LMFDB - Embedded newform 1143.2.a.k.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 269x^{12} - 1408x^{10} + 3924x^{8} - 5655x^{6} + 3886x^{4} - 1107x^{2} + 108 $ x^16 - 26x^14 + 269x^12 - 1408x^10 + 3924x^8 - 5655x^6 + 3886x^4 - 1107*x^2 + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.28010$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.28010 q^{2} +3.19885 q^{4} -3.42195 q^{5} +0.843402 q^{7} +2.73351 q^{8} +O(q^{10})$	\(q+2.28010 q^{2} +3.19885 q^{4} -3.42195 q^{5} +0.843402 q^{7} +2.73351 q^{8} -7.80238 q^{10} +4.43106 q^{11} +4.58813 q^{13} +1.92304 q^{14} -0.165040 q^{16} +1.00391 q^{17} +8.11733 q^{19} -10.9463 q^{20} +10.1033 q^{22} +1.02192 q^{23} +6.70971 q^{25} +10.4614 q^{26} +2.69792 q^{28} +1.29940 q^{29} +8.48585 q^{31} -5.84332 q^{32} +2.28902 q^{34} -2.88608 q^{35} -5.26578 q^{37} +18.5083 q^{38} -9.35391 q^{40} -9.49446 q^{41} -10.7240 q^{43} +14.1743 q^{44} +2.33008 q^{46} -4.14378 q^{47} -6.28867 q^{49} +15.2988 q^{50} +14.6767 q^{52} -3.65908 q^{53} -15.1628 q^{55} +2.30545 q^{56} +2.96276 q^{58} +10.2664 q^{59} -5.18377 q^{61} +19.3486 q^{62} -12.9933 q^{64} -15.7003 q^{65} +11.3539 q^{67} +3.21137 q^{68} -6.58054 q^{70} -3.12707 q^{71} +9.92239 q^{73} -12.0065 q^{74} +25.9662 q^{76} +3.73716 q^{77} -12.0998 q^{79} +0.564756 q^{80} -21.6483 q^{82} +1.67046 q^{83} -3.43533 q^{85} -24.4518 q^{86} +12.1123 q^{88} -7.18230 q^{89} +3.86963 q^{91} +3.26897 q^{92} -9.44824 q^{94} -27.7771 q^{95} +3.53465 q^{97} -14.3388 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 10 * q^7	$ 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

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 $

$2$	2.28010	1.61227	0.806137	−	0.591729i	$-0.201554\pi$
$2$	2.28010	1.61227	0.806137	+	0.591729i	$0.201554\pi$
$3$	0	0
$3$	0	0
$4$	3.19885	1.59943
$5$	−3.42195	−1.53034	−0.765170	−	0.643828i	$-0.777345\pi$
$5$	−3.42195	−1.53034	−0.765170	+	0.643828i	$0.777345\pi$
$6$	0	0
$7$	0.843402	0.318776	0.159388	−	0.987216i	$-0.449048\pi$
$7$	0.843402	0.318776	0.159388	+	0.987216i	$0.449048\pi$
$8$	2.73351	0.966441
$9$	0	0
$10$	−7.80238	−2.46733
$11$	4.43106	1.33601	0.668007	−	0.744155i	$-0.267147\pi$
$11$	4.43106	1.33601	0.668007	+	0.744155i	$0.267147\pi$
$12$	0	0
$13$	4.58813	1.27252	0.636259	−	0.771476i	$-0.280481\pi$
$13$	4.58813	1.27252	0.636259	+	0.771476i	$0.280481\pi$
$14$	1.92304	0.513954
$15$	0	0
$16$	−0.165040	−0.0412599
$17$	1.00391	0.243484	0.121742	−	0.992562i	$-0.461152\pi$
$17$	1.00391	0.243484	0.121742	+	0.992562i	$0.461152\pi$
$18$	0	0
$19$	8.11733	1.86224	0.931122	−	0.364708i	$-0.118831\pi$
$19$	8.11733	1.86224	0.931122	+	0.364708i	$0.118831\pi$
$20$	−10.9463	−2.44767
$21$	0	0
$22$	10.1033	2.15402
$23$	1.02192	0.213085	0.106543	−	0.994308i	$-0.466022\pi$
$23$	1.02192	0.213085	0.106543	+	0.994308i	$0.466022\pi$
$24$	0	0
$25$	6.70971	1.34194
$26$	10.4614	2.05165
$27$	0	0
$28$	2.69792	0.509859
$29$	1.29940	0.241292	0.120646	−	0.992696i	$-0.461503\pi$
$29$	1.29940	0.241292	0.120646	+	0.992696i	$0.461503\pi$
$30$	0	0
$31$	8.48585	1.52410	0.762051	−	0.647516i	$-0.224192\pi$
$31$	8.48585	1.52410	0.762051	+	0.647516i	$0.224192\pi$
$32$	−5.84332	−1.03296
$33$	0	0
$34$	2.28902	0.392563
$35$	−2.88608	−0.487836
$36$	0	0
$37$	−5.26578	−0.865688	−0.432844	−	0.901469i	$-0.642490\pi$
$37$	−5.26578	−0.865688	−0.432844	+	0.901469i	$0.642490\pi$
$38$	18.5083	3.00245
$39$	0	0
$40$	−9.35391	−1.47898
$41$	−9.49446	−1.48279	−0.741393	−	0.671071i	$-0.765835\pi$
$41$	−9.49446	−1.48279	−0.741393	+	0.671071i	$0.765835\pi$
$42$	0	0
$43$	−10.7240	−1.63540	−0.817698	−	0.575648i	$-0.804750\pi$
$43$	−10.7240	−1.63540	−0.817698	+	0.575648i	$0.804750\pi$
$44$	14.1743	2.13686
$45$	0	0
$46$	2.33008	0.343551
$47$	−4.14378	−0.604433	−0.302216	−	0.953239i	$-0.597727\pi$
$47$	−4.14378	−0.604433	−0.302216	+	0.953239i	$0.597727\pi$
$48$	0	0
$49$	−6.28867	−0.898382
$50$	15.2988	2.16358
$51$	0	0
$52$	14.6767	2.03530
$53$	−3.65908	−0.502613	−0.251307	−	0.967908i	$-0.580860\pi$
$53$	−3.65908	−0.502613	−0.251307	+	0.967908i	$0.580860\pi$
$54$	0	0
$55$	−15.1628	−2.04456
$56$	2.30545	0.308078
$57$	0	0
$58$	2.96276	0.389029
$59$	10.2664	1.33658	0.668289	−	0.743902i	$-0.267027\pi$
$59$	10.2664	1.33658	0.668289	+	0.743902i	$0.267027\pi$
$60$	0	0
$61$	−5.18377	−0.663713	−0.331857	−	0.943330i	$-0.607675\pi$
$61$	−5.18377	−0.663713	−0.331857	+	0.943330i	$0.607675\pi$
$62$	19.3486	2.45727
$63$	0	0
$64$	−12.9933	−1.62416
$65$	−15.7003	−1.94738
$66$	0	0
$67$	11.3539	1.38710	0.693551	−	0.720408i	$-0.256045\pi$
$67$	11.3539	1.38710	0.693551	+	0.720408i	$0.256045\pi$
$68$	3.21137	0.389435
$69$	0	0
$70$	−6.58054	−0.786525
$71$	−3.12707	−0.371114	−0.185557	−	0.982633i	$-0.559409\pi$
$71$	−3.12707	−0.371114	−0.185557	+	0.982633i	$0.559409\pi$
$72$	0	0
$73$	9.92239	1.16133	0.580664	−	0.814144i	$-0.302793\pi$
$73$	9.92239	1.16133	0.580664	+	0.814144i	$0.302793\pi$
$74$	−12.0065	−1.39573
$75$	0	0
$76$	25.9662	2.97852
$77$	3.73716	0.425889
$78$	0	0
$79$	−12.0998	−1.36134	−0.680669	−	0.732591i	$-0.738311\pi$
$79$	−12.0998	−1.36134	−0.680669	+	0.732591i	$0.738311\pi$
$80$	0.564756	0.0631417
$81$	0	0
$82$	−21.6483	−2.39066
$83$	1.67046	0.183356	0.0916782	−	0.995789i	$-0.470777\pi$
$83$	1.67046	0.183356	0.0916782	+	0.995789i	$0.470777\pi$
$84$	0	0
$85$	−3.43533	−0.372614
$86$	−24.4518	−2.63670
$87$	0	0
$88$	12.1123	1.29118
$89$	−7.18230	−0.761322	−0.380661	−	0.924715i	$-0.624304\pi$
$89$	−7.18230	−0.761322	−0.380661	+	0.924715i	$0.624304\pi$
$90$	0	0
$91$	3.86963	0.405648
$92$	3.26897	0.340814
$93$	0	0
$94$	−9.44824	−0.974511
$95$	−27.7771	−2.84987
$96$	0	0
$97$	3.53465	0.358890	0.179445	−	0.983768i	$-0.442570\pi$
$97$	3.53465	0.358890	0.179445	+	0.983768i	$0.442570\pi$
$98$	−14.3388	−1.44844
$99$	0	0
$100$	21.4634	2.14634
$101$	−9.99977	−0.995014	−0.497507	−	0.867460i	$-0.665751\pi$
$101$	−9.99977	−0.995014	−0.497507	+	0.867460i	$0.665751\pi$
$102$	0	0
$103$	14.6135	1.43991	0.719957	−	0.694018i	$-0.244161\pi$
$103$	14.6135	1.43991	0.719957	+	0.694018i	$0.244161\pi$
$104$	12.5417	1.22981
$105$	0	0
$106$	−8.34306	−0.810350
$107$	9.59054	0.927152	0.463576	−	0.886057i	$-0.346566\pi$
$107$	9.59054	0.927152	0.463576	+	0.886057i	$0.346566\pi$
$108$	0	0
$109$	2.77854	0.266136	0.133068	−	0.991107i	$-0.457517\pi$
$109$	2.77854	0.266136	0.133068	+	0.991107i	$0.457517\pi$
$110$	−34.5728	−3.29639
$111$	0	0
$112$	−0.139195	−0.0131527
$113$	3.53603	0.332641	0.166321	−	0.986072i	$-0.446811\pi$
$113$	3.53603	0.332641	0.166321	+	0.986072i	$0.446811\pi$
$114$	0	0
$115$	−3.49695	−0.326093
$116$	4.15659	0.385930
$117$	0	0
$118$	23.4085	2.15493
$119$	0.846701	0.0776169
$120$	0	0
$121$	8.63429	0.784935
$122$	−11.8195	−1.07009
$123$	0	0
$124$	27.1450	2.43769
$125$	−5.85053	−0.523288
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−17.9393	−1.58563
$129$	0	0
$130$	−35.7983	−3.13972
$131$	6.50183	0.568068	0.284034	−	0.958814i	$-0.408327\pi$
$131$	6.50183	0.568068	0.284034	+	0.958814i	$0.408327\pi$
$132$	0	0
$133$	6.84617	0.593638
$134$	25.8881	2.23639
$135$	0	0
$136$	2.74420	0.235313
$137$	−17.3515	−1.48244	−0.741218	−	0.671265i	$-0.765751\pi$
$137$	−17.3515	−1.48244	−0.741218	+	0.671265i	$0.765751\pi$
$138$	0	0
$139$	−19.3018	−1.63715	−0.818577	−	0.574396i	$-0.805237\pi$
$139$	−19.3018	−1.63715	−0.818577	+	0.574396i	$0.805237\pi$
$140$	−9.23213	−0.780258
$141$	0	0
$142$	−7.13002	−0.598338
$143$	20.3303	1.70010
$144$	0	0
$145$	−4.44647	−0.369259
$146$	22.6240	1.87238
$147$	0	0
$148$	−16.8444	−1.38461
$149$	−13.7377	−1.12544	−0.562719	−	0.826648i	$-0.690245\pi$
$149$	−13.7377	−1.12544	−0.562719	+	0.826648i	$0.690245\pi$
$150$	0	0
$151$	−5.18739	−0.422144	−0.211072	−	0.977471i	$-0.567695\pi$
$151$	−5.18739	−0.422144	−0.211072	+	0.977471i	$0.567695\pi$
$152$	22.1888	1.79975
$153$	0	0
$154$	8.52111	0.686650
$155$	−29.0381	−2.33240
$156$	0	0
$157$	16.5431	1.32029	0.660143	−	0.751140i	$-0.270496\pi$
$157$	16.5431	1.32029	0.660143	+	0.751140i	$0.270496\pi$
$158$	−27.5888	−2.19485
$159$	0	0
$160$	19.9955	1.58079
$161$	0.861889	0.0679264
$162$	0	0
$163$	13.2296	1.03622	0.518110	−	0.855314i	$-0.326636\pi$
$163$	13.2296	1.03622	0.518110	+	0.855314i	$0.326636\pi$
$164$	−30.3714	−2.37161
$165$	0	0
$166$	3.80881	0.295621
$167$	17.2254	1.33294	0.666472	−	0.745530i	$-0.267803\pi$
$167$	17.2254	1.33294	0.666472	+	0.745530i	$0.267803\pi$
$168$	0	0
$169$	8.05090	0.619300
$170$	−7.83289	−0.600756
$171$	0	0
$172$	−34.3045	−2.61570
$173$	−25.6623	−1.95107	−0.975535	−	0.219843i	$-0.929446\pi$
$173$	−25.6623	−1.95107	−0.975535	+	0.219843i	$0.929446\pi$
$174$	0	0
$175$	5.65898	0.427779
$176$	−0.731300	−0.0551238
$177$	0	0
$178$	−16.3764	−1.22746
$179$	11.9704	0.894707	0.447353	−	0.894357i	$-0.352367\pi$
$179$	11.9704	0.894707	0.447353	+	0.894357i	$0.352367\pi$
$180$	0	0
$181$	2.53298	0.188275	0.0941373	−	0.995559i	$-0.469991\pi$
$181$	2.53298	0.188275	0.0941373	+	0.995559i	$0.469991\pi$
$182$	8.82315	0.654015
$183$	0	0
$184$	2.79343	0.205934
$185$	18.0192	1.32480
$186$	0	0
$187$	4.44839	0.325299
$188$	−13.2554	−0.966746
$189$	0	0
$190$	−63.3345	−4.59477
$191$	10.8901	0.787979	0.393989	−	0.919115i	$-0.371095\pi$
$191$	10.8901	0.787979	0.393989	+	0.919115i	$0.371095\pi$
$192$	0	0
$193$	0.678458	0.0488365	0.0244182	−	0.999702i	$-0.492227\pi$
$193$	0.678458	0.0488365	0.0244182	+	0.999702i	$0.492227\pi$
$194$	8.05936	0.578629
$195$	0	0
$196$	−20.1166	−1.43690
$197$	−16.0801	−1.14566	−0.572829	−	0.819675i	$-0.694154\pi$
$197$	−16.0801	−1.14566	−0.572829	+	0.819675i	$0.694154\pi$
$198$	0	0
$199$	−2.43571	−0.172663	−0.0863314	−	0.996266i	$-0.527514\pi$
$199$	−2.43571	−0.172663	−0.0863314	+	0.996266i	$0.527514\pi$
$200$	18.3410	1.29691
$201$	0	0
$202$	−22.8005	−1.60423
$203$	1.09592	0.0769182
$204$	0	0
$205$	32.4895	2.26917
$206$	33.3203	2.32154
$207$	0	0
$208$	−0.757222	−0.0525039
$209$	35.9684	2.48798
$210$	0	0
$211$	−15.0019	−1.03277	−0.516386	−	0.856356i	$-0.672723\pi$
$211$	−15.0019	−1.03277	−0.516386	+	0.856356i	$0.672723\pi$
$212$	−11.7049	−0.803893
$213$	0	0
$214$	21.8674	1.49482
$215$	36.6970	2.50271
$216$	0	0
$217$	7.15698	0.485847
$218$	6.33535	0.429084
$219$	0	0
$220$	−48.5037	−3.27012
$221$	4.60607	0.309838
$222$	0	0
$223$	−2.95620	−0.197962	−0.0989810	−	0.995089i	$-0.531558\pi$
$223$	−2.95620	−0.197962	−0.0989810	+	0.995089i	$0.531558\pi$
$224$	−4.92827	−0.329284
$225$	0	0
$226$	8.06249	0.536309
$227$	3.61576	0.239986	0.119993	−	0.992775i	$-0.461713\pi$
$227$	3.61576	0.239986	0.119993	+	0.992775i	$0.461713\pi$
$228$	0	0
$229$	−8.89550	−0.587831	−0.293916	−	0.955831i	$-0.594958\pi$
$229$	−8.89550	−0.587831	−0.293916	+	0.955831i	$0.594958\pi$
$230$	−7.97340	−0.525751
$231$	0	0
$232$	3.55192	0.233195
$233$	−17.8335	−1.16831	−0.584156	−	0.811641i	$-0.698574\pi$
$233$	−17.8335	−1.16831	−0.584156	+	0.811641i	$0.698574\pi$
$234$	0	0
$235$	14.1798	0.924988
$236$	32.8409	2.13776
$237$	0	0
$238$	1.93056	0.125140
$239$	−1.83162	−0.118478	−0.0592388	−	0.998244i	$-0.518867\pi$
$239$	−1.83162	−0.118478	−0.0592388	+	0.998244i	$0.518867\pi$
$240$	0	0
$241$	−8.43682	−0.543464	−0.271732	−	0.962373i	$-0.587596\pi$
$241$	−8.43682	−0.543464	−0.271732	+	0.962373i	$0.587596\pi$
$242$	19.6870	1.26553
$243$	0	0
$244$	−16.5821	−1.06156
$245$	21.5195	1.37483
$246$	0	0
$247$	37.2433	2.36974
$248$	23.1961	1.47296
$249$	0	0
$250$	−13.3398	−0.843683
$251$	−23.8762	−1.50705	−0.753527	−	0.657417i	$-0.771649\pi$
$251$	−23.8762	−1.50705	−0.753527	+	0.657417i	$0.771649\pi$
$252$	0	0
$253$	4.52819	0.284685
$254$	2.28010	0.143066
$255$	0	0
$256$	−14.9169	−0.932305
$257$	28.0472	1.74954	0.874769	−	0.484540i	$-0.161013\pi$
$257$	28.0472	1.74954	0.874769	+	0.484540i	$0.161013\pi$
$258$	0	0
$259$	−4.44116	−0.275961
$260$	−50.2230	−3.11470
$261$	0	0
$262$	14.8248	0.915881
$263$	−15.7047	−0.968394	−0.484197	−	0.874959i	$-0.660888\pi$
$263$	−15.7047	−0.968394	−0.484197	+	0.874959i	$0.660888\pi$
$264$	0	0
$265$	12.5212	0.769169
$266$	15.6100	0.957108
$267$	0	0
$268$	36.3195	2.21857
$269$	−19.7346	−1.20324	−0.601619	−	0.798783i	$-0.705477\pi$
$269$	−19.7346	−1.20324	−0.601619	+	0.798783i	$0.705477\pi$
$270$	0	0
$271$	1.05681	0.0641968	0.0320984	−	0.999485i	$-0.489781\pi$
$271$	1.05681	0.0641968	0.0320984	+	0.999485i	$0.489781\pi$
$272$	−0.165685	−0.0100461
$273$	0	0
$274$	−39.5631	−2.39009
$275$	29.7311	1.79285
$276$	0	0
$277$	−10.4961	−0.630651	−0.315326	−	0.948984i	$-0.602114\pi$
$277$	−10.4961	−0.630651	−0.315326	+	0.948984i	$0.602114\pi$
$278$	−44.0100	−2.63954
$279$	0	0
$280$	−7.88911	−0.471464
$281$	14.7154	0.877848	0.438924	−	0.898524i	$-0.355360\pi$
$281$	14.7154	0.877848	0.438924	+	0.898524i	$0.355360\pi$
$282$	0	0
$283$	−27.2460	−1.61960	−0.809802	−	0.586703i	$-0.800426\pi$
$283$	−27.2460	−1.61960	−0.809802	+	0.586703i	$0.800426\pi$
$284$	−10.0030	−0.593571
$285$	0	0
$286$	46.3550	2.74103
$287$	−8.00765	−0.472677
$288$	0	0
$289$	−15.9922	−0.940715
$290$	−10.1384	−0.595347
$291$	0	0
$292$	31.7403	1.85746
$293$	18.2408	1.06564	0.532821	−	0.846228i	$-0.321132\pi$
$293$	18.2408	1.06564	0.532821	+	0.846228i	$0.321132\pi$
$294$	0	0
$295$	−35.1312	−2.04542
$296$	−14.3940	−0.836636
$297$	0	0
$298$	−31.3234	−1.81451
$299$	4.68870	0.271154
$300$	0	0
$301$	−9.04464	−0.521325
$302$	−11.8278	−0.680612
$303$	0	0
$304$	−1.33968	−0.0768359
$305$	17.7386	1.01571
$306$	0	0
$307$	14.7408	0.841300	0.420650	−	0.907223i	$-0.361802\pi$
$307$	14.7408	0.841300	0.420650	+	0.907223i	$0.361802\pi$
$308$	11.9546	0.681179
$309$	0	0
$310$	−66.2098	−3.76046
$311$	−6.05209	−0.343182	−0.171591	−	0.985168i	$-0.554891\pi$
$311$	−6.05209	−0.343182	−0.171591	+	0.985168i	$0.554891\pi$
$312$	0	0
$313$	−1.10936	−0.0627046	−0.0313523	−	0.999508i	$-0.509981\pi$
$313$	−1.10936	−0.0627046	−0.0313523	+	0.999508i	$0.509981\pi$
$314$	37.7200	2.12866
$315$	0	0
$316$	−38.7056	−2.17736
$317$	−28.9380	−1.62532	−0.812659	−	0.582740i	$-0.801981\pi$
$317$	−28.9380	−1.62532	−0.812659	+	0.582740i	$0.801981\pi$
$318$	0	0
$319$	5.75771	0.322370
$320$	44.4623	2.48552
$321$	0	0
$322$	1.96519	0.109516
$323$	8.14908	0.453427
$324$	0	0
$325$	30.7850	1.70764
$326$	30.1647	1.67067
$327$	0	0
$328$	−25.9532	−1.43302
$329$	−3.49487	−0.192679
$330$	0	0
$331$	6.86344	0.377249	0.188624	−	0.982049i	$-0.439597\pi$
$331$	6.86344	0.377249	0.188624	+	0.982049i	$0.439597\pi$
$332$	5.34355	0.293265
$333$	0	0
$334$	39.2757	2.14907
$335$	−38.8525	−2.12274
$336$	0	0
$337$	−34.9097	−1.90165	−0.950826	−	0.309727i	$-0.899762\pi$
$337$	−34.9097	−1.90165	−0.950826	+	0.309727i	$0.899762\pi$
$338$	18.3568	0.998481
$339$	0	0
$340$	−10.9891	−0.595969
$341$	37.6013	2.03622
$342$	0	0
$343$	−11.2077	−0.605158
$344$	−29.3141	−1.58051
$345$	0	0
$346$	−58.5127	−3.14566
$347$	−7.58695	−0.407289	−0.203644	−	0.979045i	$-0.565279\pi$
$347$	−7.58695	−0.407289	−0.203644	+	0.979045i	$0.565279\pi$
$348$	0	0
$349$	−19.5095	−1.04432	−0.522159	−	0.852848i	$-0.674873\pi$
$349$	−19.5095	−1.04432	−0.522159	+	0.852848i	$0.674873\pi$
$350$	12.9030	0.689697
$351$	0	0
$352$	−25.8921	−1.38005
$353$	12.9279	0.688084	0.344042	−	0.938954i	$-0.388204\pi$
$353$	12.9279	0.688084	0.344042	+	0.938954i	$0.388204\pi$
$354$	0	0
$355$	10.7006	0.567931
$356$	−22.9751	−1.21768
$357$	0	0
$358$	27.2936	1.44251
$359$	6.43941	0.339859	0.169930	−	0.985456i	$-0.445646\pi$
$359$	6.43941	0.339859	0.169930	+	0.985456i	$0.445646\pi$
$360$	0	0
$361$	46.8911	2.46795
$362$	5.77544	0.303550
$363$	0	0
$364$	12.3784	0.648804
$365$	−33.9539	−1.77723
$366$	0	0
$367$	9.74215	0.508536	0.254268	−	0.967134i	$-0.418165\pi$
$367$	9.74215	0.508536	0.254268	+	0.967134i	$0.418165\pi$
$368$	−0.168657	−0.00879186
$369$	0	0
$370$	41.0856	2.13594
$371$	−3.08607	−0.160221
$372$	0	0
$373$	−13.5879	−0.703553	−0.351777	−	0.936084i	$-0.614422\pi$
$373$	−13.5879	−0.703553	−0.351777	+	0.936084i	$0.614422\pi$
$374$	10.1428	0.524470
$375$	0	0
$376$	−11.3271	−0.584149
$377$	5.96181	0.307049
$378$	0	0
$379$	−32.5941	−1.67425	−0.837123	−	0.547014i	$-0.815764\pi$
$379$	−32.5941	−1.67425	−0.837123	+	0.547014i	$0.815764\pi$
$380$	−88.8548	−4.55815
$381$	0	0
$382$	24.8305	1.27044
$383$	−6.79032	−0.346969	−0.173485	−	0.984837i	$-0.555503\pi$
$383$	−6.79032	−0.346969	−0.173485	+	0.984837i	$0.555503\pi$
$384$	0	0
$385$	−12.7884	−0.651756
$386$	1.54695	0.0787378
$387$	0	0
$388$	11.3068	0.574018
$389$	−18.4103	−0.933437	−0.466719	−	0.884406i	$-0.654564\pi$
$389$	−18.4103	−0.933437	−0.466719	+	0.884406i	$0.654564\pi$
$390$	0	0
$391$	1.02592	0.0518828
$392$	−17.1901	−0.868233
$393$	0	0
$394$	−36.6642	−1.84711
$395$	41.4050	2.08331
$396$	0	0
$397$	−1.57204	−0.0788982	−0.0394491	−	0.999222i	$-0.512560\pi$
$397$	−1.57204	−0.0788982	−0.0394491	+	0.999222i	$0.512560\pi$
$398$	−5.55366	−0.278380
$399$	0	0
$400$	−1.10737	−0.0553684
$401$	24.2028	1.20863	0.604314	−	0.796746i	$-0.293447\pi$
$401$	24.2028	1.20863	0.604314	+	0.796746i	$0.293447\pi$
$402$	0	0
$403$	38.9341	1.93945
$404$	−31.9878	−1.59145
$405$	0	0
$406$	2.49880	0.124013
$407$	−23.3330	−1.15657
$408$	0	0
$409$	1.02904	0.0508829	0.0254415	−	0.999676i	$-0.491901\pi$
$409$	1.02904	0.0508829	0.0254415	+	0.999676i	$0.491901\pi$
$410$	74.0794	3.65852
$411$	0	0
$412$	46.7466	2.30304
$413$	8.65874	0.426069
$414$	0	0
$415$	−5.71621	−0.280598
$416$	−26.8099	−1.31446
$417$	0	0
$418$	82.0115	4.01131
$419$	8.64674	0.422421	0.211210	−	0.977441i	$-0.432259\pi$
$419$	8.64674	0.422421	0.211210	+	0.977441i	$0.432259\pi$
$420$	0	0
$421$	20.6633	1.00707	0.503534	−	0.863976i	$-0.332033\pi$
$421$	20.6633	1.00707	0.503534	+	0.863976i	$0.332033\pi$
$422$	−34.2058	−1.66511
$423$	0	0
$424$	−10.0021	−0.485746
$425$	6.73595	0.326742
$426$	0	0
$427$	−4.37200	−0.211576
$428$	30.6787	1.48291
$429$	0	0
$430$	83.6727	4.03506
$431$	17.9957	0.866823	0.433412	−	0.901196i	$-0.357310\pi$
$431$	17.9957	0.866823	0.433412	+	0.901196i	$0.357310\pi$
$432$	0	0
$433$	39.4539	1.89603	0.948017	−	0.318219i	$-0.103085\pi$
$433$	39.4539	1.89603	0.948017	+	0.318219i	$0.103085\pi$
$434$	16.3186	0.783319
$435$	0	0
$436$	8.88815	0.425665
$437$	8.29526	0.396816
$438$	0	0
$439$	−32.1475	−1.53432	−0.767159	−	0.641457i	$-0.778330\pi$
$439$	−32.1475	−1.53432	−0.767159	+	0.641457i	$0.778330\pi$
$440$	−41.4477	−1.97594
$441$	0	0
$442$	10.5023	0.499544
$443$	12.2589	0.582436	0.291218	−	0.956657i	$-0.405940\pi$
$443$	12.2589	0.582436	0.291218	+	0.956657i	$0.405940\pi$
$444$	0	0
$445$	24.5774	1.16508
$446$	−6.74044	−0.319169
$447$	0	0
$448$	−10.9586	−0.517743
$449$	−11.9744	−0.565105	−0.282552	−	0.959252i	$-0.591181\pi$
$449$	−11.9744	−0.565105	−0.282552	+	0.959252i	$0.591181\pi$
$450$	0	0
$451$	−42.0705	−1.98102
$452$	11.3112	0.532036
$453$	0	0
$454$	8.24428	0.386923
$455$	−13.2417	−0.620779
$456$	0	0
$457$	19.3855	0.906814	0.453407	−	0.891303i	$-0.350208\pi$
$457$	19.3855	0.906814	0.453407	+	0.891303i	$0.350208\pi$
$458$	−20.2826	−0.947745
$459$	0	0
$460$	−11.1862	−0.521561
$461$	16.3404	0.761047	0.380524	−	0.924771i	$-0.375744\pi$
$461$	16.3404	0.761047	0.380524	+	0.924771i	$0.375744\pi$
$462$	0	0
$463$	−4.05178	−0.188302	−0.0941510	−	0.995558i	$-0.530014\pi$
$463$	−4.05178	−0.188302	−0.0941510	+	0.995558i	$0.530014\pi$
$464$	−0.214452	−0.00995569
$465$	0	0
$466$	−40.6622	−1.88364
$467$	28.6388	1.32525	0.662624	−	0.748952i	$-0.269443\pi$
$467$	28.6388	1.32525	0.662624	+	0.748952i	$0.269443\pi$
$468$	0	0
$469$	9.57591	0.442174
$470$	32.3314	1.49133
$471$	0	0
$472$	28.0634	1.29172
$473$	−47.5187	−2.18491
$474$	0	0
$475$	54.4649	2.49902
$476$	2.70847	0.124143
$477$	0	0
$478$	−4.17627	−0.191018
$479$	−30.8364	−1.40895	−0.704477	−	0.709727i	$-0.748818\pi$
$479$	−30.8364	−1.40895	−0.704477	+	0.709727i	$0.748818\pi$
$480$	0	0
$481$	−24.1600	−1.10160
$482$	−19.2368	−0.876212
$483$	0	0
$484$	27.6198	1.25545
$485$	−12.0954	−0.549223
$486$	0	0
$487$	22.7741	1.03199	0.515996	−	0.856591i	$-0.327422\pi$
$487$	22.7741	1.03199	0.515996	+	0.856591i	$0.327422\pi$
$488$	−14.1699	−0.641440
$489$	0	0
$490$	49.0666	2.21660
$491$	18.9802	0.856563	0.428282	−	0.903645i	$-0.359119\pi$
$491$	18.9802	0.856563	0.428282	+	0.903645i	$0.359119\pi$
$492$	0	0
$493$	1.30448	0.0587509
$494$	84.9185	3.82066
$495$	0	0
$496$	−1.40050	−0.0628843
$497$	−2.63737	−0.118302
$498$	0	0
$499$	6.42460	0.287605	0.143802	−	0.989606i	$-0.454067\pi$
$499$	6.42460	0.287605	0.143802	+	0.989606i	$0.454067\pi$
$500$	−18.7150	−0.836961
$501$	0	0
$502$	−54.4402	−2.42978
$503$	−31.7049	−1.41365	−0.706826	−	0.707387i	$-0.749874\pi$
$503$	−31.7049	−1.41365	−0.706826	+	0.707387i	$0.749874\pi$
$504$	0	0
$505$	34.2186	1.52271
$506$	10.3247	0.458990
$507$	0	0
$508$	3.19885	0.141926
$509$	29.8311	1.32224	0.661119	−	0.750281i	$-0.270082\pi$
$509$	29.8311	1.32224	0.661119	+	0.750281i	$0.270082\pi$
$510$	0	0
$511$	8.36856	0.370203
$512$	1.86665	0.0824952
$513$	0	0
$514$	63.9505	2.82074
$515$	−50.0067	−2.20356
$516$	0	0
$517$	−18.3613	−0.807531
$518$	−10.1263	−0.444924
$519$	0	0
$520$	−42.9169	−1.88203
$521$	41.0993	1.80059	0.900296	−	0.435277i	$-0.143350\pi$
$521$	41.0993	1.80059	0.900296	+	0.435277i	$0.143350\pi$
$522$	0	0
$523$	−15.2319	−0.666044	−0.333022	−	0.942919i	$-0.608068\pi$
$523$	−15.2319	−0.666044	−0.333022	+	0.942919i	$0.608068\pi$
$524$	20.7984	0.908583
$525$	0	0
$526$	−35.8083	−1.56132
$527$	8.51904	0.371095
$528$	0	0
$529$	−21.9557	−0.954595
$530$	28.5495	1.24011
$531$	0	0
$532$	21.8999	0.949481
$533$	−43.5618	−1.88687
$534$	0	0
$535$	−32.8183	−1.41886
$536$	31.0360	1.34055
$537$	0	0
$538$	−44.9968	−1.93995
$539$	−27.8655	−1.20025
$540$	0	0
$541$	−35.8911	−1.54308	−0.771540	−	0.636181i	$-0.780513\pi$
$541$	−35.8911	−1.54308	−0.771540	+	0.636181i	$0.780513\pi$
$542$	2.40964	0.103503
$543$	0	0
$544$	−5.86618	−0.251510
$545$	−9.50802	−0.407279
$546$	0	0
$547$	10.4130	0.445227	0.222613	−	0.974907i	$-0.428541\pi$
$547$	10.4130	0.445227	0.222613	+	0.974907i	$0.428541\pi$
$548$	−55.5048	−2.37105
$549$	0	0
$550$	67.7899	2.89057
$551$	10.5477	0.449345
$552$	0	0
$553$	−10.2050	−0.433962
$554$	−23.9322	−1.01678
$555$	0	0
$556$	−61.7436	−2.61851
$557$	−10.4915	−0.444540	−0.222270	−	0.974985i	$-0.571347\pi$
$557$	−10.4915	−0.444540	−0.222270	+	0.974985i	$0.571347\pi$
$558$	0	0
$559$	−49.2031	−2.08107
$560$	0.476316	0.0201280
$561$	0	0
$562$	33.5526	1.41533
$563$	−23.1051	−0.973766	−0.486883	−	0.873467i	$-0.661866\pi$
$563$	−23.1051	−0.973766	−0.486883	+	0.873467i	$0.661866\pi$
$564$	0	0
$565$	−12.1001	−0.509055
$566$	−62.1235	−2.61125
$567$	0	0
$568$	−8.54786	−0.358660
$569$	33.8836	1.42047	0.710237	−	0.703962i	$-0.248588\pi$
$569$	33.8836	1.42047	0.710237	+	0.703962i	$0.248588\pi$
$570$	0	0
$571$	−2.76744	−0.115814	−0.0579070	−	0.998322i	$-0.518443\pi$
$571$	−2.76744	−0.115814	−0.0579070	+	0.998322i	$0.518443\pi$
$572$	65.0335	2.71919
$573$	0	0
$574$	−18.2582	−0.762084
$575$	6.85679	0.285948
$576$	0	0
$577$	14.4484	0.601494	0.300747	−	0.953704i	$-0.402764\pi$
$577$	14.4484	0.601494	0.300747	+	0.953704i	$0.402764\pi$
$578$	−36.4637	−1.51669
$579$	0	0
$580$	−14.2236	−0.590604
$581$	1.40887	0.0584496
$582$	0	0
$583$	−16.2136	−0.671499
$584$	27.1229	1.12235
$585$	0	0
$586$	41.5909	1.71811
$587$	19.3600	0.799074	0.399537	−	0.916717i	$-0.369171\pi$
$587$	19.3600	0.799074	0.399537	+	0.916717i	$0.369171\pi$
$588$	0	0
$589$	68.8824	2.83825
$590$	−80.1027	−3.29777
$591$	0	0
$592$	0.869061	0.0357182
$593$	33.0002	1.35516	0.677579	−	0.735450i	$-0.263029\pi$
$593$	33.0002	1.35516	0.677579	+	0.735450i	$0.263029\pi$
$594$	0	0
$595$	−2.89736	−0.118780
$596$	−43.9450	−1.80006
$597$	0	0
$598$	10.6907	0.437175
$599$	−22.8782	−0.934779	−0.467390	−	0.884051i	$-0.654805\pi$
$599$	−22.8782	−0.934779	−0.467390	+	0.884051i	$0.654805\pi$
$600$	0	0
$601$	−20.9333	−0.853889	−0.426944	−	0.904278i	$-0.640410\pi$
$601$	−20.9333	−0.853889	−0.426944	+	0.904278i	$0.640410\pi$
$602$	−20.6227	−0.840518
$603$	0	0
$604$	−16.5937	−0.675188
$605$	−29.5461	−1.20122
$606$	0	0
$607$	−32.4483	−1.31703	−0.658517	−	0.752566i	$-0.728816\pi$
$607$	−32.4483	−1.31703	−0.658517	+	0.752566i	$0.728816\pi$
$608$	−47.4322	−1.92363
$609$	0	0
$610$	40.4457	1.63760
$611$	−19.0122	−0.769151
$612$	0	0
$613$	−15.1939	−0.613675	−0.306838	−	0.951762i	$-0.599271\pi$
$613$	−15.1939	−0.613675	−0.306838	+	0.951762i	$0.599271\pi$
$614$	33.6104	1.35641
$615$	0	0
$616$	10.2156	0.411597
$617$	22.2945	0.897543	0.448772	−	0.893646i	$-0.351862\pi$
$617$	22.2945	0.897543	0.448772	+	0.893646i	$0.351862\pi$
$618$	0	0
$619$	−36.5001	−1.46706	−0.733531	−	0.679656i	$-0.762129\pi$
$619$	−36.5001	−1.46706	−0.733531	+	0.679656i	$0.762129\pi$
$620$	−92.8887	−3.73050
$621$	0	0
$622$	−13.7994	−0.553304
$623$	−6.05757	−0.242691
$624$	0	0
$625$	−13.5283	−0.541134
$626$	−2.52945	−0.101097
$627$	0	0
$628$	52.9191	2.11170
$629$	−5.28637	−0.210781
$630$	0	0
$631$	0.487686	0.0194145	0.00970724	−	0.999953i	$-0.496910\pi$
$631$	0.487686	0.0194145	0.00970724	+	0.999953i	$0.496910\pi$
$632$	−33.0750	−1.31565
$633$	0	0
$634$	−65.9814	−2.62046
$635$	−3.42195	−0.135796
$636$	0	0
$637$	−28.8532	−1.14321
$638$	13.1282	0.519749
$639$	0	0
$640$	61.3874	2.42655
$641$	36.0988	1.42582	0.712908	−	0.701258i	$-0.247378\pi$
$641$	36.0988	1.42582	0.712908	+	0.701258i	$0.247378\pi$
$642$	0	0
$643$	3.44043	0.135677	0.0678387	−	0.997696i	$-0.478390\pi$
$643$	3.44043	0.135677	0.0678387	+	0.997696i	$0.478390\pi$
$644$	2.75706	0.108643
$645$	0	0
$646$	18.5807	0.731049
$647$	31.6022	1.24241	0.621205	−	0.783648i	$-0.286644\pi$
$647$	31.6022	1.24241	0.621205	+	0.783648i	$0.286644\pi$
$648$	0	0
$649$	45.4912	1.78569
$650$	70.1928	2.75319
$651$	0	0
$652$	42.3195	1.65736
$653$	−8.68247	−0.339771	−0.169886	−	0.985464i	$-0.554340\pi$
$653$	−8.68247	−0.339771	−0.169886	+	0.985464i	$0.554340\pi$
$654$	0	0
$655$	−22.2489	−0.869337
$656$	1.56696	0.0611796
$657$	0	0
$658$	−7.96866	−0.310651
$659$	8.70588	0.339133	0.169566	−	0.985519i	$-0.445763\pi$
$659$	8.70588	0.339133	0.169566	+	0.985519i	$0.445763\pi$
$660$	0	0
$661$	37.2632	1.44937	0.724686	−	0.689079i	$-0.241985\pi$
$661$	37.2632	1.44937	0.724686	+	0.689079i	$0.241985\pi$
$662$	15.6493	0.608228
$663$	0	0
$664$	4.56620	0.177203
$665$	−23.4272	−0.908469
$666$	0	0
$667$	1.32788	0.0514158
$668$	55.1017	2.13195
$669$	0	0
$670$	−88.5875	−3.42243
$671$	−22.9696	−0.886731
$672$	0	0
$673$	9.24609	0.356411	0.178205	−	0.983993i	$-0.442971\pi$
$673$	9.24609	0.356411	0.178205	+	0.983993i	$0.442971\pi$
$674$	−79.5975	−3.06598
$675$	0	0
$676$	25.7536	0.990525
$677$	42.5020	1.63348	0.816742	−	0.577003i	$-0.195778\pi$
$677$	42.5020	1.63348	0.816742	+	0.577003i	$0.195778\pi$
$678$	0	0
$679$	2.98113	0.114405
$680$	−9.39050	−0.360109
$681$	0	0
$682$	85.7347	3.28295
$683$	30.6917	1.17439	0.587193	−	0.809447i	$-0.300233\pi$
$683$	30.6917	1.17439	0.587193	+	0.809447i	$0.300233\pi$
$684$	0	0
$685$	59.3757	2.26863
$686$	−25.5547	−0.975681
$687$	0	0
$688$	1.76988	0.0674762
$689$	−16.7883	−0.639584
$690$	0	0
$691$	31.6998	1.20592	0.602959	−	0.797772i	$-0.293988\pi$
$691$	31.6998	1.20592	0.602959	+	0.797772i	$0.293988\pi$
$692$	−82.0900	−3.12060
$693$	0	0
$694$	−17.2990	−0.656661
$695$	66.0496	2.50540
$696$	0	0
$697$	−9.53160	−0.361035
$698$	−44.4835	−1.68373
$699$	0	0
$700$	18.1023	0.684201
$701$	23.6946	0.894934	0.447467	−	0.894301i	$-0.352326\pi$
$701$	23.6946	0.894934	0.447467	+	0.894301i	$0.352326\pi$
$702$	0	0
$703$	−42.7440	−1.61212
$704$	−57.5740	−2.16990
$705$	0	0
$706$	29.4770	1.10938
$707$	−8.43382	−0.317186
$708$	0	0
$709$	19.3501	0.726710	0.363355	−	0.931651i	$-0.381631\pi$
$709$	19.3501	0.726710	0.363355	+	0.931651i	$0.381631\pi$
$710$	24.3985	0.915661
$711$	0	0
$712$	−19.6329	−0.735773
$713$	8.67185	0.324764
$714$	0	0
$715$	−69.5690	−2.60173
$716$	38.2914	1.43102
$717$	0	0
$718$	14.6825	0.547946
$719$	−8.43417	−0.314541	−0.157271	−	0.987556i	$-0.550270\pi$
$719$	−8.43417	−0.314541	−0.157271	+	0.987556i	$0.550270\pi$
$720$	0	0
$721$	12.3251	0.459010
$722$	106.916	3.97901
$723$	0	0
$724$	8.10262	0.301132
$725$	8.71859	0.323800
$726$	0	0
$727$	26.5083	0.983140	0.491570	−	0.870838i	$-0.336423\pi$
$727$	26.5083	0.983140	0.491570	+	0.870838i	$0.336423\pi$
$728$	10.5777	0.392035
$729$	0	0
$730$	−77.4182	−2.86538
$731$	−10.7659	−0.398193
$732$	0	0
$733$	−15.8683	−0.586108	−0.293054	−	0.956096i	$-0.594672\pi$
$733$	−15.8683	−0.586108	−0.293054	+	0.956096i	$0.594672\pi$
$734$	22.2131	0.819899
$735$	0	0
$736$	−5.97141	−0.220109
$737$	50.3099	1.85319
$738$	0	0
$739$	10.3088	0.379214	0.189607	−	0.981860i	$-0.439279\pi$
$739$	10.3088	0.379214	0.189607	+	0.981860i	$0.439279\pi$
$740$	57.6408	2.11892
$741$	0	0
$742$	−7.03656	−0.258320
$743$	4.75002	0.174261	0.0871307	−	0.996197i	$-0.472230\pi$
$743$	4.75002	0.174261	0.0871307	+	0.996197i	$0.472230\pi$
$744$	0	0
$745$	47.0097	1.72230
$746$	−30.9817	−1.13432
$747$	0	0
$748$	14.2298	0.520291
$749$	8.08868	0.295554
$750$	0	0
$751$	34.9195	1.27423	0.637115	−	0.770768i	$-0.280127\pi$
$751$	34.9195	1.27423	0.637115	+	0.770768i	$0.280127\pi$
$752$	0.683888	0.0249388
$753$	0	0
$754$	13.5935	0.495046
$755$	17.7510	0.646024
$756$	0	0
$757$	−32.9950	−1.19923	−0.599613	−	0.800290i	$-0.704679\pi$
$757$	−32.9950	−1.19923	−0.599613	+	0.800290i	$0.704679\pi$
$758$	−74.3178	−2.69934
$759$	0	0
$760$	−75.9288	−2.75423
$761$	9.23688	0.334837	0.167418	−	0.985886i	$-0.446457\pi$
$761$	9.23688	0.334837	0.167418	+	0.985886i	$0.446457\pi$
$762$	0	0
$763$	2.34343	0.0848378
$764$	34.8358	1.26031
$765$	0	0
$766$	−15.4826	−0.559410
$767$	47.1037	1.70082
$768$	0	0
$769$	50.8696	1.83441	0.917203	−	0.398420i	$-0.130441\pi$
$769$	50.8696	1.83441	0.917203	+	0.398420i	$0.130441\pi$
$770$	−29.1588	−1.05081
$771$	0	0
$772$	2.17029	0.0781104
$773$	0.845737	0.0304190	0.0152095	−	0.999884i	$-0.495158\pi$
$773$	0.845737	0.0304190	0.0152095	+	0.999884i	$0.495158\pi$
$774$	0	0
$775$	56.9376	2.04526
$776$	9.66200	0.346846
$777$	0	0
$778$	−41.9772	−1.50496
$779$	−77.0697	−2.76131
$780$	0	0
$781$	−13.8562	−0.495814
$782$	2.33919	0.0836494
$783$	0	0
$784$	1.03788	0.0370671
$785$	−56.6097	−2.02049
$786$	0	0
$787$	16.3785	0.583830	0.291915	−	0.956444i	$-0.405708\pi$
$787$	16.3785	0.583830	0.291915	+	0.956444i	$0.405708\pi$
$788$	−51.4378	−1.83240
$789$	0	0
$790$	94.4074	3.35887
$791$	2.98229	0.106038
$792$	0	0
$793$	−23.7838	−0.844586
$794$	−3.58440	−0.127206
$795$	0	0
$796$	−7.79148	−0.276162
$797$	27.0890	0.959541	0.479771	−	0.877394i	$-0.340720\pi$
$797$	27.0890	0.959541	0.479771	+	0.877394i	$0.340720\pi$
$798$	0	0
$799$	−4.15999	−0.147170
$800$	−39.2070	−1.38618
$801$	0	0
$802$	55.1847	1.94864
$803$	43.9667	1.55155
$804$	0	0
$805$	−2.94934	−0.103950
$806$	88.7737	3.12692
$807$	0	0
$808$	−27.3344	−0.961622
$809$	−4.80770	−0.169030	−0.0845149	−	0.996422i	$-0.526934\pi$
$809$	−4.80770	−0.169030	−0.0845149	+	0.996422i	$0.526934\pi$
$810$	0	0
$811$	−32.3468	−1.13585	−0.567925	−	0.823080i	$-0.692254\pi$
$811$	−32.3468	−1.13585	−0.567925	+	0.823080i	$0.692254\pi$
$812$	3.50567	0.123025
$813$	0	0
$814$	−53.2015	−1.86471
$815$	−45.2709	−1.58577
$816$	0	0
$817$	−87.0503	−3.04550
$818$	2.34632	0.0820372
$819$	0	0
$820$	103.929	3.62937
$821$	49.6228	1.73185	0.865923	−	0.500177i	$-0.166732\pi$
$821$	49.6228	1.73185	0.865923	+	0.500177i	$0.166732\pi$
$822$	0	0
$823$	8.53450	0.297494	0.148747	−	0.988875i	$-0.452476\pi$
$823$	8.53450	0.297494	0.148747	+	0.988875i	$0.452476\pi$
$824$	39.9462	1.39159
$825$	0	0
$826$	19.7428	0.686939
$827$	44.5701	1.54986	0.774928	−	0.632050i	$-0.217786\pi$
$827$	44.5701	1.54986	0.774928	+	0.632050i	$0.217786\pi$
$828$	0	0
$829$	−19.7256	−0.685099	−0.342549	−	0.939500i	$-0.611290\pi$
$829$	−19.7256	−0.685099	−0.342549	+	0.939500i	$0.611290\pi$
$830$	−13.0335	−0.452400
$831$	0	0
$832$	−59.6148	−2.06677
$833$	−6.31327	−0.218742
$834$	0	0
$835$	−58.9445	−2.03986
$836$	115.058	3.97935
$837$	0	0
$838$	19.7154	0.681058
$839$	7.96808	0.275089	0.137544	−	0.990496i	$-0.456079\pi$
$839$	7.96808	0.275089	0.137544	+	0.990496i	$0.456079\pi$
$840$	0	0
$841$	−27.3116	−0.941778
$842$	47.1144	1.62367
$843$	0	0
$844$	−47.9888	−1.65184
$845$	−27.5497	−0.947739
$846$	0	0
$847$	7.28218	0.250219
$848$	0.603893	0.0207378
$849$	0	0
$850$	15.3586	0.526797
$851$	−5.38120	−0.184465
$852$	0	0
$853$	15.0002	0.513598	0.256799	−	0.966465i	$-0.417332\pi$
$853$	15.0002	0.513598	0.256799	+	0.966465i	$0.417332\pi$
$854$	−9.96859	−0.341118
$855$	0	0
$856$	26.2158	0.896038
$857$	28.5785	0.976223	0.488111	−	0.872781i	$-0.337686\pi$
$857$	28.5785	0.976223	0.488111	+	0.872781i	$0.337686\pi$
$858$	0	0
$859$	−28.3584	−0.967578	−0.483789	−	0.875185i	$-0.660740\pi$
$859$	−28.3584	−0.967578	−0.483789	+	0.875185i	$0.660740\pi$
$860$	117.388	4.00290
$861$	0	0
$862$	41.0320	1.39756
$863$	−32.9021	−1.12000	−0.560001	−	0.828492i	$-0.689199\pi$
$863$	−32.9021	−1.12000	−0.560001	+	0.828492i	$0.689199\pi$
$864$	0	0
$865$	87.8151	2.98580
$866$	89.9589	3.05693
$867$	0	0
$868$	22.8941	0.777077
$869$	−53.6151	−1.81877
$870$	0	0
$871$	52.0932	1.76511
$872$	7.59517	0.257205
$873$	0	0
$874$	18.9140	0.639776
$875$	−4.93435	−0.166812
$876$	0	0
$877$	9.60992	0.324504	0.162252	−	0.986749i	$-0.448124\pi$
$877$	9.60992	0.324504	0.162252	+	0.986749i	$0.448124\pi$
$878$	−73.2996	−2.47374
$879$	0	0
$880$	2.50247	0.0843582
$881$	−29.2388	−0.985082	−0.492541	−	0.870289i	$-0.663932\pi$
$881$	−29.2388	−0.985082	−0.492541	+	0.870289i	$0.663932\pi$
$882$	0	0
$883$	−33.3469	−1.12221	−0.561106	−	0.827744i	$-0.689624\pi$
$883$	−33.3469	−1.12221	−0.561106	+	0.827744i	$0.689624\pi$
$884$	14.7342	0.495563
$885$	0	0
$886$	27.9514	0.939046
$887$	17.8059	0.597863	0.298931	−	0.954275i	$-0.403370\pi$
$887$	17.8059	0.597863	0.298931	+	0.954275i	$0.403370\pi$
$888$	0	0
$889$	0.843402	0.0282868
$890$	56.0390	1.87843
$891$	0	0
$892$	−9.45647	−0.316626
$893$	−33.6365	−1.12560
$894$	0	0
$895$	−40.9619	−1.36921
$896$	−15.1301	−0.505460
$897$	0	0
$898$	−27.3027	−0.911103
$899$	11.0265	0.367754
$900$	0	0
$901$	−3.67339	−0.122378
$902$	−95.9250	−3.19395
$903$	0	0
$904$	9.66576	0.321478
$905$	−8.66771	−0.288124
$906$	0	0
$907$	−11.4876	−0.381439	−0.190719	−	0.981645i	$-0.561082\pi$
$907$	−11.4876	−0.381439	−0.190719	+	0.981645i	$0.561082\pi$
$908$	11.5663	0.383840
$909$	0	0
$910$	−30.1923	−1.00087
$911$	12.9345	0.428540	0.214270	−	0.976774i	$-0.431263\pi$
$911$	12.9345	0.428540	0.214270	+	0.976774i	$0.431263\pi$
$912$	0	0
$913$	7.40189	0.244967
$914$	44.2008	1.46203
$915$	0	0
$916$	−28.4554	−0.940193
$917$	5.48366	0.181086
$918$	0	0
$919$	−12.8641	−0.424347	−0.212174	−	0.977232i	$-0.568054\pi$
$919$	−12.8641	−0.424347	−0.212174	+	0.977232i	$0.568054\pi$
$920$	−9.55895	−0.315149
$921$	0	0
$922$	37.2577	1.22702
$923$	−14.3474	−0.472249
$924$	0	0
$925$	−35.3318	−1.16170
$926$	−9.23845	−0.303594
$927$	0	0
$928$	−7.59281	−0.249246
$929$	10.1630	0.333436	0.166718	−	0.986005i	$-0.446683\pi$
$929$	10.1630	0.333436	0.166718	+	0.986005i	$0.446683\pi$
$930$	0	0
$931$	−51.0472	−1.67301
$932$	−57.0468	−1.86863
$933$	0	0
$934$	65.2994	2.13666
$935$	−15.2222	−0.497818
$936$	0	0
$937$	−24.3487	−0.795438	−0.397719	−	0.917507i	$-0.630198\pi$
$937$	−24.3487	−0.795438	−0.397719	+	0.917507i	$0.630198\pi$
$938$	21.8340	0.712906
$939$	0	0
$940$	45.3591	1.47945
$941$	−15.2554	−0.497313	−0.248657	−	0.968592i	$-0.579989\pi$
$941$	−15.2554	−0.497313	−0.248657	+	0.968592i	$0.579989\pi$
$942$	0	0
$943$	−9.70258	−0.315959
$944$	−1.69437	−0.0551470
$945$	0	0
$946$	−108.347	−3.52268
$947$	−21.6481	−0.703468	−0.351734	−	0.936100i	$-0.614408\pi$
$947$	−21.6481	−0.703468	−0.351734	+	0.936100i	$0.614408\pi$
$948$	0	0
$949$	45.5252	1.47781
$950$	124.185	4.02911
$951$	0	0
$952$	2.31446	0.0750122
$953$	40.6548	1.31694	0.658469	−	0.752608i	$-0.271204\pi$
$953$	40.6548	1.31694	0.658469	+	0.752608i	$0.271204\pi$
$954$	0	0
$955$	−37.2653	−1.20588
$956$	−5.85908	−0.189496
$957$	0	0
$958$	−70.3102	−2.27162
$959$	−14.6343	−0.472565
$960$	0	0
$961$	41.0096	1.32289
$962$	−55.0873	−1.77609
$963$	0	0
$964$	−26.9882	−0.869230
$965$	−2.32165	−0.0747364
$966$	0	0
$967$	39.4603	1.26896	0.634480	−	0.772940i	$-0.281214\pi$
$967$	39.4603	1.26896	0.634480	+	0.772940i	$0.281214\pi$
$968$	23.6019	0.758594
$969$	0	0
$970$	−27.5787	−0.885499
$971$	−55.5779	−1.78358	−0.891791	−	0.452448i	$-0.850551\pi$
$971$	−55.5779	−1.78358	−0.891791	+	0.452448i	$0.850551\pi$
$972$	0	0
$973$	−16.2791	−0.521886
$974$	51.9271	1.66385
$975$	0	0
$976$	0.855526	0.0273847
$977$	51.4584	1.64630	0.823150	−	0.567824i	$-0.192215\pi$
$977$	51.4584	1.64630	0.823150	+	0.567824i	$0.192215\pi$
$978$	0	0
$979$	−31.8252	−1.01714
$980$	68.8377	2.19894
$981$	0	0
$982$	43.2767	1.38101
$983$	25.7997	0.822883	0.411442	−	0.911436i	$-0.365025\pi$
$983$	25.7997	0.822883	0.411442	+	0.911436i	$0.365025\pi$
$984$	0	0
$985$	55.0251	1.75325
$986$	2.97435	0.0947225
$987$	0	0
$988$	119.136	3.79022
$989$	−10.9591	−0.348478
$990$	0	0
$991$	52.1234	1.65575	0.827877	−	0.560910i	$-0.189549\pi$
$991$	52.1234	1.65575	0.827877	+	0.560910i	$0.189549\pi$
$992$	−49.5855	−1.57434
$993$	0	0
$994$	−6.01347	−0.190736
$995$	8.33486	0.264233
$996$	0	0
$997$	−14.4759	−0.458456	−0.229228	−	0.973373i	$-0.573620\pi$
$997$	−14.4759	−0.458456	−0.229228	+	0.973373i	$0.573620\pi$
$998$	14.6487	0.463698
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.k.1.14	yes	16
3.2	odd	2	inner	1143.2.a.k.1.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.2.a.k.1.3	✓	16	3.2	odd	2	inner
1143.2.a.k.1.14	yes	16	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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+2.28010 q^{2} +3.19885 q^{4} -3.42195 q^{5} +0.843402 q^{7} +2.73351 q^{8} +O(q^{10})\)	\(q+2.28010 q^{2} +3.19885 q^{4} -3.42195 q^{5} +0.843402 q^{7} +2.73351 q^{8} -7.80238 q^{10} +4.43106 q^{11} +4.58813 q^{13} +1.92304 q^{14} -0.165040 q^{16} +1.00391 q^{17} +8.11733 q^{19} -10.9463 q^{20} +10.1033 q^{22} +1.02192 q^{23} +6.70971 q^{25} +10.4614 q^{26} +2.69792 q^{28} +1.29940 q^{29} +8.48585 q^{31} -5.84332 q^{32} +2.28902 q^{34} -2.88608 q^{35} -5.26578 q^{37} +18.5083 q^{38} -9.35391 q^{40} -9.49446 q^{41} -10.7240 q^{43} +14.1743 q^{44} +2.33008 q^{46} -4.14378 q^{47} -6.28867 q^{49} +15.2988 q^{50} +14.6767 q^{52} -3.65908 q^{53} -15.1628 q^{55} +2.30545 q^{56} +2.96276 q^{58} +10.2664 q^{59} -5.18377 q^{61} +19.3486 q^{62} -12.9933 q^{64} -15.7003 q^{65} +11.3539 q^{67} +3.21137 q^{68} -6.58054 q^{70} -3.12707 q^{71} +9.92239 q^{73} -12.0065 q^{74} +25.9662 q^{76} +3.73716 q^{77} -12.0998 q^{79} +0.564756 q^{80} -21.6483 q^{82} +1.67046 q^{83} -3.43533 q^{85} -24.4518 q^{86} +12.1123 q^{88} -7.18230 q^{89} +3.86963 q^{91} +3.26897 q^{92} -9.44824 q^{94} -27.7771 q^{95} +3.53465 q^{97} -14.3388 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 10 * q^7	\( 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more