−	0.993879i	$-0.464764\pi$
$101$	2.22045	0.220943	0.110472	+	0.993879i	$0.464764\pi$
$102$	0	0
$103$	−11.0971	−1.09343	−0.546715	−	0.837319i	$-0.684122\pi$
$103$	−11.0971	−1.09343	−0.546715	+	0.837319i	$0.684122\pi$
$104$	0	0
$105$	0.467138	0.0455880
$106$	0	0
$107$	3.12334	0.301945	0.150972	−	0.988538i	$-0.451760\pi$
$107$	3.12334	0.301945	0.150972	+	0.988538i	$0.451760\pi$
$108$	0	0
$109$	10.6876	1.02369	0.511843	−	0.859079i	$-0.328963\pi$
$109$	10.6876	1.02369	0.511843	+	0.859079i	$0.328963\pi$
$110$	0	0
$111$	−4.40952	−0.418533
$112$	0	0
$113$	−12.0599	−1.13450	−0.567248	−	0.823547i	$-0.691992\pi$
$113$	−12.0599	−1.13450	−0.567248	+	0.823547i	$0.691992\pi$
$114$	0	0
$115$	−1.91323	−0.178410
$116$	0	0
$117$	4.56279	0.421830
$118$	0	0
$119$	−6.09565	−0.558787
$120$	0	0
$121$	12.7818	1.16198
$122$	0	0
$123$	6.09565	0.549626
$124$	0	0
$125$	4.56944	0.408703
$126$	0	0
$127$	18.8789	1.67523	0.837615	−	0.546261i	$-0.183949\pi$
$127$	18.8789	1.67523	0.837615	+	0.546261i	$0.183949\pi$
$128$	0	0
$129$	4.15327	0.365675
$130$	0	0
$131$	4.93428	0.431110	0.215555	−	0.976492i	$-0.430844\pi$
$131$	4.93428	0.431110	0.215555	+	0.976492i	$0.430844\pi$
$132$	0	0
$133$	1.34379	0.116522
$134$	0	0
$135$	−0.467138	−0.0402048
$136$	0	0
$137$	−19.0103	−1.62416	−0.812082	−	0.583544i	$-0.801666\pi$
$137$	−19.0103	−1.62416	−0.812082	+	0.583544i	$0.801666\pi$
$138$	0	0
$139$	10.4380	0.885339	0.442669	−	0.896685i	$-0.354032\pi$
$139$	10.4380	0.885339	0.442669	+	0.896685i	$0.354032\pi$
$140$	0	0
$141$	−6.68759	−0.563197
$142$	0	0
$143$	22.2512	1.86073
$144$	0	0
$145$	3.63516	0.301884
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	6.65621	0.545298	0.272649	−	0.962114i	$-0.412100\pi$
$149$	6.65621	0.545298	0.272649	+	0.962114i	$0.412100\pi$
$150$	0	0
$151$	2.68759	0.218713	0.109356	−	0.994003i	$-0.465121\pi$
$151$	2.68759	0.218713	0.109356	+	0.994003i	$0.465121\pi$
$152$	0	0
$153$	6.09565	0.492804
$154$	0	0
$155$	−2.05985	−0.165451
$156$	0	0
$157$	−23.1351	−1.84639	−0.923193	−	0.384338i	$-0.874430\pi$
$157$	−23.1351	−1.84639	−0.923193	+	0.384338i	$0.874430\pi$
$158$	0	0
$159$	−1.34379	−0.106570
$160$	0	0
$161$	−4.09565	−0.322783
$162$	0	0
$163$	12.0380	0.942891	0.471446	−	0.881895i	$-0.343732\pi$
$163$	12.0380	0.942891	0.471446	+	0.881895i	$0.343732\pi$
$164$	0	0
$165$	−2.27807	−0.177347
$166$	0	0
$167$	−9.01034	−0.697241	−0.348621	−	0.937264i	$-0.613350\pi$
$167$	−9.01034	−0.697241	−0.348621	+	0.937264i	$0.613350\pi$
$168$	0	0
$169$	7.81904	0.601465
$170$	0	0
$171$	−1.34379	−0.102763
$172$	0	0
$173$	−9.59271	−0.729321	−0.364660	−	0.931141i	$-0.618815\pi$
$173$	−9.59271	−0.729321	−0.364660	+	0.931141i	$0.618815\pi$
$174$	0	0
$175$	4.78178	0.361469
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	7.69278	0.574985	0.287493	−	0.957783i	$-0.407178\pi$
$179$	7.69278	0.574985	0.287493	+	0.957783i	$0.407178\pi$
$180$	0	0
$181$	15.2504	1.13355	0.566776	−	0.823872i	$-0.308191\pi$
$181$	15.2504	1.13355	0.566776	+	0.823872i	$0.308191\pi$
$182$	0	0
$183$	5.49706	0.406355
$184$	0	0
$185$	2.05985	0.151443
$186$	0	0
$187$	29.7264	2.17381
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−4.72339	−0.341772	−0.170886	−	0.985291i	$-0.554663\pi$
$191$	−4.72339	−0.341772	−0.170886	+	0.985291i	$0.554663\pi$
$192$	0	0
$193$	22.3379	1.60792	0.803959	−	0.594684i	$-0.202723\pi$
$193$	22.3379	1.60792	0.803959	+	0.594684i	$0.202723\pi$
$194$	0	0
$195$	−2.13145	−0.152636
$196$	0	0
$197$	1.53510	0.109371	0.0546855	−	0.998504i	$-0.482584\pi$
$197$	1.53510	0.109371	0.0546855	+	0.998504i	$0.482584\pi$
$198$	0	0
$199$	14.6876	1.04118	0.520588	−	0.853808i	$-0.325713\pi$
$199$	14.6876	1.04118	0.520588	+	0.853808i	$0.325713\pi$
$200$	0	0
$201$	−5.90658	−0.416618
$202$	0	0
$203$	7.78178	0.546174
$204$	0	0
$205$	−2.84751	−0.198879
$206$	0	0
$207$	4.09565	0.284667
$208$	0	0
$209$	−6.55322	−0.453296
$210$	0	0
$211$	22.8409	1.57243	0.786215	−	0.617953i	$-0.212038\pi$
$211$	22.8409	1.57243	0.786215	+	0.617953i	$0.212038\pi$
$212$	0	0
$213$	4.72339	0.323641
$214$	0	0
$215$	−1.94015	−0.132317
$216$	0	0
$217$	−4.40952	−0.299338
$218$	0	0
$219$	12.0599	0.814929
$220$	0	0
$221$	27.8132	1.87092
$222$	0	0
$223$	21.4321	1.43520	0.717600	−	0.696455i	$-0.245240\pi$
$223$	21.4321	1.43520	0.717600	+	0.696455i	$0.245240\pi$
$224$	0	0
$225$	−4.78178	−0.318786
$226$	0	0
$227$	29.3169	1.94583	0.972915	−	0.231164i	$-0.0742534\pi$
$227$	29.3169	1.94583	0.972915	+	0.231164i	$0.0742534\pi$
$228$	0	0
$229$	−22.5074	−1.48733	−0.743666	−	0.668552i	$-0.766914\pi$
$229$	−22.5074	−1.48733	−0.743666	+	0.668552i	$0.766914\pi$
$230$	0	0
$231$	−4.87666	−0.320860
$232$	0	0
$233$	0.687589	0.0450454	0.0225227	−	0.999746i	$-0.492830\pi$
$233$	0.687589	0.0450454	0.0225227	+	0.999746i	$0.492830\pi$
$234$	0	0
$235$	3.12402	0.203789
$236$	0	0
$237$	−16.1913	−1.05174
$238$	0	0
$239$	−9.59936	−0.620931	−0.310466	−	0.950585i	$-0.600485\pi$
$239$	−9.59936	−0.620931	−0.310466	+	0.950585i	$0.600485\pi$
$240$	0	0
$241$	0.496287	0.0319687	0.0159843	−	0.999872i	$-0.494912\pi$
$241$	0.496287	0.0319687	0.0159843	+	0.999872i	$0.494912\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.467138	−0.0298443
$246$	0	0
$247$	−6.13145	−0.390135
$248$	0	0
$249$	13.7533	0.871581
$250$	0	0
$251$	−22.1359	−1.39721	−0.698603	−	0.715509i	$-0.746195\pi$
$251$	−22.1359	−1.39721	−0.698603	+	0.715509i	$0.746195\pi$
$252$	0	0
$253$	19.9731	1.25570
$254$	0	0
$255$	−2.84751	−0.178318
$256$	0	0
$257$	2.28695	0.142656	0.0713281	−	0.997453i	$-0.477276\pi$
$257$	2.28695	0.142656	0.0713281	+	0.997453i	$0.477276\pi$
$258$	0	0
$259$	4.40952	0.273994
$260$	0	0
$261$	−7.78178	−0.481680
$262$	0	0
$263$	30.1555	1.85947	0.929734	−	0.368232i	$-0.120037\pi$
$263$	30.1555	1.85947	0.929734	+	0.368232i	$0.120037\pi$
$264$	0	0
$265$	0.627737	0.0385616
$266$	0	0
$267$	−7.96420	−0.487401
$268$	0	0
$269$	9.47748	0.577852	0.288926	−	0.957351i	$-0.406702\pi$
$269$	9.47748	0.577852	0.288926	+	0.957351i	$0.406702\pi$
$270$	0	0
$271$	−13.2855	−0.807036	−0.403518	−	0.914972i	$-0.632213\pi$
$271$	−13.2855	−0.807036	−0.403518	+	0.914972i	$0.632213\pi$
$272$	0	0
$273$	−4.56279	−0.276153
$274$	0	0
$275$	−23.3191	−1.40620
$276$	0	0
$277$	26.8475	1.61311	0.806555	−	0.591159i	$-0.201329\pi$
$277$	26.8475	1.61311	0.806555	+	0.591159i	$0.201329\pi$
$278$	0	0
$279$	4.40952	0.263991
$280$	0	0
$281$	−11.8686	−0.708018	−0.354009	−	0.935242i	$-0.615182\pi$
$281$	−11.8686	−0.708018	−0.354009	+	0.935242i	$0.615182\pi$
$282$	0	0
$283$	−2.46937	−0.146789	−0.0733944	−	0.997303i	$-0.523383\pi$
$283$	−2.46937	−0.146789	−0.0733944	+	0.997303i	$0.523383\pi$
$284$	0	0
$285$	0.627737	0.0371839
$286$	0	0
$287$	−6.09565	−0.359815
$288$	0	0
$289$	20.1570	1.18570
$290$	0	0
$291$	−12.8789	−0.754974
$292$	0	0
$293$	−22.7899	−1.33140	−0.665700	−	0.746220i	$-0.731867\pi$
$293$	−22.7899	−1.33140	−0.665700	+	0.746220i	$0.731867\pi$
$294$	0	0
$295$	−1.86855	−0.108791
$296$	0	0
$297$	4.87666	0.282972
$298$	0	0
$299$	18.6876	1.08073
$300$	0	0
$301$	−4.15327	−0.239390
$302$	0	0
$303$	2.22045	0.127562
$304$	0	0
$305$	−2.56788	−0.147037
$306$	0	0
$307$	9.34379	0.533279	0.266639	−	0.963796i	$-0.414087\pi$
$307$	9.34379	0.533279	0.266639	+	0.963796i	$0.414087\pi$
$308$	0	0
$309$	−11.0971	−0.631292
$310$	0	0
$311$	−6.68759	−0.379218	−0.189609	−	0.981860i	$-0.560722\pi$
$311$	−6.68759	−0.379218	−0.189609	+	0.981860i	$0.560722\pi$
$312$	0	0
$313$	15.6381	0.883916	0.441958	−	0.897036i	$-0.354284\pi$
$313$	15.6381	0.883916	0.441958	+	0.897036i	$0.354284\pi$
$314$	0	0
$315$	0.467138	0.0263202
$316$	0	0
$317$	9.34379	0.524800	0.262400	−	0.964959i	$-0.415486\pi$
$317$	9.34379	0.524800	0.262400	+	0.964959i	$0.415486\pi$
$318$	0	0
$319$	−37.9491	−2.12474
$320$	0	0
$321$	3.12334	0.174328
$322$	0	0
$323$	−8.19130	−0.455776
$324$	0	0
$325$	−21.8183	−1.21026
$326$	0	0
$327$	10.6876	0.591025
$328$	0	0
$329$	6.68759	0.368699
$330$	0	0
$331$	5.40874	0.297291	0.148646	−	0.988891i	$-0.452509\pi$
$331$	5.40874	0.297291	0.148646	+	0.988891i	$0.452509\pi$
$332$	0	0
$333$	−4.40952	−0.241640
$334$	0	0
$335$	2.75919	0.150750
$336$	0	0
$337$	−6.55614	−0.357136	−0.178568	−	0.983928i	$-0.557146\pi$
$337$	−6.55614	−0.357136	−0.178568	+	0.983928i	$0.557146\pi$
$338$	0	0
$339$	−12.0599	−0.655001
$340$	0	0
$341$	21.5037	1.16449
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.91323	−0.103005
$346$	0	0
$347$	24.9964	1.34187	0.670937	−	0.741514i	$-0.265892\pi$
$347$	24.9964	1.34187	0.670937	+	0.741514i	$0.265892\pi$
$348$	0	0
$349$	−27.1921	−1.45556	−0.727779	−	0.685811i	$-0.759447\pi$
$349$	−27.1921	−1.45556	−0.727779	+	0.685811i	$0.759447\pi$
$350$	0	0
$351$	4.56279	0.243544
$352$	0	0
$353$	−30.2870	−1.61201	−0.806006	−	0.591907i	$-0.798375\pi$
$353$	−30.2870	−1.61201	−0.806006	+	0.591907i	$0.798375\pi$
$354$	0	0
$355$	−2.20647	−0.117107
$356$	0	0
$357$	−6.09565	−0.322616
$358$	0	0
$359$	−6.59194	−0.347909	−0.173955	−	0.984754i	$-0.555655\pi$
$359$	−6.59194	−0.347909	−0.173955	+	0.984754i	$0.555655\pi$
$360$	0	0
$361$	−17.1942	−0.904959
$362$	0	0
$363$	12.7818	0.670870
$364$	0	0
$365$	−5.63361	−0.294877
$366$	0	0
$367$	−6.68759	−0.349089	−0.174545	−	0.984649i	$-0.555845\pi$
$367$	−6.68759	−0.349089	−0.174545	+	0.984649i	$0.555845\pi$
$368$	0	0
$369$	6.09565	0.317327
$370$	0	0
$371$	1.34379	0.0697663
$372$	0	0
$373$	17.1256	0.886729	0.443364	−	0.896341i	$-0.353785\pi$
$373$	17.1256	0.886729	0.443364	+	0.896341i	$0.353785\pi$
$374$	0	0
$375$	4.56944	0.235965
$376$	0	0
$377$	−35.5066	−1.82868
$378$	0	0
$379$	16.7094	0.858305	0.429152	−	0.903232i	$-0.358812\pi$
$379$	16.7094	0.858305	0.429152	+	0.903232i	$0.358812\pi$
$380$	0	0
$381$	18.8789	0.967195
$382$	0	0
$383$	−9.94015	−0.507918	−0.253959	−	0.967215i	$-0.581733\pi$
$383$	−9.94015	−0.507918	−0.253959	+	0.967215i	$0.581733\pi$
$384$	0	0
$385$	2.27807	0.116101
$386$	0	0
$387$	4.15327	0.211123
$388$	0	0
$389$	−19.2884	−0.977961	−0.488981	−	0.872295i	$-0.662631\pi$
$389$	−19.2884	−0.977961	−0.488981	+	0.872295i	$0.662631\pi$
$390$	0	0
$391$	24.9657	1.26257
$392$	0	0
$393$	4.93428	0.248901
$394$	0	0
$395$	7.56357	0.380564
$396$	0	0
$397$	23.4417	1.17650	0.588252	−	0.808678i	$-0.299816\pi$
$397$	23.4417	1.17650	0.588252	+	0.808678i	$0.299816\pi$
$398$	0	0
$399$	1.34379	0.0672739
$400$	0	0
$401$	−16.0029	−0.799147	−0.399574	−	0.916701i	$-0.630842\pi$
$401$	−16.0029	−0.799147	−0.399574	+	0.916701i	$0.630842\pi$
$402$	0	0
$403$	20.1197	1.00223
$404$	0	0
$405$	−0.467138	−0.0232123
$406$	0	0
$407$	−21.5037	−1.06590
$408$	0	0
$409$	−15.5665	−0.769713	−0.384856	−	0.922976i	$-0.625749\pi$
$409$	−15.5665	−0.769713	−0.384856	+	0.922976i	$0.625749\pi$
$410$	0	0
$411$	−19.0103	−0.937711
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−6.42469	−0.315376
$416$	0	0
$417$	10.4380	0.511150
$418$	0	0
$419$	1.31241	0.0641155	0.0320577	−	0.999486i	$-0.489794\pi$
$419$	1.31241	0.0641155	0.0320577	+	0.999486i	$0.489794\pi$
$420$	0	0
$421$	3.66655	0.178697	0.0893483	−	0.996000i	$-0.471522\pi$
$421$	3.66655	0.178697	0.0893483	+	0.996000i	$0.471522\pi$
$422$	0	0
$423$	−6.68759	−0.325162
$424$	0	0
$425$	−29.1481	−1.41389
$426$	0	0
$427$	−5.49706	−0.266022
$428$	0	0
$429$	22.2512	1.07430
$430$	0	0
$431$	20.0957	0.967973	0.483987	−	0.875075i	$-0.339188\pi$
$431$	20.0957	0.967973	0.483987	+	0.875075i	$0.339188\pi$
$432$	0	0
$433$	8.68759	0.417499	0.208749	−	0.977969i	$-0.433061\pi$
$433$	8.68759	0.417499	0.208749	+	0.977969i	$0.433061\pi$
$434$	0	0
$435$	3.63516	0.174293
$436$	0	0
$437$	−5.50371	−0.263278
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.57012	0.217133	0.108566	−	0.994089i	$-0.465374\pi$
$443$	4.57012	0.217133	0.108566	+	0.994089i	$0.465374\pi$
$444$	0	0
$445$	3.72038	0.176363
$446$	0	0
$447$	6.65621	0.314828
$448$	0	0
$449$	−3.94015	−0.185947	−0.0929735	−	0.995669i	$-0.529637\pi$
$449$	−3.94015	−0.185947	−0.0929735	+	0.995669i	$0.529637\pi$
$450$	0	0
$451$	29.7264	1.39976
$452$	0	0
$453$	2.68759	0.126274
$454$	0	0
$455$	2.13145	0.0999239
$456$	0	0
$457$	−40.0207	−1.87209	−0.936044	−	0.351882i	$-0.885542\pi$
$457$	−40.0207	−1.87209	−0.936044	+	0.351882i	$0.885542\pi$
$458$	0	0
$459$	6.09565	0.284521
$460$	0	0
$461$	−34.6031	−1.61162	−0.805812	−	0.592171i	$-0.798271\pi$
$461$	−34.6031	−1.61162	−0.805812	+	0.592171i	$0.798271\pi$
$462$	0	0
$463$	0.191302	0.00889055	0.00444528	−	0.999990i	$-0.498585\pi$
$463$	0.191302	0.00889055	0.00444528	+	0.999990i	$0.498585\pi$
$464$	0	0
$465$	−2.05985	−0.0955234
$466$	0	0
$467$	4.49784	0.208135	0.104068	−	0.994570i	$-0.466814\pi$
$467$	4.49784	0.208135	0.104068	+	0.994570i	$0.466814\pi$
$468$	0	0
$469$	5.90658	0.272741
$470$	0	0
$471$	−23.1351	−1.06601
$472$	0	0
$473$	20.2541	0.931283
$474$	0	0
$475$	6.42573	0.294833
$476$	0	0
$477$	−1.34379	−0.0615281
$478$	0	0
$479$	−22.2512	−1.01668	−0.508341	−	0.861156i	$-0.669741\pi$
$479$	−22.2512	−1.01668	−0.508341	+	0.861156i	$0.669741\pi$
$480$	0	0
$481$	−20.1197	−0.917380
$482$	0	0
$483$	−4.09565	−0.186359
$484$	0	0
$485$	6.01621	0.273182
$486$	0	0
$487$	1.86855	0.0846721	0.0423360	−	0.999103i	$-0.486520\pi$
$487$	1.86855	0.0846721	0.0423360	+	0.999103i	$0.486520\pi$
$488$	0	0
$489$	12.0380	0.544379
$490$	0	0
$491$	17.6374	0.795965	0.397982	−	0.917393i	$-0.369711\pi$
$491$	17.6374	0.795965	0.397982	+	0.917393i	$0.369711\pi$
$492$	0	0
$493$	−47.4350	−2.13637
$494$	0	0
$495$	−2.27807	−0.102392
$496$	0	0
$497$	−4.72339	−0.211873
$498$	0	0
$499$	21.5854	0.966295	0.483147	−	0.875539i	$-0.339494\pi$
$499$	21.5854	0.966295	0.483147	+	0.875539i	$0.339494\pi$
$500$	0	0
$501$	−9.01034	−0.402552
$502$	0	0
$503$	−5.24081	−0.233676	−0.116838	−	0.993151i	$-0.537276\pi$
$503$	−5.24081	−0.233676	−0.116838	+	0.993151i	$0.537276\pi$
$504$	0	0
$505$	−1.03726	−0.0461573
$506$	0	0
$507$	7.81904	0.347256
$508$	0	0
$509$	18.3357	0.812715	0.406358	−	0.913714i	$-0.366799\pi$
$509$	18.3357	0.812715	0.406358	+	0.913714i	$0.366799\pi$
$510$	0	0
$511$	−12.0599	−0.533496
$512$	0	0
$513$	−1.34379	−0.0593300
$514$	0	0
$515$	5.18388	0.228429
$516$	0	0
$517$	−32.6131	−1.43432
$518$	0	0
$519$	−9.59271	−0.421073
$520$	0	0
$521$	−6.78033	−0.297051	−0.148526	−	0.988909i	$-0.547453\pi$
$521$	−6.78033	−0.297051	−0.148526	+	0.988909i	$0.547453\pi$
$522$	0	0
$523$	−23.5066	−1.02787	−0.513937	−	0.857828i	$-0.671813\pi$
$523$	−23.5066	−1.02787	−0.513937	+	0.857828i	$0.671813\pi$
$524$	0	0
$525$	4.78178	0.208694
$526$	0	0
$527$	26.8789	1.17086
$528$	0	0
$529$	−6.22564	−0.270680
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	27.8132	1.20472
$534$	0	0
$535$	−1.45903	−0.0630794
$536$	0	0
$537$	7.69278	0.331968
$538$	0	0
$539$	4.87666	0.210052
$540$	0	0
$541$	−5.21526	−0.224222	−0.112111	−	0.993696i	$-0.535761\pi$
$541$	−5.21526	−0.224222	−0.112111	+	0.993696i	$0.535761\pi$
$542$	0	0
$543$	15.2504	0.654456
$544$	0	0
$545$	−4.99257	−0.213858
$546$	0	0
$547$	−27.2951	−1.16705	−0.583526	−	0.812094i	$-0.698327\pi$
$547$	−27.2951	−1.16705	−0.583526	+	0.812094i	$0.698327\pi$
$548$	0	0
$549$	5.49706	0.234609
$550$	0	0
$551$	10.4571	0.445488
$552$	0	0
$553$	16.1913	0.688524
$554$	0	0
$555$	2.05985	0.0874359
$556$	0	0
$557$	4.78766	0.202859	0.101430	−	0.994843i	$-0.467658\pi$
$557$	4.78766	0.202859	0.101430	+	0.994843i	$0.467658\pi$
$558$	0	0
$559$	18.9505	0.801520
$560$	0	0
$561$	29.7264	1.25505
$562$	0	0
$563$	−18.7445	−0.789988	−0.394994	−	0.918684i	$-0.629253\pi$
$563$	−18.7445	−0.789988	−0.394994	+	0.918684i	$0.629253\pi$
$564$	0	0
$565$	5.63361	0.237008
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−27.6950	−1.16104	−0.580518	−	0.814248i	$-0.697150\pi$
$569$	−27.6950	−1.16104	−0.580518	+	0.814248i	$0.697150\pi$
$570$	0	0
$571$	−17.4132	−0.728720	−0.364360	−	0.931258i	$-0.618712\pi$
$571$	−17.4132	−0.728720	−0.364360	+	0.931258i	$0.618712\pi$
$572$	0	0
$573$	−4.72339	−0.197322
$574$	0	0
$575$	−19.5845	−0.816731
$576$	0	0
$577$	−6.81904	−0.283880	−0.141940	−	0.989875i	$-0.545334\pi$
$577$	−6.81904	−0.283880	−0.141940	+	0.989875i	$0.545334\pi$
$578$	0	0
$579$	22.3379	0.928332
$580$	0	0
$581$	−13.7533	−0.570584
$582$	0	0
$583$	−6.55322	−0.271407
$584$	0	0
$585$	−2.13145	−0.0881246
$586$	0	0
$587$	−9.81025	−0.404912	−0.202456	−	0.979291i	$-0.564892\pi$
$587$	−9.81025	−0.404912	−0.202456	+	0.979291i	$0.564892\pi$
$588$	0	0
$589$	−5.92549	−0.244155
$590$	0	0
$591$	1.53510	0.0631454
$592$	0	0
$593$	−39.0913	−1.60529	−0.802644	−	0.596458i	$-0.796574\pi$
$593$	−39.0913	−1.60529	−0.802644	+	0.596458i	$0.796574\pi$
$594$	0	0
$595$	2.84751	0.116736
$596$	0	0
$597$	14.6876	0.601123
$598$	0	0
$599$	−22.0270	−0.899998	−0.449999	−	0.893029i	$-0.648576\pi$
$599$	−22.0270	−0.899998	−0.449999	+	0.893029i	$0.648576\pi$
$600$	0	0
$601$	20.2512	0.826062	0.413031	−	0.910717i	$-0.364470\pi$
$601$	20.2512	0.826062	0.413031	+	0.910717i	$0.364470\pi$
$602$	0	0
$603$	−5.90658	−0.240535
$604$	0	0
$605$	−5.97085	−0.242750
$606$	0	0
$607$	44.4454	1.80398	0.901991	−	0.431755i	$-0.142105\pi$
$607$	44.4454	1.80398	0.901991	+	0.431755i	$0.142105\pi$
$608$	0	0
$609$	7.78178	0.315334
$610$	0	0
$611$	−30.5141	−1.23447
$612$	0	0
$613$	47.6263	1.92361	0.961805	−	0.273737i	$-0.0882596\pi$
$613$	47.6263	1.92361	0.961805	+	0.273737i	$0.0882596\pi$
$614$	0	0
$615$	−2.84751	−0.114823
$616$	0	0
$617$	3.01034	0.121192	0.0605959	−	0.998162i	$-0.480700\pi$
$617$	3.01034	0.121192	0.0605959	+	0.998162i	$0.480700\pi$
$618$	0	0
$619$	−18.5141	−0.744143	−0.372071	−	0.928204i	$-0.621352\pi$
$619$	−18.5141	−0.744143	−0.372071	+	0.928204i	$0.621352\pi$
$620$	0	0
$621$	4.09565	0.164353
$622$	0	0
$623$	7.96420	0.319079
$624$	0	0
$625$	21.7744	0.870974
$626$	0	0
$627$	−6.55322	−0.261711
$628$	0	0
$629$	−26.8789	−1.07173
$630$	0	0
$631$	−10.2658	−0.408676	−0.204338	−	0.978900i	$-0.565504\pi$
$631$	−10.2658	−0.408676	−0.204338	+	0.978900i	$0.565504\pi$
$632$	0	0
$633$	22.8409	0.907843
$634$	0	0
$635$	−8.81904	−0.349973
$636$	0	0
$637$	4.56279	0.180784
$638$	0	0
$639$	4.72339	0.186854
$640$	0	0
$641$	−16.9358	−0.668925	−0.334462	−	0.942409i	$-0.608555\pi$
$641$	−16.9358	−0.668925	−0.334462	+	0.942409i	$0.608555\pi$
$642$	0	0
$643$	−19.5189	−0.769750	−0.384875	−	0.922969i	$-0.625755\pi$
$643$	−19.5189	−0.769750	−0.384875	+	0.922969i	$0.625755\pi$
$644$	0	0
$645$	−1.94015	−0.0763933
$646$	0	0
$647$	23.3723	0.918858	0.459429	−	0.888214i	$-0.348054\pi$
$647$	23.3723	0.918858	0.459429	+	0.888214i	$0.348054\pi$
$648$	0	0
$649$	19.5066	0.765702
$650$	0	0
$651$	−4.40952	−0.172823
$652$	0	0
$653$	1.45903	0.0570963	0.0285481	−	0.999592i	$-0.490912\pi$
$653$	1.45903	0.0570963	0.0285481	+	0.999592i	$0.490912\pi$
$654$	0	0
$655$	−2.30499	−0.0900632
$656$	0	0
$657$	12.0599	0.470500
$658$	0	0
$659$	10.8929	0.424326	0.212163	−	0.977234i	$-0.431949\pi$
$659$	10.8929	0.424326	0.212163	+	0.977234i	$0.431949\pi$
$660$	0	0
$661$	−8.69715	−0.338280	−0.169140	−	0.985592i	$-0.554099\pi$
$661$	−8.69715	−0.338280	−0.169140	+	0.985592i	$0.554099\pi$
$662$	0	0
$663$	27.8132	1.08017
$664$	0	0
$665$	−0.627737	−0.0243426
$666$	0	0
$667$	−31.8715	−1.23407
$668$	0	0
$669$	21.4321	0.828613
$670$	0	0
$671$	26.8073	1.03488
$672$	0	0
$673$	18.0447	0.695571	0.347786	−	0.937574i	$-0.386934\pi$
$673$	18.0447	0.695571	0.347786	+	0.937574i	$0.386934\pi$
$674$	0	0
$675$	−4.78178	−0.184051
$676$	0	0
$677$	28.7781	1.10603	0.553017	−	0.833170i	$-0.313476\pi$
$677$	28.7781	1.10603	0.553017	+	0.833170i	$0.313476\pi$
$678$	0	0
$679$	12.8789	0.494246
$680$	0	0
$681$	29.3169	1.12343
$682$	0	0
$683$	−34.4431	−1.31793	−0.658965	−	0.752174i	$-0.729005\pi$
$683$	−34.4431	−1.31793	−0.658965	+	0.752174i	$0.729005\pi$
$684$	0	0
$685$	8.88044	0.339304
$686$	0	0
$687$	−22.5074	−0.858711
$688$	0	0
$689$	−6.13145	−0.233590
$690$	0	0
$691$	−41.5651	−1.58121	−0.790606	−	0.612325i	$-0.790234\pi$
$691$	−41.5651	−1.58121	−0.790606	+	0.612325i	$0.790234\pi$
$692$	0	0
$693$	−4.87666	−0.185249
$694$	0	0
$695$	−4.87598	−0.184956
$696$	0	0
$697$	37.1570	1.40742
$698$	0	0
$699$	0.687589	0.0260070
$700$	0	0
$701$	−46.7804	−1.76687	−0.883435	−	0.468553i	$-0.844775\pi$
$701$	−46.7804	−1.76687	−0.883435	+	0.468553i	$0.844775\pi$
$702$	0	0
$703$	5.92549	0.223484
$704$	0	0
$705$	3.12402	0.117658
$706$	0	0
$707$	−2.22045	−0.0835087
$708$	0	0
$709$	−8.43643	−0.316837	−0.158418	−	0.987372i	$-0.550640\pi$
$709$	−8.43643	−0.316837	−0.158418	+	0.987372i	$0.550640\pi$
$710$	0	0
$711$	−16.1913	−0.607221
$712$	0	0
$713$	18.0599	0.676347
$714$	0	0
$715$	−10.3943	−0.388727
$716$	0	0
$717$	−9.59936	−0.358495
$718$	0	0
$719$	−38.4425	−1.43366	−0.716831	−	0.697247i	$-0.754408\pi$
$719$	−38.4425	−1.43366	−0.716831	+	0.697247i	$0.754408\pi$
$720$	0	0
$721$	11.0971	0.413278
$722$	0	0
$723$	0.496287	0.0184571
$724$	0	0
$725$	37.2108	1.38197
$726$	0	0
$727$	17.5484	0.650834	0.325417	−	0.945571i	$-0.394495\pi$
$727$	17.5484	0.650834	0.325417	+	0.945571i	$0.394495\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	25.3169	0.936379
$732$	0	0
$733$	38.8272	1.43412	0.717058	−	0.697013i	$-0.245488\pi$
$733$	38.8272	1.43412	0.717058	+	0.697013i	$0.245488\pi$
$734$	0	0
$735$	−0.467138	−0.0172306
$736$	0	0
$737$	−28.8044	−1.06102
$738$	0	0
$739$	13.9066	0.511562	0.255781	−	0.966735i	$-0.417667\pi$
$739$	13.9066	0.511562	0.255781	+	0.966735i	$0.417667\pi$
$740$	0	0
$741$	−6.13145	−0.225244
$742$	0	0
$743$	−40.2300	−1.47590	−0.737948	−	0.674858i	$-0.764205\pi$
$743$	−40.2300	−1.47590	−0.737948	+	0.674858i	$0.764205\pi$
$744$	0	0
$745$	−3.10936	−0.113918
$746$	0	0
$747$	13.7533	0.503207
$748$	0	0
$749$	−3.12334	−0.114124
$750$	0	0
$751$	16.9957	0.620181	0.310091	−	0.950707i	$-0.399641\pi$
$751$	16.9957	0.620181	0.310091	+	0.950707i	$0.399641\pi$
$752$	0	0
$753$	−22.1359	−0.806678
$754$	0	0
$755$	−1.25547	−0.0456914
$756$	0	0
$757$	−9.43212	−0.342816	−0.171408	−	0.985200i	$-0.554832\pi$
$757$	−9.43212	−0.342816	−0.171408	+	0.985200i	$0.554832\pi$
$758$	0	0
$759$	19.9731	0.724977
$760$	0	0
$761$	9.41098	0.341148	0.170574	−	0.985345i	$-0.445438\pi$
$761$	9.41098	0.341148	0.170574	+	0.985345i	$0.445438\pi$
$762$	0	0
$763$	−10.6876	−0.386917
$764$	0	0
$765$	−2.84751	−0.102952
$766$	0	0
$767$	18.2512	0.659011
$768$	0	0
$769$	27.6950	0.998708	0.499354	−	0.866398i	$-0.333571\pi$
$769$	27.6950	0.998708	0.499354	+	0.866398i	$0.333571\pi$
$770$	0	0
$771$	2.28695	0.0823626
$772$	0	0
$773$	−33.8993	−1.21927	−0.609636	−	0.792682i	$-0.708684\pi$
$773$	−33.8993	−1.21927	−0.609636	+	0.792682i	$0.708684\pi$
$774$	0	0
$775$	−21.0854	−0.757409
$776$	0	0
$777$	4.40952	0.158191
$778$	0	0
$779$	−8.19130	−0.293484
$780$	0	0
$781$	23.0343	0.824234
$782$	0	0
$783$	−7.78178	−0.278098
$784$	0	0
$785$	10.8073	0.385729
$786$	0	0
$787$	−31.2001	−1.11216	−0.556082	−	0.831128i	$-0.687696\pi$
$787$	−31.2001	−1.11216	−0.556082	+	0.831128i	$0.687696\pi$
$788$	0	0
$789$	30.1555	1.07356
$790$	0	0
$791$	12.0599	0.428799
$792$	0	0
$793$	25.0819	0.890686
$794$	0	0
$795$	0.627737	0.0222635
$796$	0	0
$797$	32.2848	1.14359	0.571793	−	0.820398i	$-0.306248\pi$
$797$	32.2848	1.14359	0.571793	+	0.820398i	$0.306248\pi$
$798$	0	0
$799$	−40.7652	−1.44217
$800$	0	0
$801$	−7.96420	−0.281401
$802$	0	0
$803$	58.8118	2.07542
$804$	0	0
$805$	1.91323	0.0674326
$806$	0	0
$807$	9.47748	0.333623
$808$	0	0
$809$	−8.49629	−0.298714	−0.149357	−	0.988783i	$-0.547720\pi$
$809$	−8.49629	−0.298714	−0.149357	+	0.988783i	$0.547720\pi$
$810$	0	0
$811$	12.3826	0.434812	0.217406	−	0.976081i	$-0.430240\pi$
$811$	12.3826	0.434812	0.217406	+	0.976081i	$0.430240\pi$
$812$	0	0
$813$	−13.2855	−0.465943
$814$	0	0
$815$	−5.62342	−0.196980
$816$	0	0
$817$	−5.58114	−0.195259
$818$	0	0
$819$	−4.56279	−0.159437
$820$	0	0
$821$	21.8999	0.764313	0.382156	−	0.924098i	$-0.375182\pi$
$821$	21.8999	0.764313	0.382156	+	0.924098i	$0.375182\pi$
$822$	0	0
$823$	1.88321	0.0656446	0.0328223	−	0.999461i	$-0.489550\pi$
$823$	1.88321	0.0656446	0.0328223	+	0.999461i	$0.489550\pi$
$824$	0	0
$825$	−23.3191	−0.811867
$826$	0	0
$827$	−32.8051	−1.14074	−0.570372	−	0.821387i	$-0.693201\pi$
$827$	−32.8051	−1.14074	−0.570372	+	0.821387i	$0.693201\pi$
$828$	0	0
$829$	9.63143	0.334513	0.167257	−	0.985913i	$-0.446509\pi$
$829$	9.63143	0.334513	0.167257	+	0.985913i	$0.446509\pi$
$830$	0	0
$831$	26.8475	0.931330
$832$	0	0
$833$	6.09565	0.211202
$834$	0	0
$835$	4.20907	0.145661
$836$	0	0
$837$	4.40952	0.152415
$838$	0	0
$839$	48.0481	1.65880	0.829402	−	0.558652i	$-0.188681\pi$
$839$	48.0481	1.65880	0.829402	+	0.558652i	$0.188681\pi$
$840$	0	0
$841$	31.5561	1.08814
$842$	0	0
$843$	−11.8686	−0.408775
$844$	0	0
$845$	−3.65257	−0.125652
$846$	0	0
$847$	−12.7818	−0.439187
$848$	0	0
$849$	−2.46937	−0.0847486
$850$	0	0
$851$	−18.0599	−0.619084
$852$	0	0
$853$	−1.44013	−0.0493090	−0.0246545	−	0.999696i	$-0.507849\pi$
$853$	−1.44013	−0.0493090	−0.0246545	+	0.999696i	$0.507849\pi$
$854$	0	0
$855$	0.627737	0.0214681
$856$	0	0
$857$	−34.9775	−1.19481	−0.597404	−	0.801941i	$-0.703801\pi$
$857$	−34.9775	−1.19481	−0.597404	+	0.801941i	$0.703801\pi$
$858$	0	0
$859$	−6.45024	−0.220079	−0.110040	−	0.993927i	$-0.535098\pi$
$859$	−6.45024	−0.220079	−0.110040	+	0.993927i	$0.535098\pi$
$860$	0	0
$861$	−6.09565	−0.207739
$862$	0	0
$863$	−34.4037	−1.17112	−0.585559	−	0.810630i	$-0.699125\pi$
$863$	−34.4037	−1.17112	−0.585559	+	0.810630i	$0.699125\pi$
$864$	0	0
$865$	4.48112	0.152363
$866$	0	0
$867$	20.1570	0.684566
$868$	0	0
$869$	−78.9594	−2.67852
$870$	0	0
$871$	−26.9505	−0.913182
$872$	0	0
$873$	−12.8789	−0.435884
$874$	0	0
$875$	−4.56944	−0.154475
$876$	0	0
$877$	−40.3826	−1.36362	−0.681812	−	0.731528i	$-0.738808\pi$
$877$	−40.3826	−1.36362	−0.681812	+	0.731528i	$0.738808\pi$
$878$	0	0
$879$	−22.7899	−0.768684
$880$	0	0
$881$	15.8926	0.535435	0.267718	−	0.963497i	$-0.413731\pi$
$881$	15.8926	0.535435	0.267718	+	0.963497i	$0.413731\pi$
$882$	0	0
$883$	−25.3113	−0.851792	−0.425896	−	0.904772i	$-0.640041\pi$
$883$	−25.3113	−0.851792	−0.425896	+	0.904772i	$0.640041\pi$
$884$	0	0
$885$	−1.86855	−0.0628106
$886$	0	0
$887$	−19.6352	−0.659284	−0.329642	−	0.944106i	$-0.606928\pi$
$887$	−19.6352	−0.659284	−0.329642	+	0.944106i	$0.606928\pi$
$888$	0	0
$889$	−18.8789	−0.633178
$890$	0	0
$891$	4.87666	0.163374
$892$	0	0
$893$	8.98674	0.300730
$894$	0	0
$895$	−3.59359	−0.120120
$896$	0	0
$897$	18.6876	0.623960
$898$	0	0
$899$	−34.3139	−1.14443
$900$	0	0
$901$	−8.19130	−0.272892
$902$	0	0
$903$	−4.15327	−0.138212
$904$	0	0
$905$	−7.12402	−0.236811
$906$	0	0
$907$	4.84241	0.160790	0.0803948	−	0.996763i	$-0.474382\pi$
$907$	4.84241	0.160790	0.0803948	+	0.996763i	$0.474382\pi$
$908$	0	0
$909$	2.22045	0.0736477
$910$	0	0
$911$	−18.9176	−0.626768	−0.313384	−	0.949626i	$-0.601463\pi$
$911$	−18.9176	−0.626768	−0.313384	+	0.949626i	$0.601463\pi$
$912$	0	0
$913$	67.0702	2.21970
$914$	0	0
$915$	−2.56788	−0.0848917
$916$	0	0
$917$	−4.93428	−0.162944
$918$	0	0
$919$	9.11228	0.300586	0.150293	−	0.988641i	$-0.451978\pi$
$919$	9.11228	0.300586	0.150293	+	0.988641i	$0.451978\pi$
$920$	0	0
$921$	9.34379	0.307888
$922$	0	0
$923$	21.5518	0.709387
$924$	0	0
$925$	21.0854	0.693282
$926$	0	0
$927$	−11.0971	−0.364477
$928$	0	0
$929$	32.7294	1.07382	0.536909	−	0.843640i	$-0.319592\pi$
$929$	32.7294	1.07382	0.536909	+	0.843640i	$0.319592\pi$
$930$	0	0
$931$	−1.34379	−0.0440411
$932$	0	0
$933$	−6.68759	−0.218942
$934$	0	0
$935$	−13.8863	−0.454131
$936$	0	0
$937$	21.5066	0.702591	0.351295	−	0.936265i	$-0.385741\pi$
$937$	21.5066	0.702591	0.351295	+	0.936265i	$0.385741\pi$
$938$	0	0
$939$	15.6381	0.510329
$940$	0	0
$941$	−1.85115	−0.0603456	−0.0301728	−	0.999545i	$-0.509606\pi$
$941$	−1.85115	−0.0603456	−0.0301728	+	0.999545i	$0.509606\pi$
$942$	0	0
$943$	24.9657	0.812994
$944$	0	0
$945$	0.467138	0.0151960
$946$	0	0
$947$	45.3353	1.47320	0.736600	−	0.676329i	$-0.236430\pi$
$947$	45.3353	1.47320	0.736600	+	0.676329i	$0.236430\pi$
$948$	0	0
$949$	55.0266	1.78624
$950$	0	0
$951$	9.34379	0.302993
$952$	0	0
$953$	−40.5768	−1.31441	−0.657206	−	0.753711i	$-0.728262\pi$
$953$	−40.5768	−1.31441	−0.657206	+	0.753711i	$0.728262\pi$
$954$	0	0
$955$	2.20647	0.0713997
$956$	0	0
$957$	−37.9491	−1.22672
$958$	0	0
$959$	19.0103	0.613876
$960$	0	0
$961$	−11.5561	−0.372779
$962$	0	0
$963$	3.12334	0.100648
$964$	0	0
$965$	−10.4349	−0.335911
$966$	0	0
$967$	−15.8715	−0.510392	−0.255196	−	0.966889i	$-0.582140\pi$
$967$	−15.8715	−0.510392	−0.255196	+	0.966889i	$0.582140\pi$
$968$	0	0
$969$	−8.19130	−0.263143
$970$	0	0
$971$	−14.6876	−0.471347	−0.235674	−	0.971832i	$-0.575730\pi$
$971$	−14.6876	−0.471347	−0.235674	+	0.971832i	$0.575730\pi$
$972$	0	0
$973$	−10.4380	−0.334627
$974$	0	0
$975$	−21.8183	−0.698744
$976$	0	0
$977$	−21.2437	−0.679647	−0.339824	−	0.940489i	$-0.610367\pi$
$977$	−21.2437	−0.679647	−0.339824	+	0.940489i	$0.610367\pi$
$978$	0	0
$979$	−38.8387	−1.24129
$980$	0	0
$981$	10.6876	0.341228
$982$	0	0
$983$	−11.8265	−0.377206	−0.188603	−	0.982053i	$-0.560396\pi$
$983$	−11.8265	−0.377206	−0.188603	+	0.982053i	$0.560396\pi$
$984$	0	0
$985$	−0.717101	−0.0228487
$986$	0	0
$987$	6.68759	0.212868
$988$	0	0
$989$	17.0103	0.540897
$990$	0	0
$991$	28.9387	0.919269	0.459635	−	0.888108i	$-0.347980\pi$
$991$	28.9387	0.919269	0.459635	+	0.888108i	$0.347980\pi$
$992$	0	0
$993$	5.40874	0.171641
$994$	0	0
$995$	−6.86112	−0.217512
$996$	0	0
$997$	−56.0548	−1.77527	−0.887636	−	0.460546i	$-0.847654\pi$
$997$	−56.0548	−1.77527	−0.887636	+	0.460546i	$0.847654\pi$
$998$	0	0
$999$	−4.40952	−0.139511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.a.bp.1.3		4
4.3	odd	2		5376.2.a.bl.1.3		4
8.3	odd	2		5376.2.a.bq.1.2		4
8.5	even	2		5376.2.a.bm.1.2		4
16.3	odd	4		672.2.c.b.337.2		8
16.5	even	4		168.2.c.b.85.5	✓	8
16.11	odd	4		672.2.c.b.337.7		8
16.13	even	4		168.2.c.b.85.6	yes	8
48.5	odd	4		504.2.c.f.253.4		8
48.11	even	4		2016.2.c.e.1009.5		8
48.29	odd	4		504.2.c.f.253.3		8
48.35	even	4		2016.2.c.e.1009.4		8
112.13	odd	4		1176.2.c.c.589.6		8
112.27	even	4		4704.2.c.c.2353.2		8
112.69	odd	4		1176.2.c.c.589.5		8
112.83	even	4		4704.2.c.c.2353.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.b.85.5	✓	8	16.5	even	4
168.2.c.b.85.6	yes	8	16.13	even	4
504.2.c.f.253.3		8	48.29	odd	4
504.2.c.f.253.4		8	48.5	odd	4
672.2.c.b.337.2		8	16.3	odd	4
672.2.c.b.337.7		8	16.11	odd	4
1176.2.c.c.589.5		8	112.69	odd	4
1176.2.c.c.589.6		8	112.13	odd	4
2016.2.c.e.1009.4		8	48.35	even	4
2016.2.c.e.1009.5		8	48.11	even	4
4704.2.c.c.2353.2		8	112.27	even	4
4704.2.c.c.2353.7		8	112.83	even	4
5376.2.a.bl.1.3		4	4.3	odd	2
5376.2.a.bm.1.2		4	8.5	even	2
5376.2.a.bp.1.3		4	1.1	even	1	trivial
5376.2.a.bq.1.2		4	8.3	odd	2

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:	\( N \)	\(=\)	\( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5376.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.467138 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.467138 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.87666 q^{11} +4.56279 q^{13} -0.467138 q^{15} +6.09565 q^{17} -1.34379 q^{19} -1.00000 q^{21} +4.09565 q^{23} -4.78178 q^{25} +1.00000 q^{27} -7.78178 q^{29} +4.40952 q^{31} +4.87666 q^{33} +0.467138 q^{35} -4.40952 q^{37} +4.56279 q^{39} +6.09565 q^{41} +4.15327 q^{43} -0.467138 q^{45} -6.68759 q^{47} +1.00000 q^{49} +6.09565 q^{51} -1.34379 q^{53} -2.27807 q^{55} -1.34379 q^{57} +4.00000 q^{59} +5.49706 q^{61} -1.00000 q^{63} -2.13145 q^{65} -5.90658 q^{67} +4.09565 q^{69} +4.72339 q^{71} +12.0599 q^{73} -4.78178 q^{75} -4.87666 q^{77} -16.1913 q^{79} +1.00000 q^{81} +13.7533 q^{83} -2.84751 q^{85} -7.78178 q^{87} -7.96420 q^{89} -4.56279 q^{91} +4.40952 q^{93} +0.627737 q^{95} -12.8789 q^{97} +4.87666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 6 q^{11} - 4 q^{13} - 2 q^{15} + 2 q^{17} + 8 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} + 4 q^{27} + 4 q^{31} + 6 q^{33} + 2 q^{35} - 4 q^{37} - 4 q^{39} + 2 q^{41} + 8 q^{43} - 2 q^{45} + 4 q^{49} + 2 q^{51} + 8 q^{53} + 4 q^{55} + 8 q^{57} + 16 q^{59} - 4 q^{63} - 8 q^{65} + 12 q^{67} - 6 q^{69} + 14 q^{71} + 4 q^{73} + 12 q^{75} - 6 q^{77} - 20 q^{79} + 4 q^{81} + 28 q^{83} + 20 q^{85} - 10 q^{89} + 4 q^{91} + 4 q^{93} + 20 q^{95} + 20 q^{97} + 6 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9 + 6 * q^11 - 4 * q^13 - 2 * q^15 + 2 * q^17 + 8 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 + 4 * q^27 + 4 * q^31 + 6 * q^33 + 2 * q^35 - 4 * q^37 - 4 * q^39 + 2 * q^41 + 8 * q^43 - 2 * q^45 + 4 * q^49 + 2 * q^51 + 8 * q^53 + 4 * q^55 + 8 * q^57 + 16 * q^59 - 4 * q^63 - 8 * q^65 + 12 * q^67 - 6 * q^69 + 14 * q^71 + 4 * q^73 + 12 * q^75 - 6 * q^77 - 20 * q^79 + 4 * q^81 + 28 * q^83 + 20 * q^85 - 10 * q^89 + 4 * q^91 + 4 * q^93 + 20 * q^95 + 20 * q^97 + 6 * q^99

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