LMFDB - Embedded newform 8048.2.a.v.1.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.28
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.02746 q^{3} -1.92479 q^{5} -4.18734 q^{7} +6.16549 q^{9} +O(q^{10})$	$q+3.02746 q^{3} -1.92479 q^{5} -4.18734 q^{7} +6.16549 q^{9} +1.99884 q^{11} +1.74693 q^{13} -5.82721 q^{15} -2.18205 q^{17} +1.78148 q^{19} -12.6770 q^{21} +3.66792 q^{23} -1.29519 q^{25} +9.58339 q^{27} -4.48076 q^{29} -10.3311 q^{31} +6.05140 q^{33} +8.05974 q^{35} +1.81048 q^{37} +5.28874 q^{39} -5.92703 q^{41} -3.30352 q^{43} -11.8673 q^{45} -0.751293 q^{47} +10.5338 q^{49} -6.60606 q^{51} -1.29824 q^{53} -3.84734 q^{55} +5.39334 q^{57} +2.89024 q^{59} -0.765430 q^{61} -25.8170 q^{63} -3.36246 q^{65} +0.202708 q^{67} +11.1045 q^{69} -10.7293 q^{71} +11.3425 q^{73} -3.92114 q^{75} -8.36982 q^{77} +0.211446 q^{79} +10.5168 q^{81} +9.33406 q^{83} +4.19998 q^{85} -13.5653 q^{87} +0.912638 q^{89} -7.31497 q^{91} -31.2769 q^{93} -3.42896 q^{95} -7.66341 q^{97} +12.3238 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	3.02746	1.74790	0.873951	−	0.486013i	$-0.161549\pi$
$3$	3.02746	1.74790	0.873951	+	0.486013i	$0.161549\pi$
$4$	0	0
$5$	−1.92479	−0.860791	−0.430396	−	0.902640i	$-0.641626\pi$
$5$	−1.92479	−0.860791	−0.430396	+	0.902640i	$0.641626\pi$
$6$	0	0
$7$	−4.18734	−1.58266	−0.791332	−	0.611386i	$-0.790612\pi$
$7$	−4.18734	−1.58266	−0.791332	+	0.611386i	$0.790612\pi$
$8$	0	0
$9$	6.16549	2.05516
$10$	0	0
$11$	1.99884	0.602673	0.301337	−	0.953518i	$-0.402567\pi$
$11$	1.99884	0.602673	0.301337	+	0.953518i	$0.402567\pi$
$12$	0	0
$13$	1.74693	0.484510	0.242255	−	0.970213i	$-0.422113\pi$
$13$	1.74693	0.484510	0.242255	+	0.970213i	$0.422113\pi$
$14$	0	0
$15$	−5.82721	−1.50458
$16$	0	0
$17$	−2.18205	−0.529225	−0.264612	−	0.964355i	$-0.585244\pi$
$17$	−2.18205	−0.529225	−0.264612	+	0.964355i	$0.585244\pi$
$18$	0	0
$19$	1.78148	0.408698	0.204349	−	0.978898i	$-0.434492\pi$
$19$	1.78148	0.408698	0.204349	+	0.978898i	$0.434492\pi$
$20$	0	0
$21$	−12.6770	−2.76634
$22$	0	0
$23$	3.66792	0.764814	0.382407	−	0.923994i	$-0.375095\pi$
$23$	3.66792	0.764814	0.382407	+	0.923994i	$0.375095\pi$
$24$	0	0
$25$	−1.29519	−0.259038
$26$	0	0
$27$	9.58339	1.84432
$28$	0	0
$29$	−4.48076	−0.832057	−0.416029	−	0.909352i	$-0.636578\pi$
$29$	−4.48076	−0.832057	−0.416029	+	0.909352i	$0.636578\pi$
$30$	0	0
$31$	−10.3311	−1.85552	−0.927759	−	0.373179i	$-0.878268\pi$
$31$	−10.3311	−1.85552	−0.927759	+	0.373179i	$0.878268\pi$
$32$	0	0
$33$	6.05140	1.05341
$34$	0	0
$35$	8.05974	1.36234
$36$	0	0
$37$	1.81048	0.297642	0.148821	−	0.988864i	$-0.452452\pi$
$37$	1.81048	0.297642	0.148821	+	0.988864i	$0.452452\pi$
$38$	0	0
$39$	5.28874	0.846877
$40$	0	0
$41$	−5.92703	−0.925647	−0.462824	−	0.886450i	$-0.653164\pi$
$41$	−5.92703	−0.925647	−0.462824	+	0.886450i	$0.653164\pi$
$42$	0	0
$43$	−3.30352	−0.503782	−0.251891	−	0.967756i	$-0.581052\pi$
$43$	−3.30352	−0.503782	−0.251891	+	0.967756i	$0.581052\pi$
$44$	0	0
$45$	−11.8673	−1.76907
$46$	0	0
$47$	−0.751293	−0.109587	−0.0547937	−	0.998498i	$-0.517450\pi$
$47$	−0.751293	−0.109587	−0.0547937	+	0.998498i	$0.517450\pi$
$48$	0	0
$49$	10.5338	1.50483
$50$	0	0
$51$	−6.60606	−0.925033
$52$	0	0
$53$	−1.29824	−0.178326	−0.0891632	−	0.996017i	$-0.528419\pi$
$53$	−1.29824	−0.178326	−0.0891632	+	0.996017i	$0.528419\pi$
$54$	0	0
$55$	−3.84734	−0.518776
$56$	0	0
$57$	5.39334	0.714365
$58$	0	0
$59$	2.89024	0.376277	0.188139	−	0.982142i	$-0.439755\pi$
$59$	2.89024	0.376277	0.188139	+	0.982142i	$0.439755\pi$
$60$	0	0
$61$	−0.765430	−0.0980033	−0.0490017	−	0.998799i	$-0.515604\pi$
$61$	−0.765430	−0.0980033	−0.0490017	+	0.998799i	$0.515604\pi$
$62$	0	0
$63$	−25.8170	−3.25264
$64$	0	0
$65$	−3.36246	−0.417062
$66$	0	0
$67$	0.202708	0.0247647	0.0123823	−	0.999923i	$-0.496058\pi$
$67$	0.202708	0.0247647	0.0123823	+	0.999923i	$0.496058\pi$
$68$	0	0
$69$	11.1045	1.33682
$70$	0	0
$71$	−10.7293	−1.27333	−0.636667	−	0.771139i	$-0.719687\pi$
$71$	−10.7293	−1.27333	−0.636667	+	0.771139i	$0.719687\pi$
$72$	0	0
$73$	11.3425	1.32754	0.663768	−	0.747938i	$-0.268956\pi$
$73$	11.3425	1.32754	0.663768	+	0.747938i	$0.268956\pi$
$74$	0	0
$75$	−3.92114	−0.452774
$76$	0	0
$77$	−8.36982	−0.953830
$78$	0	0
$79$	0.211446	0.0237896	0.0118948	−	0.999929i	$-0.496214\pi$
$79$	0.211446	0.0237896	0.0118948	+	0.999929i	$0.496214\pi$
$80$	0	0
$81$	10.5168	1.16853
$82$	0	0
$83$	9.33406	1.02455	0.512273	−	0.858823i	$-0.328804\pi$
$83$	9.33406	1.02455	0.512273	+	0.858823i	$0.328804\pi$
$84$	0	0
$85$	4.19998	0.455552
$86$	0	0
$87$	−13.5653	−1.45435
$88$	0	0
$89$	0.912638	0.0967394	0.0483697	−	0.998830i	$-0.484597\pi$
$89$	0.912638	0.0967394	0.0483697	+	0.998830i	$0.484597\pi$
$90$	0	0
$91$	−7.31497	−0.766817
$92$	0	0
$93$	−31.2769	−3.24327
$94$	0	0
$95$	−3.42896	−0.351804
$96$	0	0
$97$	−7.66341	−0.778101	−0.389051	−	0.921216i	$-0.627197\pi$
$97$	−7.66341	−0.778101	−0.389051	+	0.921216i	$0.627197\pi$
$98$	0	0
$99$	12.3238	1.23859
$100$	0	0
$101$	3.38623	0.336943	0.168471	−	0.985707i	$-0.446117\pi$
$101$	3.38623	0.336943	0.168471	+	0.985707i	$0.446117\pi$
$102$	0	0
$103$	−8.59859	−0.847244	−0.423622	−	0.905839i	$-0.639241\pi$
$103$	−8.59859	−0.847244	−0.423622	+	0.905839i	$0.639241\pi$
$104$	0	0
$105$	24.4005	2.38124
$106$	0	0
$107$	−7.86354	−0.760197	−0.380099	−	0.924946i	$-0.624110\pi$
$107$	−7.86354	−0.760197	−0.380099	+	0.924946i	$0.624110\pi$
$108$	0	0
$109$	−0.237337	−0.0227328	−0.0113664	−	0.999935i	$-0.503618\pi$
$109$	−0.237337	−0.0227328	−0.0113664	+	0.999935i	$0.503618\pi$
$110$	0	0
$111$	5.48116	0.520248
$112$	0	0
$113$	−18.5081	−1.74110	−0.870549	−	0.492082i	$-0.836236\pi$
$113$	−18.5081	−1.74110	−0.870549	+	0.492082i	$0.836236\pi$
$114$	0	0
$115$	−7.05997	−0.658345
$116$	0	0
$117$	10.7707	0.995748
$118$	0	0
$119$	9.13697	0.837585
$120$	0	0
$121$	−7.00463	−0.636785
$122$	0	0
$123$	−17.9438	−1.61794
$124$	0	0
$125$	12.1169	1.08377
$126$	0	0
$127$	−12.2050	−1.08302	−0.541508	−	0.840696i	$-0.682146\pi$
$127$	−12.2050	−1.08302	−0.541508	+	0.840696i	$0.682146\pi$
$128$	0	0
$129$	−10.0013	−0.880562
$130$	0	0
$131$	−20.6911	−1.80779	−0.903896	−	0.427753i	$-0.859305\pi$
$131$	−20.6911	−1.80779	−0.903896	+	0.427753i	$0.859305\pi$
$132$	0	0
$133$	−7.45964	−0.646833
$134$	0	0
$135$	−18.4460	−1.58758
$136$	0	0
$137$	−15.3166	−1.30858	−0.654292	−	0.756242i	$-0.727033\pi$
$137$	−15.3166	−1.30858	−0.654292	+	0.756242i	$0.727033\pi$
$138$	0	0
$139$	12.8022	1.08587	0.542935	−	0.839775i	$-0.317313\pi$
$139$	12.8022	1.08587	0.542935	+	0.839775i	$0.317313\pi$
$140$	0	0
$141$	−2.27451	−0.191548
$142$	0	0
$143$	3.49183	0.292001
$144$	0	0
$145$	8.62452	0.716227
$146$	0	0
$147$	31.8906	2.63029
$148$	0	0
$149$	12.0920	0.990619	0.495309	−	0.868717i	$-0.335055\pi$
$149$	12.0920	0.990619	0.495309	+	0.868717i	$0.335055\pi$
$150$	0	0
$151$	7.09209	0.577146	0.288573	−	0.957458i	$-0.406819\pi$
$151$	7.09209	0.577146	0.288573	+	0.957458i	$0.406819\pi$
$152$	0	0
$153$	−13.4534	−1.08764
$154$	0	0
$155$	19.8852	1.59721
$156$	0	0
$157$	−17.1026	−1.36494	−0.682469	−	0.730915i	$-0.739094\pi$
$157$	−17.1026	−1.36494	−0.682469	+	0.730915i	$0.739094\pi$
$158$	0	0
$159$	−3.93035	−0.311697
$160$	0	0
$161$	−15.3588	−1.21044
$162$	0	0
$163$	9.56130	0.748899	0.374449	−	0.927247i	$-0.377832\pi$
$163$	9.56130	0.748899	0.374449	+	0.927247i	$0.377832\pi$
$164$	0	0
$165$	−11.6477	−0.906770
$166$	0	0
$167$	−8.74744	−0.676897	−0.338449	−	0.940985i	$-0.609902\pi$
$167$	−8.74744	−0.676897	−0.338449	+	0.940985i	$0.609902\pi$
$168$	0	0
$169$	−9.94825	−0.765250
$170$	0	0
$171$	10.9837	0.839942
$172$	0	0
$173$	−9.59340	−0.729373	−0.364686	−	0.931130i	$-0.618824\pi$
$173$	−9.59340	−0.729373	−0.364686	+	0.931130i	$0.618824\pi$
$174$	0	0
$175$	5.42341	0.409971
$176$	0	0
$177$	8.75008	0.657696
$178$	0	0
$179$	17.2593	1.29002	0.645009	−	0.764175i	$-0.276853\pi$
$179$	17.2593	1.29002	0.645009	+	0.764175i	$0.276853\pi$
$180$	0	0
$181$	−17.4707	−1.29859	−0.649293	−	0.760538i	$-0.724935\pi$
$181$	−17.4707	−1.29859	−0.649293	+	0.760538i	$0.724935\pi$
$182$	0	0
$183$	−2.31731	−0.171300
$184$	0	0
$185$	−3.48480	−0.256207
$186$	0	0
$187$	−4.36157	−0.318949
$188$	0	0
$189$	−40.1289	−2.91895
$190$	0	0
$191$	−0.0970807	−0.00702451	−0.00351225	−	0.999994i	$-0.501118\pi$
$191$	−0.0970807	−0.00702451	−0.00351225	+	0.999994i	$0.501118\pi$
$192$	0	0
$193$	3.31384	0.238536	0.119268	−	0.992862i	$-0.461945\pi$
$193$	3.31384	0.238536	0.119268	+	0.992862i	$0.461945\pi$
$194$	0	0
$195$	−10.1797	−0.728984
$196$	0	0
$197$	−0.444125	−0.0316426	−0.0158213	−	0.999875i	$-0.505036\pi$
$197$	−0.444125	−0.0316426	−0.0158213	+	0.999875i	$0.505036\pi$
$198$	0	0
$199$	17.3971	1.23325	0.616623	−	0.787259i	$-0.288500\pi$
$199$	17.3971	1.23325	0.616623	+	0.787259i	$0.288500\pi$
$200$	0	0
$201$	0.613688	0.0432862
$202$	0	0
$203$	18.7625	1.31687
$204$	0	0
$205$	11.4083	0.796789
$206$	0	0
$207$	22.6145	1.57182
$208$	0	0
$209$	3.56089	0.246312
$210$	0	0
$211$	−1.95704	−0.134728	−0.0673641	−	0.997728i	$-0.521459\pi$
$211$	−1.95704	−0.134728	−0.0673641	+	0.997728i	$0.521459\pi$
$212$	0	0
$213$	−32.4825	−2.22566
$214$	0	0
$215$	6.35857	0.433651
$216$	0	0
$217$	43.2598	2.93666
$218$	0	0
$219$	34.3388	2.32040
$220$	0	0
$221$	−3.81188	−0.256415
$222$	0	0
$223$	14.3677	0.962130	0.481065	−	0.876685i	$-0.340250\pi$
$223$	14.3677	0.962130	0.481065	+	0.876685i	$0.340250\pi$
$224$	0	0
$225$	−7.98550	−0.532366
$226$	0	0
$227$	−14.4923	−0.961886	−0.480943	−	0.876752i	$-0.659706\pi$
$227$	−14.4923	−0.961886	−0.480943	+	0.876752i	$0.659706\pi$
$228$	0	0
$229$	1.66945	0.110320	0.0551602	−	0.998478i	$-0.482433\pi$
$229$	1.66945	0.110320	0.0551602	+	0.998478i	$0.482433\pi$
$230$	0	0
$231$	−25.3393	−1.66720
$232$	0	0
$233$	−14.5719	−0.954638	−0.477319	−	0.878730i	$-0.658391\pi$
$233$	−14.5719	−0.954638	−0.477319	+	0.878730i	$0.658391\pi$
$234$	0	0
$235$	1.44608	0.0943318
$236$	0	0
$237$	0.640144	0.0415818
$238$	0	0
$239$	−11.4900	−0.743229	−0.371614	−	0.928387i	$-0.621196\pi$
$239$	−11.4900	−0.743229	−0.371614	+	0.928387i	$0.621196\pi$
$240$	0	0
$241$	−22.8075	−1.46916	−0.734579	−	0.678523i	$-0.762620\pi$
$241$	−22.8075	−1.46916	−0.734579	+	0.678523i	$0.762620\pi$
$242$	0	0
$243$	3.08902	0.198160
$244$	0	0
$245$	−20.2753	−1.29534
$246$	0	0
$247$	3.11211	0.198019
$248$	0	0
$249$	28.2584	1.79081
$250$	0	0
$251$	20.6236	1.30175	0.650874	−	0.759185i	$-0.274402\pi$
$251$	20.6236	1.30175	0.650874	+	0.759185i	$0.274402\pi$
$252$	0	0
$253$	7.33159	0.460933
$254$	0	0
$255$	12.7153	0.796260
$256$	0	0
$257$	−7.15484	−0.446307	−0.223153	−	0.974783i	$-0.571635\pi$
$257$	−7.15484	−0.446307	−0.223153	+	0.974783i	$0.571635\pi$
$258$	0	0
$259$	−7.58110	−0.471067
$260$	0	0
$261$	−27.6261	−1.71001
$262$	0	0
$263$	5.18369	0.319640	0.159820	−	0.987146i	$-0.448909\pi$
$263$	5.18369	0.319640	0.159820	+	0.987146i	$0.448909\pi$
$264$	0	0
$265$	2.49883	0.153502
$266$	0	0
$267$	2.76297	0.169091
$268$	0	0
$269$	−3.72454	−0.227089	−0.113545	−	0.993533i	$-0.536220\pi$
$269$	−3.72454	−0.227089	−0.113545	+	0.993533i	$0.536220\pi$
$270$	0	0
$271$	−19.9464	−1.21166	−0.605829	−	0.795595i	$-0.707158\pi$
$271$	−19.9464	−1.21166	−0.605829	+	0.795595i	$0.707158\pi$
$272$	0	0
$273$	−22.1457	−1.34032
$274$	0	0
$275$	−2.58888	−0.156116
$276$	0	0
$277$	7.64942	0.459609	0.229805	−	0.973237i	$-0.426191\pi$
$277$	7.64942	0.459609	0.229805	+	0.973237i	$0.426191\pi$
$278$	0	0
$279$	−63.6963	−3.81339
$280$	0	0
$281$	7.39477	0.441135	0.220568	−	0.975372i	$-0.429209\pi$
$281$	7.39477	0.441135	0.220568	+	0.975372i	$0.429209\pi$
$282$	0	0
$283$	2.75281	0.163637	0.0818187	−	0.996647i	$-0.473927\pi$
$283$	2.75281	0.163637	0.0818187	+	0.996647i	$0.473927\pi$
$284$	0	0
$285$	−10.3810	−0.614919
$286$	0	0
$287$	24.8185	1.46499
$288$	0	0
$289$	−12.2387	−0.719921
$290$	0	0
$291$	−23.2006	−1.36004
$292$	0	0
$293$	17.1629	1.00267	0.501333	−	0.865255i	$-0.332843\pi$
$293$	17.1629	1.00267	0.501333	+	0.865255i	$0.332843\pi$
$294$	0	0
$295$	−5.56310	−0.323896
$296$	0	0
$297$	19.1557	1.11152
$298$	0	0
$299$	6.40759	0.370560
$300$	0	0
$301$	13.8330	0.797318
$302$	0	0
$303$	10.2517	0.588943
$304$	0	0
$305$	1.47329	0.0843604
$306$	0	0
$307$	20.0008	1.14150	0.570752	−	0.821122i	$-0.306652\pi$
$307$	20.0008	1.14150	0.570752	+	0.821122i	$0.306652\pi$
$308$	0	0
$309$	−26.0318	−1.48090
$310$	0	0
$311$	−20.4690	−1.16069	−0.580344	−	0.814371i	$-0.697082\pi$
$311$	−20.4690	−1.16069	−0.580344	+	0.814371i	$0.697082\pi$
$312$	0	0
$313$	−29.0165	−1.64011	−0.820054	−	0.572286i	$-0.806057\pi$
$313$	−29.0165	−1.64011	−0.820054	+	0.572286i	$0.806057\pi$
$314$	0	0
$315$	49.6922	2.79984
$316$	0	0
$317$	−28.4199	−1.59622	−0.798109	−	0.602513i	$-0.794166\pi$
$317$	−28.4199	−1.59622	−0.798109	+	0.602513i	$0.794166\pi$
$318$	0	0
$319$	−8.95634	−0.501459
$320$	0	0
$321$	−23.8065	−1.32875
$322$	0	0
$323$	−3.88727	−0.216293
$324$	0	0
$325$	−2.26261	−0.125507
$326$	0	0
$327$	−0.718528	−0.0397347
$328$	0	0
$329$	3.14592	0.173440
$330$	0	0
$331$	19.6047	1.07757	0.538785	−	0.842443i	$-0.318883\pi$
$331$	19.6047	1.07757	0.538785	+	0.842443i	$0.318883\pi$
$332$	0	0
$333$	11.1625	0.611702
$334$	0	0
$335$	−0.390169	−0.0213172
$336$	0	0
$337$	−0.371952	−0.0202615	−0.0101308	−	0.999949i	$-0.503225\pi$
$337$	−0.371952	−0.0202615	−0.0101308	+	0.999949i	$0.503225\pi$
$338$	0	0
$339$	−56.0325	−3.04327
$340$	0	0
$341$	−20.6502	−1.11827
$342$	0	0
$343$	−14.7972	−0.798973
$344$	0	0
$345$	−21.3737	−1.15072
$346$	0	0
$347$	−18.8832	−1.01370	−0.506851	−	0.862034i	$-0.669191\pi$
$347$	−18.8832	−1.01370	−0.506851	+	0.862034i	$0.669191\pi$
$348$	0	0
$349$	8.11925	0.434614	0.217307	−	0.976103i	$-0.430273\pi$
$349$	8.11925	0.434614	0.217307	+	0.976103i	$0.430273\pi$
$350$	0	0
$351$	16.7415	0.893593
$352$	0	0
$353$	24.8693	1.32366	0.661830	−	0.749654i	$-0.269780\pi$
$353$	24.8693	1.32366	0.661830	+	0.749654i	$0.269780\pi$
$354$	0	0
$355$	20.6516	1.09607
$356$	0	0
$357$	27.6618	1.46402
$358$	0	0
$359$	9.58819	0.506045	0.253023	−	0.967460i	$-0.418575\pi$
$359$	9.58819	0.506045	0.253023	+	0.967460i	$0.418575\pi$
$360$	0	0
$361$	−15.8263	−0.832966
$362$	0	0
$363$	−21.2062	−1.11304
$364$	0	0
$365$	−21.8319	−1.14273
$366$	0	0
$367$	−22.1854	−1.15807	−0.579035	−	0.815302i	$-0.696571\pi$
$367$	−22.1854	−1.15807	−0.579035	+	0.815302i	$0.696571\pi$
$368$	0	0
$369$	−36.5431	−1.90236
$370$	0	0
$371$	5.43615	0.282231
$372$	0	0
$373$	−11.0914	−0.574289	−0.287145	−	0.957887i	$-0.592706\pi$
$373$	−11.0914	−0.574289	−0.287145	+	0.957887i	$0.592706\pi$
$374$	0	0
$375$	36.6834	1.89432
$376$	0	0
$377$	−7.82756	−0.403140
$378$	0	0
$379$	23.1111	1.18714	0.593570	−	0.804782i	$-0.297718\pi$
$379$	23.1111	1.18714	0.593570	+	0.804782i	$0.297718\pi$
$380$	0	0
$381$	−36.9500	−1.89301
$382$	0	0
$383$	10.3570	0.529218	0.264609	−	0.964356i	$-0.414757\pi$
$383$	10.3570	0.529218	0.264609	+	0.964356i	$0.414757\pi$
$384$	0	0
$385$	16.1101	0.821048
$386$	0	0
$387$	−20.3678	−1.03535
$388$	0	0
$389$	0.405843	0.0205770	0.0102885	−	0.999947i	$-0.496725\pi$
$389$	0.405843	0.0205770	0.0102885	+	0.999947i	$0.496725\pi$
$390$	0	0
$391$	−8.00358	−0.404759
$392$	0	0
$393$	−62.6415	−3.15984
$394$	0	0
$395$	−0.406989	−0.0204778
$396$	0	0
$397$	17.1489	0.860677	0.430338	−	0.902668i	$-0.358394\pi$
$397$	17.1489	0.860677	0.430338	+	0.902668i	$0.358394\pi$
$398$	0	0
$399$	−22.5837	−1.13060
$400$	0	0
$401$	−22.6203	−1.12960	−0.564802	−	0.825227i	$-0.691047\pi$
$401$	−22.6203	−1.12960	−0.564802	+	0.825227i	$0.691047\pi$
$402$	0	0
$403$	−18.0477	−0.899018
$404$	0	0
$405$	−20.2426	−1.00586
$406$	0	0
$407$	3.61887	0.179381
$408$	0	0
$409$	−31.4398	−1.55460	−0.777299	−	0.629131i	$-0.783411\pi$
$409$	−31.4398	−1.55460	−0.777299	+	0.629131i	$0.783411\pi$
$410$	0	0
$411$	−46.3703	−2.28728
$412$	0	0
$413$	−12.1024	−0.595521
$414$	0	0
$415$	−17.9661	−0.881920
$416$	0	0
$417$	38.7581	1.89799
$418$	0	0
$419$	−8.70623	−0.425327	−0.212664	−	0.977125i	$-0.568214\pi$
$419$	−8.70623	−0.425327	−0.212664	+	0.977125i	$0.568214\pi$
$420$	0	0
$421$	10.1795	0.496120	0.248060	−	0.968745i	$-0.420207\pi$
$421$	10.1795	0.496120	0.248060	+	0.968745i	$0.420207\pi$
$422$	0	0
$423$	−4.63209	−0.225220
$424$	0	0
$425$	2.82617	0.137090
$426$	0	0
$427$	3.20511	0.155106
$428$	0	0
$429$	10.5714	0.510390
$430$	0	0
$431$	39.7010	1.91233	0.956164	−	0.292832i	$-0.0945978\pi$
$431$	39.7010	1.91233	0.956164	+	0.292832i	$0.0945978\pi$
$432$	0	0
$433$	2.99144	0.143760	0.0718798	−	0.997413i	$-0.477100\pi$
$433$	2.99144	0.143760	0.0718798	+	0.997413i	$0.477100\pi$
$434$	0	0
$435$	26.1104	1.25190
$436$	0	0
$437$	6.53431	0.312578
$438$	0	0
$439$	−19.9125	−0.950373	−0.475187	−	0.879885i	$-0.657619\pi$
$439$	−19.9125	−0.950373	−0.475187	+	0.879885i	$0.657619\pi$
$440$	0	0
$441$	64.9460	3.09267
$442$	0	0
$443$	20.0397	0.952117	0.476058	−	0.879414i	$-0.342065\pi$
$443$	20.0397	0.952117	0.476058	+	0.879414i	$0.342065\pi$
$444$	0	0
$445$	−1.75663	−0.0832724
$446$	0	0
$447$	36.6081	1.73151
$448$	0	0
$449$	−10.2247	−0.482533	−0.241267	−	0.970459i	$-0.577563\pi$
$449$	−10.2247	−0.482533	−0.241267	+	0.970459i	$0.577563\pi$
$450$	0	0
$451$	−11.8472	−0.557863
$452$	0	0
$453$	21.4710	1.00880
$454$	0	0
$455$	14.0798	0.660069
$456$	0	0
$457$	21.8282	1.02108	0.510541	−	0.859854i	$-0.329445\pi$
$457$	21.8282	1.02108	0.510541	+	0.859854i	$0.329445\pi$
$458$	0	0
$459$	−20.9114	−0.976061
$460$	0	0
$461$	−29.6711	−1.38192	−0.690961	−	0.722892i	$-0.742812\pi$
$461$	−29.6711	−1.38192	−0.690961	+	0.722892i	$0.742812\pi$
$462$	0	0
$463$	22.7523	1.05739	0.528695	−	0.848812i	$-0.322681\pi$
$463$	22.7523	1.05739	0.528695	+	0.848812i	$0.322681\pi$
$464$	0	0
$465$	60.2014	2.79177
$466$	0	0
$467$	−30.5638	−1.41432	−0.707162	−	0.707052i	$-0.750025\pi$
$467$	−30.5638	−1.41432	−0.707162	+	0.707052i	$0.750025\pi$
$468$	0	0
$469$	−0.848805	−0.0391942
$470$	0	0
$471$	−51.7774	−2.38578
$472$	0	0
$473$	−6.60321	−0.303616
$474$	0	0
$475$	−2.30735	−0.105869
$476$	0	0
$477$	−8.00427	−0.366490
$478$	0	0
$479$	−5.93081	−0.270986	−0.135493	−	0.990778i	$-0.543262\pi$
$479$	−5.93081	−0.270986	−0.135493	+	0.990778i	$0.543262\pi$
$480$	0	0
$481$	3.16278	0.144210
$482$	0	0
$483$	−46.4982	−2.11574
$484$	0	0
$485$	14.7504	0.669783
$486$	0	0
$487$	−9.77824	−0.443094	−0.221547	−	0.975150i	$-0.571111\pi$
$487$	−9.77824	−0.443094	−0.221547	+	0.975150i	$0.571111\pi$
$488$	0	0
$489$	28.9464	1.30900
$490$	0	0
$491$	23.2607	1.04974	0.524870	−	0.851183i	$-0.324114\pi$
$491$	23.2607	1.04974	0.524870	+	0.851183i	$0.324114\pi$
$492$	0	0
$493$	9.77725	0.440345
$494$	0	0
$495$	−23.7208	−1.06617
$496$	0	0
$497$	44.9272	2.01526
$498$	0	0
$499$	−4.32957	−0.193818	−0.0969091	−	0.995293i	$-0.530896\pi$
$499$	−4.32957	−0.193818	−0.0969091	+	0.995293i	$0.530896\pi$
$500$	0	0
$501$	−26.4825	−1.18315
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−6.51778	−0.290037
$506$	0	0
$507$	−30.1179	−1.33758
$508$	0	0
$509$	31.2848	1.38668	0.693338	−	0.720612i	$-0.256139\pi$
$509$	31.2848	1.38668	0.693338	+	0.720612i	$0.256139\pi$
$510$	0	0
$511$	−47.4948	−2.10105
$512$	0	0
$513$	17.0726	0.753772
$514$	0	0
$515$	16.5505	0.729300
$516$	0	0
$517$	−1.50172	−0.0660454
$518$	0	0
$519$	−29.0436	−1.27487
$520$	0	0
$521$	−14.1680	−0.620710	−0.310355	−	0.950621i	$-0.600448\pi$
$521$	−14.1680	−0.620710	−0.310355	+	0.950621i	$0.600448\pi$
$522$	0	0
$523$	−1.46629	−0.0641163	−0.0320582	−	0.999486i	$-0.510206\pi$
$523$	−1.46629	−0.0641163	−0.0320582	+	0.999486i	$0.510206\pi$
$524$	0	0
$525$	16.4191	0.716590
$526$	0	0
$527$	22.5429	0.981986
$528$	0	0
$529$	−9.54636	−0.415059
$530$	0	0
$531$	17.8198	0.773312
$532$	0	0
$533$	−10.3541	−0.448485
$534$	0	0
$535$	15.1356	0.654371
$536$	0	0
$537$	52.2517	2.25483
$538$	0	0
$539$	21.0554	0.906920
$540$	0	0
$541$	−35.6990	−1.53482	−0.767410	−	0.641157i	$-0.778455\pi$
$541$	−35.6990	−1.53482	−0.767410	+	0.641157i	$0.778455\pi$
$542$	0	0
$543$	−52.8918	−2.26980
$544$	0	0
$545$	0.456824	0.0195682
$546$	0	0
$547$	−7.60742	−0.325270	−0.162635	−	0.986686i	$-0.551999\pi$
$547$	−7.60742	−0.325270	−0.162635	+	0.986686i	$0.551999\pi$
$548$	0	0
$549$	−4.71925	−0.201413
$550$	0	0
$551$	−7.98237	−0.340060
$552$	0	0
$553$	−0.885397	−0.0376509
$554$	0	0
$555$	−10.5501	−0.447825
$556$	0	0
$557$	9.75762	0.413444	0.206722	−	0.978400i	$-0.433721\pi$
$557$	9.75762	0.413444	0.206722	+	0.978400i	$0.433721\pi$
$558$	0	0
$559$	−5.77100	−0.244087
$560$	0	0
$561$	−13.2045	−0.557493
$562$	0	0
$563$	25.2165	1.06275	0.531374	−	0.847137i	$-0.321676\pi$
$563$	25.2165	1.06275	0.531374	+	0.847137i	$0.321676\pi$
$564$	0	0
$565$	35.6242	1.49872
$566$	0	0
$567$	−44.0374	−1.84940
$568$	0	0
$569$	40.0890	1.68062	0.840310	−	0.542106i	$-0.182373\pi$
$569$	40.0890	1.68062	0.840310	+	0.542106i	$0.182373\pi$
$570$	0	0
$571$	19.0147	0.795741	0.397871	−	0.917441i	$-0.369749\pi$
$571$	19.0147	0.795741	0.397871	+	0.917441i	$0.369749\pi$
$572$	0	0
$573$	−0.293907	−0.0122782
$574$	0	0
$575$	−4.75066	−0.198116
$576$	0	0
$577$	31.2490	1.30091	0.650456	−	0.759544i	$-0.274578\pi$
$577$	31.2490	1.30091	0.650456	+	0.759544i	$0.274578\pi$
$578$	0	0
$579$	10.0325	0.416937
$580$	0	0
$581$	−39.0848	−1.62151
$582$	0	0
$583$	−2.59497	−0.107473
$584$	0	0
$585$	−20.7312	−0.857131
$586$	0	0
$587$	7.54656	0.311480	0.155740	−	0.987798i	$-0.450224\pi$
$587$	7.54656	0.311480	0.155740	+	0.987798i	$0.450224\pi$
$588$	0	0
$589$	−18.4046	−0.758348
$590$	0	0
$591$	−1.34457	−0.0553082
$592$	0	0
$593$	31.8797	1.30914	0.654570	−	0.756001i	$-0.272850\pi$
$593$	31.8797	1.30914	0.654570	+	0.756001i	$0.272850\pi$
$594$	0	0
$595$	−17.5867	−0.720986
$596$	0	0
$597$	52.6689	2.15559
$598$	0	0
$599$	26.6431	1.08861	0.544303	−	0.838889i	$-0.316794\pi$
$599$	26.6431	1.08861	0.544303	+	0.838889i	$0.316794\pi$
$600$	0	0
$601$	−10.2262	−0.417134	−0.208567	−	0.978008i	$-0.566880\pi$
$601$	−10.2262	−0.417134	−0.208567	+	0.978008i	$0.566880\pi$
$602$	0	0
$603$	1.24979	0.0508955
$604$	0	0
$605$	13.4824	0.548139
$606$	0	0
$607$	−11.0198	−0.447281	−0.223640	−	0.974672i	$-0.571794\pi$
$607$	−11.0198	−0.447281	−0.223640	+	0.974672i	$0.571794\pi$
$608$	0	0
$609$	56.8026	2.30176
$610$	0	0
$611$	−1.31245	−0.0530962
$612$	0	0
$613$	22.9207	0.925759	0.462880	−	0.886421i	$-0.346816\pi$
$613$	22.9207	0.925759	0.462880	+	0.886421i	$0.346816\pi$
$614$	0	0
$615$	34.5381	1.39271
$616$	0	0
$617$	−29.0044	−1.16767	−0.583836	−	0.811871i	$-0.698449\pi$
$617$	−29.0044	−1.16767	−0.583836	+	0.811871i	$0.698449\pi$
$618$	0	0
$619$	38.2556	1.53762	0.768812	−	0.639475i	$-0.220848\pi$
$619$	38.2556	1.53762	0.768812	+	0.639475i	$0.220848\pi$
$620$	0	0
$621$	35.1511	1.41057
$622$	0	0
$623$	−3.82152	−0.153106
$624$	0	0
$625$	−16.8465	−0.673861
$626$	0	0
$627$	10.7804	0.430529
$628$	0	0
$629$	−3.95056	−0.157519
$630$	0	0
$631$	0.945492	0.0376394	0.0188197	−	0.999823i	$-0.494009\pi$
$631$	0.945492	0.0376394	0.0188197	+	0.999823i	$0.494009\pi$
$632$	0	0
$633$	−5.92485	−0.235492
$634$	0	0
$635$	23.4920	0.932250
$636$	0	0
$637$	18.4018	0.729104
$638$	0	0
$639$	−66.1514	−2.61691
$640$	0	0
$641$	19.8943	0.785777	0.392888	−	0.919586i	$-0.371476\pi$
$641$	19.8943	0.785777	0.392888	+	0.919586i	$0.371476\pi$
$642$	0	0
$643$	−1.15954	−0.0457279	−0.0228639	−	0.999739i	$-0.507278\pi$
$643$	−1.15954	−0.0457279	−0.0228639	+	0.999739i	$0.507278\pi$
$644$	0	0
$645$	19.2503	0.757980
$646$	0	0
$647$	24.4790	0.962368	0.481184	−	0.876620i	$-0.340207\pi$
$647$	24.4790	0.962368	0.481184	+	0.876620i	$0.340207\pi$
$648$	0	0
$649$	5.77713	0.226772
$650$	0	0
$651$	130.967	5.13300
$652$	0	0
$653$	−1.56713	−0.0613267	−0.0306633	−	0.999530i	$-0.509762\pi$
$653$	−1.56713	−0.0613267	−0.0306633	+	0.999530i	$0.509762\pi$
$654$	0	0
$655$	39.8260	1.55613
$656$	0	0
$657$	69.9319	2.72830
$658$	0	0
$659$	28.4072	1.10659	0.553295	−	0.832986i	$-0.313370\pi$
$659$	28.4072	1.10659	0.553295	+	0.832986i	$0.313370\pi$
$660$	0	0
$661$	27.5261	1.07064	0.535321	−	0.844648i	$-0.320190\pi$
$661$	27.5261	1.07064	0.535321	+	0.844648i	$0.320190\pi$
$662$	0	0
$663$	−11.5403	−0.448188
$664$	0	0
$665$	14.3582	0.556788
$666$	0	0
$667$	−16.4351	−0.636369
$668$	0	0
$669$	43.4975	1.68171
$670$	0	0
$671$	−1.52997	−0.0590640
$672$	0	0
$673$	−34.7101	−1.33797	−0.668987	−	0.743274i	$-0.733272\pi$
$673$	−34.7101	−1.33797	−0.668987	+	0.743274i	$0.733272\pi$
$674$	0	0
$675$	−12.4123	−0.477751
$676$	0	0
$677$	11.6423	0.447451	0.223725	−	0.974652i	$-0.428178\pi$
$677$	11.6423	0.447451	0.223725	+	0.974652i	$0.428178\pi$
$678$	0	0
$679$	32.0893	1.23147
$680$	0	0
$681$	−43.8747	−1.68128
$682$	0	0
$683$	39.9131	1.52723	0.763617	−	0.645669i	$-0.223422\pi$
$683$	39.9131	1.52723	0.763617	+	0.645669i	$0.223422\pi$
$684$	0	0
$685$	29.4812	1.12642
$686$	0	0
$687$	5.05419	0.192829
$688$	0	0
$689$	−2.26792	−0.0864010
$690$	0	0
$691$	7.60823	0.289431	0.144715	−	0.989473i	$-0.453773\pi$
$691$	7.60823	0.289431	0.144715	+	0.989473i	$0.453773\pi$
$692$	0	0
$693$	−51.6041	−1.96028
$694$	0	0
$695$	−24.6415	−0.934707
$696$	0	0
$697$	12.9331	0.489875
$698$	0	0
$699$	−44.1159	−1.66862
$700$	0	0
$701$	4.18523	0.158074	0.0790370	−	0.996872i	$-0.474815\pi$
$701$	4.18523	0.158074	0.0790370	+	0.996872i	$0.474815\pi$
$702$	0	0
$703$	3.22533	0.121646
$704$	0	0
$705$	4.37794	0.164883
$706$	0	0
$707$	−14.1793	−0.533268
$708$	0	0
$709$	−26.2363	−0.985326	−0.492663	−	0.870220i	$-0.663976\pi$
$709$	−26.2363	−0.985326	−0.492663	+	0.870220i	$0.663976\pi$
$710$	0	0
$711$	1.30367	0.0488914
$712$	0	0
$713$	−37.8936	−1.41913
$714$	0	0
$715$	−6.72103	−0.251352
$716$	0	0
$717$	−34.7856	−1.29909
$718$	0	0
$719$	−36.3421	−1.35533	−0.677665	−	0.735371i	$-0.737008\pi$
$719$	−36.3421	−1.35533	−0.677665	+	0.735371i	$0.737008\pi$
$720$	0	0
$721$	36.0052	1.34090
$722$	0	0
$723$	−69.0486	−2.56795
$724$	0	0
$725$	5.80345	0.215535
$726$	0	0
$727$	24.2159	0.898120	0.449060	−	0.893502i	$-0.351759\pi$
$727$	24.2159	0.898120	0.449060	+	0.893502i	$0.351759\pi$
$728$	0	0
$729$	−22.1986	−0.822169
$730$	0	0
$731$	7.20844	0.266614
$732$	0	0
$733$	21.8245	0.806107	0.403053	−	0.915176i	$-0.367949\pi$
$733$	21.8245	0.806107	0.403053	+	0.915176i	$0.367949\pi$
$734$	0	0
$735$	−61.3826	−2.26413
$736$	0	0
$737$	0.405180	0.0149250
$738$	0	0
$739$	−44.5321	−1.63814	−0.819069	−	0.573695i	$-0.805510\pi$
$739$	−44.5321	−1.63814	−0.819069	+	0.573695i	$0.805510\pi$
$740$	0	0
$741$	9.42176	0.346117
$742$	0	0
$743$	28.1627	1.03319	0.516594	−	0.856230i	$-0.327200\pi$
$743$	28.1627	1.03319	0.516594	+	0.856230i	$0.327200\pi$
$744$	0	0
$745$	−23.2746	−0.852716
$746$	0	0
$747$	57.5490	2.10561
$748$	0	0
$749$	32.9273	1.20314
$750$	0	0
$751$	45.0726	1.64472	0.822361	−	0.568966i	$-0.192656\pi$
$751$	45.0726	1.64472	0.822361	+	0.568966i	$0.192656\pi$
$752$	0	0
$753$	62.4370	2.27533
$754$	0	0
$755$	−13.6508	−0.496803
$756$	0	0
$757$	−15.2957	−0.555932	−0.277966	−	0.960591i	$-0.589660\pi$
$757$	−15.2957	−0.555932	−0.277966	+	0.960591i	$0.589660\pi$
$758$	0	0
$759$	22.1961	0.805666
$760$	0	0
$761$	13.2795	0.481382	0.240691	−	0.970602i	$-0.422626\pi$
$761$	13.2795	0.481382	0.240691	+	0.970602i	$0.422626\pi$
$762$	0	0
$763$	0.993811	0.0359784
$764$	0	0
$765$	25.8949	0.936234
$766$	0	0
$767$	5.04904	0.182310
$768$	0	0
$769$	0.214114	0.00772116	0.00386058	−	0.999993i	$-0.498771\pi$
$769$	0.214114	0.00772116	0.00386058	+	0.999993i	$0.498771\pi$
$770$	0	0
$771$	−21.6610	−0.780101
$772$	0	0
$773$	−0.552400	−0.0198684	−0.00993422	−	0.999951i	$-0.503162\pi$
$773$	−0.552400	−0.0198684	−0.00993422	+	0.999951i	$0.503162\pi$
$774$	0	0
$775$	13.3808	0.480651
$776$	0	0
$777$	−22.9515	−0.823379
$778$	0	0
$779$	−10.5589	−0.378311
$780$	0	0
$781$	−21.4462	−0.767404
$782$	0	0
$783$	−42.9409	−1.53458
$784$	0	0
$785$	32.9189	1.17493
$786$	0	0
$787$	−11.5564	−0.411940	−0.205970	−	0.978558i	$-0.566035\pi$
$787$	−11.5564	−0.411940	−0.205970	+	0.978558i	$0.566035\pi$
$788$	0	0
$789$	15.6934	0.558699
$790$	0	0
$791$	77.4998	2.75557
$792$	0	0
$793$	−1.33715	−0.0474836
$794$	0	0
$795$	7.56510	0.268306
$796$	0	0
$797$	27.2766	0.966188	0.483094	−	0.875569i	$-0.339513\pi$
$797$	27.2766	0.966188	0.483094	+	0.875569i	$0.339513\pi$
$798$	0	0
$799$	1.63936	0.0579963
$800$	0	0
$801$	5.62686	0.198815
$802$	0	0
$803$	22.6718	0.800071
$804$	0	0
$805$	29.5625	1.04194
$806$	0	0
$807$	−11.2759	−0.396930
$808$	0	0
$809$	−28.5509	−1.00380	−0.501898	−	0.864927i	$-0.667365\pi$
$809$	−28.5509	−1.00380	−0.501898	+	0.864927i	$0.667365\pi$
$810$	0	0
$811$	54.7477	1.92245	0.961226	−	0.275762i	$-0.0889301\pi$
$811$	54.7477	1.92245	0.961226	+	0.275762i	$0.0889301\pi$
$812$	0	0
$813$	−60.3869	−2.11786
$814$	0	0
$815$	−18.4035	−0.644645
$816$	0	0
$817$	−5.88514	−0.205895
$818$	0	0
$819$	−45.1004	−1.57593
$820$	0	0
$821$	19.5798	0.683339	0.341670	−	0.939820i	$-0.389008\pi$
$821$	19.5798	0.683339	0.341670	+	0.939820i	$0.389008\pi$
$822$	0	0
$823$	25.8671	0.901669	0.450834	−	0.892608i	$-0.351126\pi$
$823$	25.8671	0.901669	0.450834	+	0.892608i	$0.351126\pi$
$824$	0	0
$825$	−7.83773	−0.272875
$826$	0	0
$827$	−24.0220	−0.835327	−0.417663	−	0.908602i	$-0.637151\pi$
$827$	−24.0220	−0.835327	−0.417663	+	0.908602i	$0.637151\pi$
$828$	0	0
$829$	−13.2092	−0.458774	−0.229387	−	0.973335i	$-0.573672\pi$
$829$	−13.2092	−0.458774	−0.229387	+	0.973335i	$0.573672\pi$
$830$	0	0
$831$	23.1583	0.803352
$832$	0	0
$833$	−22.9853	−0.796392
$834$	0	0
$835$	16.8370	0.582667
$836$	0	0
$837$	−99.0068	−3.42218
$838$	0	0
$839$	11.8939	0.410625	0.205312	−	0.978697i	$-0.434179\pi$
$839$	11.8939	0.410625	0.205312	+	0.978697i	$0.434179\pi$
$840$	0	0
$841$	−8.92275	−0.307681
$842$	0	0
$843$	22.3874	0.771062
$844$	0	0
$845$	19.1483	0.658720
$846$	0	0
$847$	29.3308	1.00782
$848$	0	0
$849$	8.33400	0.286022
$850$	0	0
$851$	6.64071	0.227641
$852$	0	0
$853$	39.5243	1.35329	0.676643	−	0.736311i	$-0.263434\pi$
$853$	39.5243	1.35329	0.676643	+	0.736311i	$0.263434\pi$
$854$	0	0
$855$	−21.1412	−0.723015
$856$	0	0
$857$	43.0691	1.47121	0.735606	−	0.677410i	$-0.236898\pi$
$857$	43.0691	1.47121	0.735606	+	0.677410i	$0.236898\pi$
$858$	0	0
$859$	24.2573	0.827648	0.413824	−	0.910357i	$-0.364193\pi$
$859$	24.2573	0.827648	0.413824	+	0.910357i	$0.364193\pi$
$860$	0	0
$861$	75.1369	2.56066
$862$	0	0
$863$	14.7300	0.501415	0.250707	−	0.968063i	$-0.419337\pi$
$863$	14.7300	0.501415	0.250707	+	0.968063i	$0.419337\pi$
$864$	0	0
$865$	18.4653	0.627838
$866$	0	0
$867$	−37.0520	−1.25835
$868$	0	0
$869$	0.422647	0.0143373
$870$	0	0
$871$	0.354115	0.0119987
$872$	0	0
$873$	−47.2487	−1.59912
$874$	0	0
$875$	−50.7376	−1.71524
$876$	0	0
$877$	−25.8480	−0.872825	−0.436413	−	0.899747i	$-0.643751\pi$
$877$	−25.8480	−0.872825	−0.436413	+	0.899747i	$0.643751\pi$
$878$	0	0
$879$	51.9598	1.75256
$880$	0	0
$881$	−45.7809	−1.54240	−0.771199	−	0.636595i	$-0.780342\pi$
$881$	−45.7809	−1.54240	−0.771199	+	0.636595i	$0.780342\pi$
$882$	0	0
$883$	29.1017	0.979351	0.489675	−	0.871905i	$-0.337115\pi$
$883$	29.1017	0.979351	0.489675	+	0.871905i	$0.337115\pi$
$884$	0	0
$885$	−16.8420	−0.566139
$886$	0	0
$887$	32.1311	1.07886	0.539428	−	0.842031i	$-0.318640\pi$
$887$	32.1311	1.07886	0.539428	+	0.842031i	$0.318640\pi$
$888$	0	0
$889$	51.1063	1.71405
$890$	0	0
$891$	21.0214	0.704244
$892$	0	0
$893$	−1.33841	−0.0447882
$894$	0	0
$895$	−33.2204	−1.11044
$896$	0	0
$897$	19.3987	0.647703
$898$	0	0
$899$	46.2912	1.54390
$900$	0	0
$901$	2.83282	0.0943748
$902$	0	0
$903$	41.8787	1.39363
$904$	0	0
$905$	33.6274	1.11781
$906$	0	0
$907$	−16.9899	−0.564139	−0.282070	−	0.959394i	$-0.591021\pi$
$907$	−16.9899	−0.564139	−0.282070	+	0.959394i	$0.591021\pi$
$908$	0	0
$909$	20.8778	0.692473
$910$	0	0
$911$	42.6649	1.41355	0.706776	−	0.707438i	$-0.250149\pi$
$911$	42.6649	1.41355	0.706776	+	0.707438i	$0.250149\pi$
$912$	0	0
$913$	18.6573	0.617466
$914$	0	0
$915$	4.46032	0.147454
$916$	0	0
$917$	86.6407	2.86113
$918$	0	0
$919$	11.7437	0.387389	0.193694	−	0.981062i	$-0.437953\pi$
$919$	11.7437	0.387389	0.193694	+	0.981062i	$0.437953\pi$
$920$	0	0
$921$	60.5515	1.99524
$922$	0	0
$923$	−18.7433	−0.616943
$924$	0	0
$925$	−2.34492	−0.0771006
$926$	0	0
$927$	−53.0145	−1.74122
$928$	0	0
$929$	−58.6215	−1.92331	−0.961655	−	0.274263i	$-0.911566\pi$
$929$	−58.6215	−1.92331	−0.961655	+	0.274263i	$0.911566\pi$
$930$	0	0
$931$	18.7657	0.615021
$932$	0	0
$933$	−61.9689	−2.02877
$934$	0	0
$935$	8.39509	0.274549
$936$	0	0
$937$	−48.7133	−1.59140	−0.795698	−	0.605694i	$-0.792895\pi$
$937$	−48.7133	−1.59140	−0.795698	+	0.605694i	$0.792895\pi$
$938$	0	0
$939$	−87.8461	−2.86675
$940$	0	0
$941$	52.6479	1.71627	0.858137	−	0.513421i	$-0.171622\pi$
$941$	52.6479	1.71627	0.858137	+	0.513421i	$0.171622\pi$
$942$	0	0
$943$	−21.7399	−0.707948
$944$	0	0
$945$	77.2396	2.51260
$946$	0	0
$947$	21.2551	0.690698	0.345349	−	0.938474i	$-0.387761\pi$
$947$	21.2551	0.690698	0.345349	+	0.938474i	$0.387761\pi$
$948$	0	0
$949$	19.8145	0.643205
$950$	0	0
$951$	−86.0399	−2.79003
$952$	0	0
$953$	8.14389	0.263807	0.131903	−	0.991263i	$-0.457891\pi$
$953$	8.14389	0.263807	0.131903	+	0.991263i	$0.457891\pi$
$954$	0	0
$955$	0.186860	0.00604664
$956$	0	0
$957$	−27.1149	−0.876501
$958$	0	0
$959$	64.1357	2.07105
$960$	0	0
$961$	75.7314	2.44295
$962$	0	0
$963$	−48.4826	−1.56233
$964$	0	0
$965$	−6.37844	−0.205329
$966$	0	0
$967$	−3.44544	−0.110798	−0.0553990	−	0.998464i	$-0.517643\pi$
$967$	−3.44544	−0.110798	−0.0553990	+	0.998464i	$0.517643\pi$
$968$	0	0
$969$	−11.7685	−0.378060
$970$	0	0
$971$	−47.5402	−1.52564	−0.762819	−	0.646612i	$-0.776185\pi$
$971$	−47.5402	−1.52564	−0.762819	+	0.646612i	$0.776185\pi$
$972$	0	0
$973$	−53.6072	−1.71857
$974$	0	0
$975$	−6.84994	−0.219374
$976$	0	0
$977$	56.7589	1.81588	0.907940	−	0.419101i	$-0.137655\pi$
$977$	56.7589	1.81588	0.907940	+	0.419101i	$0.137655\pi$
$978$	0	0
$979$	1.82422	0.0583022
$980$	0	0
$981$	−1.46330	−0.0467196
$982$	0	0
$983$	−41.0543	−1.30943	−0.654714	−	0.755877i	$-0.727211\pi$
$983$	−41.0543	−1.30943	−0.654714	+	0.755877i	$0.727211\pi$
$984$	0	0
$985$	0.854847	0.0272377
$986$	0	0
$987$	9.52413	0.303156
$988$	0	0
$989$	−12.1170	−0.385300
$990$	0	0
$991$	32.0872	1.01928	0.509642	−	0.860386i	$-0.329778\pi$
$991$	32.0872	1.01928	0.509642	+	0.860386i	$0.329778\pi$
$992$	0	0
$993$	59.3523	1.88349
$994$	0	0
$995$	−33.4857	−1.06157
$996$	0	0
$997$	16.4819	0.521986	0.260993	−	0.965341i	$-0.415950\pi$
$997$	16.4819	0.521986	0.260993	+	0.965341i	$0.415950\pi$
$998$	0	0
$999$	17.3506	0.548947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.28		28
4.3	odd	2		4024.2.a.d.1.1	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.1	✓	28	4.3	odd	2
8048.2.a.v.1.28		28	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+3.02746 q^{3} -1.92479 q^{5} -4.18734 q^{7} +6.16549 q^{9} +O(q^{10})\)	\(q+3.02746 q^{3} -1.92479 q^{5} -4.18734 q^{7} +6.16549 q^{9} +1.99884 q^{11} +1.74693 q^{13} -5.82721 q^{15} -2.18205 q^{17} +1.78148 q^{19} -12.6770 q^{21} +3.66792 q^{23} -1.29519 q^{25} +9.58339 q^{27} -4.48076 q^{29} -10.3311 q^{31} +6.05140 q^{33} +8.05974 q^{35} +1.81048 q^{37} +5.28874 q^{39} -5.92703 q^{41} -3.30352 q^{43} -11.8673 q^{45} -0.751293 q^{47} +10.5338 q^{49} -6.60606 q^{51} -1.29824 q^{53} -3.84734 q^{55} +5.39334 q^{57} +2.89024 q^{59} -0.765430 q^{61} -25.8170 q^{63} -3.36246 q^{65} +0.202708 q^{67} +11.1045 q^{69} -10.7293 q^{71} +11.3425 q^{73} -3.92114 q^{75} -8.36982 q^{77} +0.211446 q^{79} +10.5168 q^{81} +9.33406 q^{83} +4.19998 q^{85} -13.5653 q^{87} +0.912638 q^{89} -7.31497 q^{91} -31.2769 q^{93} -3.42896 q^{95} -7.66341 q^{97} +12.3238 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

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