LMFDB - Embedded newform 8048.2.a.t.1.17

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:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+2.48667 q^{3} -2.90154 q^{5} +4.76998 q^{7} +3.18353 q^{9} +O(q^{10})$	$q+2.48667 q^{3} -2.90154 q^{5} +4.76998 q^{7} +3.18353 q^{9} -1.39880 q^{11} +5.92030 q^{13} -7.21518 q^{15} -0.806345 q^{17} -3.02004 q^{19} +11.8614 q^{21} -3.95904 q^{23} +3.41894 q^{25} +0.456381 q^{27} +1.70635 q^{29} +7.91836 q^{31} -3.47837 q^{33} -13.8403 q^{35} +0.267401 q^{37} +14.7218 q^{39} +9.75823 q^{41} +0.977947 q^{43} -9.23714 q^{45} +5.93952 q^{47} +15.7528 q^{49} -2.00511 q^{51} -10.8567 q^{53} +4.05869 q^{55} -7.50985 q^{57} -0.574963 q^{59} -4.46807 q^{61} +15.1854 q^{63} -17.1780 q^{65} +15.4152 q^{67} -9.84482 q^{69} +8.80449 q^{71} +8.74581 q^{73} +8.50177 q^{75} -6.67227 q^{77} -14.4701 q^{79} -8.41572 q^{81} -3.50674 q^{83} +2.33964 q^{85} +4.24314 q^{87} +16.8495 q^{89} +28.2397 q^{91} +19.6904 q^{93} +8.76278 q^{95} +2.08018 q^{97} -4.45314 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * 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
$3$	2.48667	1.43568	0.717840	−	0.696208i	$-0.245131\pi$
$3$	2.48667	1.43568	0.717840	+	0.696208i	$0.245131\pi$
$4$	0	0
$5$	−2.90154	−1.29761	−0.648804	−	0.760955i	$-0.724731\pi$
$5$	−2.90154	−1.29761	−0.648804	+	0.760955i	$0.724731\pi$
$6$	0	0
$7$	4.76998	1.80288	0.901442	−	0.432899i	$-0.142509\pi$
$7$	4.76998	1.80288	0.901442	+	0.432899i	$0.142509\pi$
$8$	0	0
$9$	3.18353	1.06118
$10$	0	0
$11$	−1.39880	−0.421755	−0.210878	−	0.977512i	$-0.567632\pi$
$11$	−1.39880	−0.421755	−0.210878	+	0.977512i	$0.567632\pi$
$12$	0	0
$13$	5.92030	1.64200	0.820998	−	0.570931i	$-0.193418\pi$
$13$	5.92030	1.64200	0.820998	+	0.570931i	$0.193418\pi$
$14$	0	0
$15$	−7.21518	−1.86295
$16$	0	0
$17$	−0.806345	−0.195567	−0.0977836	−	0.995208i	$-0.531175\pi$
$17$	−0.806345	−0.195567	−0.0977836	+	0.995208i	$0.531175\pi$
$18$	0	0
$19$	−3.02004	−0.692845	−0.346423	−	0.938079i	$-0.612604\pi$
$19$	−3.02004	−0.692845	−0.346423	+	0.938079i	$0.612604\pi$
$20$	0	0
$21$	11.8614	2.58837
$22$	0	0
$23$	−3.95904	−0.825516	−0.412758	−	0.910841i	$-0.635435\pi$
$23$	−3.95904	−0.825516	−0.412758	+	0.910841i	$0.635435\pi$
$24$	0	0
$25$	3.41894	0.683787
$26$	0	0
$27$	0.456381	0.0878307
$28$	0	0
$29$	1.70635	0.316862	0.158431	−	0.987370i	$-0.449356\pi$
$29$	1.70635	0.316862	0.158431	+	0.987370i	$0.449356\pi$
$30$	0	0
$31$	7.91836	1.42218	0.711090	−	0.703101i	$-0.248202\pi$
$31$	7.91836	1.42218	0.711090	+	0.703101i	$0.248202\pi$
$32$	0	0
$33$	−3.47837	−0.605506
$34$	0	0
$35$	−13.8403	−2.33944
$36$	0	0
$37$	0.267401	0.0439605	0.0219802	−	0.999758i	$-0.493003\pi$
$37$	0.267401	0.0439605	0.0219802	+	0.999758i	$0.493003\pi$
$38$	0	0
$39$	14.7218	2.35738
$40$	0	0
$41$	9.75823	1.52398	0.761989	−	0.647589i	$-0.224223\pi$
$41$	9.75823	1.52398	0.761989	+	0.647589i	$0.224223\pi$
$42$	0	0
$43$	0.977947	0.149136	0.0745678	−	0.997216i	$-0.476242\pi$
$43$	0.977947	0.149136	0.0745678	+	0.997216i	$0.476242\pi$
$44$	0	0
$45$	−9.23714	−1.37699
$46$	0	0
$47$	5.93952	0.866368	0.433184	−	0.901305i	$-0.357390\pi$
$47$	5.93952	0.866368	0.433184	+	0.901305i	$0.357390\pi$
$48$	0	0
$49$	15.7528	2.25039
$50$	0	0
$51$	−2.00511	−0.280772
$52$	0	0
$53$	−10.8567	−1.49128	−0.745639	−	0.666350i	$-0.767856\pi$
$53$	−10.8567	−1.49128	−0.745639	+	0.666350i	$0.767856\pi$
$54$	0	0
$55$	4.05869	0.547273
$56$	0	0
$57$	−7.50985	−0.994704
$58$	0	0
$59$	−0.574963	−0.0748539	−0.0374269	−	0.999299i	$-0.511916\pi$
$59$	−0.574963	−0.0748539	−0.0374269	+	0.999299i	$0.511916\pi$
$60$	0	0
$61$	−4.46807	−0.572077	−0.286039	−	0.958218i	$-0.592339\pi$
$61$	−4.46807	−0.572077	−0.286039	+	0.958218i	$0.592339\pi$
$62$	0	0
$63$	15.1854	1.91318
$64$	0	0
$65$	−17.1780	−2.13067
$66$	0	0
$67$	15.4152	1.88327	0.941634	−	0.336639i	$-0.109290\pi$
$67$	15.4152	1.88327	0.941634	+	0.336639i	$0.109290\pi$
$68$	0	0
$69$	−9.84482	−1.18518
$70$	0	0
$71$	8.80449	1.04490	0.522450	−	0.852670i	$-0.325018\pi$
$71$	8.80449	1.04490	0.522450	+	0.852670i	$0.325018\pi$
$72$	0	0
$73$	8.74581	1.02362	0.511810	−	0.859099i	$-0.328975\pi$
$73$	8.74581	1.02362	0.511810	+	0.859099i	$0.328975\pi$
$74$	0	0
$75$	8.50177	0.981700
$76$	0	0
$77$	−6.67227	−0.760376
$78$	0	0
$79$	−14.4701	−1.62801	−0.814007	−	0.580855i	$-0.802718\pi$
$79$	−14.4701	−1.62801	−0.814007	+	0.580855i	$0.802718\pi$
$80$	0	0
$81$	−8.41572	−0.935080
$82$	0	0
$83$	−3.50674	−0.384915	−0.192458	−	0.981305i	$-0.561646\pi$
$83$	−3.50674	−0.384915	−0.192458	+	0.981305i	$0.561646\pi$
$84$	0	0
$85$	2.33964	0.253770
$86$	0	0
$87$	4.24314	0.454912
$88$	0	0
$89$	16.8495	1.78605	0.893023	−	0.450010i	$-0.148580\pi$
$89$	16.8495	1.78605	0.893023	+	0.450010i	$0.148580\pi$
$90$	0	0
$91$	28.2397	2.96033
$92$	0	0
$93$	19.6904	2.04180
$94$	0	0
$95$	8.76278	0.899042
$96$	0	0
$97$	2.08018	0.211211	0.105605	−	0.994408i	$-0.466322\pi$
$97$	2.08018	0.211211	0.105605	+	0.994408i	$0.466322\pi$
$98$	0	0
$99$	−4.45314	−0.447557
$100$	0	0
$101$	0.293452	0.0291995	0.0145998	−	0.999893i	$-0.495353\pi$
$101$	0.293452	0.0291995	0.0145998	+	0.999893i	$0.495353\pi$
$102$	0	0
$103$	−1.16817	−0.115103	−0.0575515	−	0.998343i	$-0.518329\pi$
$103$	−1.16817	−0.115103	−0.0575515	+	0.998343i	$0.518329\pi$
$104$	0	0
$105$	−34.4163	−3.35868
$106$	0	0
$107$	−8.31770	−0.804103	−0.402051	−	0.915617i	$-0.631703\pi$
$107$	−8.31770	−0.804103	−0.402051	+	0.915617i	$0.631703\pi$
$108$	0	0
$109$	−7.68463	−0.736054	−0.368027	−	0.929815i	$-0.619967\pi$
$109$	−7.68463	−0.736054	−0.368027	+	0.929815i	$0.619967\pi$
$110$	0	0
$111$	0.664939	0.0631132
$112$	0	0
$113$	−4.62148	−0.434752	−0.217376	−	0.976088i	$-0.569750\pi$
$113$	−4.62148	−0.434752	−0.217376	+	0.976088i	$0.569750\pi$
$114$	0	0
$115$	11.4873	1.07120
$116$	0	0
$117$	18.8475	1.74245
$118$	0	0
$119$	−3.84625	−0.352585
$120$	0	0
$121$	−9.04335	−0.822122
$122$	0	0
$123$	24.2655	2.18795
$124$	0	0
$125$	4.58752	0.410320
$126$	0	0
$127$	12.6306	1.12078	0.560391	−	0.828228i	$-0.310651\pi$
$127$	12.6306	1.12078	0.560391	+	0.828228i	$0.310651\pi$
$128$	0	0
$129$	2.43183	0.214111
$130$	0	0
$131$	−9.47867	−0.828155	−0.414078	−	0.910242i	$-0.635896\pi$
$131$	−9.47867	−0.828155	−0.414078	+	0.910242i	$0.635896\pi$
$132$	0	0
$133$	−14.4056	−1.24912
$134$	0	0
$135$	−1.32421	−0.113970
$136$	0	0
$137$	2.86645	0.244898	0.122449	−	0.992475i	$-0.460925\pi$
$137$	2.86645	0.244898	0.122449	+	0.992475i	$0.460925\pi$
$138$	0	0
$139$	7.07740	0.600297	0.300149	−	0.953892i	$-0.402964\pi$
$139$	7.07740	0.600297	0.300149	+	0.953892i	$0.402964\pi$
$140$	0	0
$141$	14.7696	1.24383
$142$	0	0
$143$	−8.28134	−0.692521
$144$	0	0
$145$	−4.95105	−0.411163
$146$	0	0
$147$	39.1719	3.23084
$148$	0	0
$149$	20.7031	1.69606	0.848031	−	0.529947i	$-0.177788\pi$
$149$	20.7031	1.69606	0.848031	+	0.529947i	$0.177788\pi$
$150$	0	0
$151$	−14.2273	−1.15780	−0.578901	−	0.815398i	$-0.696518\pi$
$151$	−14.2273	−1.15780	−0.578901	+	0.815398i	$0.696518\pi$
$152$	0	0
$153$	−2.56702	−0.207532
$154$	0	0
$155$	−22.9754	−1.84543
$156$	0	0
$157$	20.3648	1.62529	0.812645	−	0.582759i	$-0.198027\pi$
$157$	20.3648	1.62529	0.812645	+	0.582759i	$0.198027\pi$
$158$	0	0
$159$	−26.9970	−2.14100
$160$	0	0
$161$	−18.8845	−1.48831
$162$	0	0
$163$	16.6247	1.30215	0.651074	−	0.759014i	$-0.274319\pi$
$163$	16.6247	1.30215	0.651074	+	0.759014i	$0.274319\pi$
$164$	0	0
$165$	10.0926	0.785709
$166$	0	0
$167$	14.6966	1.13726	0.568629	−	0.822594i	$-0.307474\pi$
$167$	14.6966	1.13726	0.568629	+	0.822594i	$0.307474\pi$
$168$	0	0
$169$	22.0500	1.69615
$170$	0	0
$171$	−9.61440	−0.735231
$172$	0	0
$173$	−19.6706	−1.49553	−0.747765	−	0.663963i	$-0.768873\pi$
$173$	−19.6706	−1.49553	−0.747765	+	0.663963i	$0.768873\pi$
$174$	0	0
$175$	16.3083	1.23279
$176$	0	0
$177$	−1.42974	−0.107466
$178$	0	0
$179$	10.6664	0.797242	0.398621	−	0.917116i	$-0.369489\pi$
$179$	10.6664	0.797242	0.398621	+	0.917116i	$0.369489\pi$
$180$	0	0
$181$	−14.2682	−1.06054	−0.530272	−	0.847828i	$-0.677910\pi$
$181$	−14.2682	−1.06054	−0.530272	+	0.847828i	$0.677910\pi$
$182$	0	0
$183$	−11.1106	−0.821320
$184$	0	0
$185$	−0.775876	−0.0570435
$186$	0	0
$187$	1.12792	0.0824816
$188$	0	0
$189$	2.17693	0.158349
$190$	0	0
$191$	−21.0283	−1.52155	−0.760776	−	0.649014i	$-0.775182\pi$
$191$	−21.0283	−1.52155	−0.760776	+	0.649014i	$0.775182\pi$
$192$	0	0
$193$	−12.3350	−0.887889	−0.443945	−	0.896054i	$-0.646421\pi$
$193$	−12.3350	−0.887889	−0.443945	+	0.896054i	$0.646421\pi$
$194$	0	0
$195$	−42.7160	−3.05896
$196$	0	0
$197$	−17.6252	−1.25574	−0.627871	−	0.778317i	$-0.716073\pi$
$197$	−17.6252	−1.25574	−0.627871	+	0.778317i	$0.716073\pi$
$198$	0	0
$199$	12.4517	0.882681	0.441340	−	0.897340i	$-0.354503\pi$
$199$	12.4517	0.882681	0.441340	+	0.897340i	$0.354503\pi$
$200$	0	0
$201$	38.3326	2.70377
$202$	0	0
$203$	8.13928	0.571266
$204$	0	0
$205$	−28.3139	−1.97753
$206$	0	0
$207$	−12.6037	−0.876019
$208$	0	0
$209$	4.22445	0.292211
$210$	0	0
$211$	18.7804	1.29290	0.646448	−	0.762958i	$-0.276254\pi$
$211$	18.7804	1.29290	0.646448	+	0.762958i	$0.276254\pi$
$212$	0	0
$213$	21.8939	1.50014
$214$	0	0
$215$	−2.83755	−0.193520
$216$	0	0
$217$	37.7705	2.56403
$218$	0	0
$219$	21.7480	1.46959
$220$	0	0
$221$	−4.77380	−0.321121
$222$	0	0
$223$	1.92933	0.129197	0.0645986	−	0.997911i	$-0.479423\pi$
$223$	1.92933	0.129197	0.0645986	+	0.997911i	$0.479423\pi$
$224$	0	0
$225$	10.8843	0.725620
$226$	0	0
$227$	6.54632	0.434495	0.217247	−	0.976117i	$-0.430292\pi$
$227$	6.54632	0.434495	0.217247	+	0.976117i	$0.430292\pi$
$228$	0	0
$229$	7.39619	0.488754	0.244377	−	0.969680i	$-0.421417\pi$
$229$	7.39619	0.488754	0.244377	+	0.969680i	$0.421417\pi$
$230$	0	0
$231$	−16.5917	−1.09166
$232$	0	0
$233$	−1.43968	−0.0943165	−0.0471582	−	0.998887i	$-0.515017\pi$
$233$	−1.43968	−0.0943165	−0.0471582	+	0.998887i	$0.515017\pi$
$234$	0	0
$235$	−17.2338	−1.12421
$236$	0	0
$237$	−35.9824	−2.33731
$238$	0	0
$239$	−0.965551	−0.0624563	−0.0312282	−	0.999512i	$-0.509942\pi$
$239$	−0.965551	−0.0624563	−0.0312282	+	0.999512i	$0.509942\pi$
$240$	0	0
$241$	−7.20544	−0.464143	−0.232072	−	0.972699i	$-0.574550\pi$
$241$	−7.20544	−0.464143	−0.232072	+	0.972699i	$0.574550\pi$
$242$	0	0
$243$	−22.2963	−1.43031
$244$	0	0
$245$	−45.7072	−2.92013
$246$	0	0
$247$	−17.8796	−1.13765
$248$	0	0
$249$	−8.72011	−0.552615
$250$	0	0
$251$	11.6696	0.736581	0.368291	−	0.929711i	$-0.379943\pi$
$251$	11.6696	0.736581	0.368291	+	0.929711i	$0.379943\pi$
$252$	0	0
$253$	5.53792	0.348166
$254$	0	0
$255$	5.81792	0.364332
$256$	0	0
$257$	14.9792	0.934376	0.467188	−	0.884158i	$-0.345267\pi$
$257$	14.9792	0.934376	0.467188	+	0.884158i	$0.345267\pi$
$258$	0	0
$259$	1.27550	0.0792557
$260$	0	0
$261$	5.43223	0.336247
$262$	0	0
$263$	10.4459	0.644123	0.322061	−	0.946719i	$-0.395624\pi$
$263$	10.4459	0.644123	0.322061	+	0.946719i	$0.395624\pi$
$264$	0	0
$265$	31.5011	1.93510
$266$	0	0
$267$	41.8992	2.56419
$268$	0	0
$269$	−23.3342	−1.42271	−0.711356	−	0.702832i	$-0.751918\pi$
$269$	−23.3342	−1.42271	−0.711356	+	0.702832i	$0.751918\pi$
$270$	0	0
$271$	−2.20440	−0.133908	−0.0669540	−	0.997756i	$-0.521328\pi$
$271$	−2.20440	−0.133908	−0.0669540	+	0.997756i	$0.521328\pi$
$272$	0	0
$273$	70.2229	4.25009
$274$	0	0
$275$	−4.78242	−0.288391
$276$	0	0
$277$	−21.3838	−1.28483	−0.642413	−	0.766358i	$-0.722067\pi$
$277$	−21.3838	−1.28483	−0.642413	+	0.766358i	$0.722067\pi$
$278$	0	0
$279$	25.2084	1.50918
$280$	0	0
$281$	−24.9733	−1.48978	−0.744890	−	0.667187i	$-0.767498\pi$
$281$	−24.9733	−1.48978	−0.744890	+	0.667187i	$0.767498\pi$
$282$	0	0
$283$	18.6105	1.10628	0.553140	−	0.833088i	$-0.313430\pi$
$283$	18.6105	1.10628	0.553140	+	0.833088i	$0.313430\pi$
$284$	0	0
$285$	21.7901	1.29074
$286$	0	0
$287$	46.5466	2.74756
$288$	0	0
$289$	−16.3498	−0.961753
$290$	0	0
$291$	5.17273	0.303231
$292$	0	0
$293$	20.9272	1.22258	0.611290	−	0.791406i	$-0.290651\pi$
$293$	20.9272	1.22258	0.611290	+	0.791406i	$0.290651\pi$
$294$	0	0
$295$	1.66828	0.0971310
$296$	0	0
$297$	−0.638388	−0.0370431
$298$	0	0
$299$	−23.4387	−1.35549
$300$	0	0
$301$	4.66479	0.268874
$302$	0	0
$303$	0.729718	0.0419212
$304$	0	0
$305$	12.9643	0.742332
$306$	0	0
$307$	−16.5716	−0.945792	−0.472896	−	0.881118i	$-0.656791\pi$
$307$	−16.5716	−0.945792	−0.472896	+	0.881118i	$0.656791\pi$
$308$	0	0
$309$	−2.90485	−0.165251
$310$	0	0
$311$	−13.2333	−0.750389	−0.375195	−	0.926946i	$-0.622424\pi$
$311$	−13.2333	−0.750389	−0.375195	+	0.926946i	$0.622424\pi$
$312$	0	0
$313$	7.71096	0.435849	0.217925	−	0.975966i	$-0.430071\pi$
$313$	7.71096	0.435849	0.217925	+	0.975966i	$0.430071\pi$
$314$	0	0
$315$	−44.0610	−2.48256
$316$	0	0
$317$	−10.6378	−0.597478	−0.298739	−	0.954335i	$-0.596566\pi$
$317$	−10.6378	−0.597478	−0.298739	+	0.954335i	$0.596566\pi$
$318$	0	0
$319$	−2.38686	−0.133638
$320$	0	0
$321$	−20.6834	−1.15443
$322$	0	0
$323$	2.43519	0.135498
$324$	0	0
$325$	20.2411	1.12278
$326$	0	0
$327$	−19.1091	−1.05674
$328$	0	0
$329$	28.3314	1.56196
$330$	0	0
$331$	−2.98352	−0.163989	−0.0819944	−	0.996633i	$-0.526129\pi$
$331$	−2.98352	−0.163989	−0.0819944	+	0.996633i	$0.526129\pi$
$332$	0	0
$333$	0.851280	0.0466499
$334$	0	0
$335$	−44.7279	−2.44374
$336$	0	0
$337$	−6.99519	−0.381052	−0.190526	−	0.981682i	$-0.561019\pi$
$337$	−6.99519	−0.381052	−0.190526	+	0.981682i	$0.561019\pi$
$338$	0	0
$339$	−11.4921	−0.624165
$340$	0	0
$341$	−11.0762	−0.599812
$342$	0	0
$343$	41.7505	2.25431
$344$	0	0
$345$	28.5652	1.53790
$346$	0	0
$347$	6.40731	0.343962	0.171981	−	0.985100i	$-0.444983\pi$
$347$	6.40731	0.343962	0.171981	+	0.985100i	$0.444983\pi$
$348$	0	0
$349$	−17.7123	−0.948118	−0.474059	−	0.880493i	$-0.657212\pi$
$349$	−17.7123	−0.948118	−0.474059	+	0.880493i	$0.657212\pi$
$350$	0	0
$351$	2.70192	0.144218
$352$	0	0
$353$	34.7422	1.84914	0.924570	−	0.381012i	$-0.124424\pi$
$353$	34.7422	1.84914	0.924570	+	0.381012i	$0.124424\pi$
$354$	0	0
$355$	−25.5466	−1.35587
$356$	0	0
$357$	−9.56436	−0.506200
$358$	0	0
$359$	8.46341	0.446682	0.223341	−	0.974740i	$-0.428304\pi$
$359$	8.46341	0.446682	0.223341	+	0.974740i	$0.428304\pi$
$360$	0	0
$361$	−9.87934	−0.519965
$362$	0	0
$363$	−22.4878	−1.18030
$364$	0	0
$365$	−25.3763	−1.32826
$366$	0	0
$367$	−15.8395	−0.826814	−0.413407	−	0.910546i	$-0.635661\pi$
$367$	−15.8395	−0.826814	−0.413407	+	0.910546i	$0.635661\pi$
$368$	0	0
$369$	31.0656	1.61721
$370$	0	0
$371$	−51.7862	−2.68860
$372$	0	0
$373$	−21.3990	−1.10800	−0.553999	−	0.832517i	$-0.686899\pi$
$373$	−21.3990	−1.10800	−0.553999	+	0.832517i	$0.686899\pi$
$374$	0	0
$375$	11.4076	0.589088
$376$	0	0
$377$	10.1021	0.520286
$378$	0	0
$379$	−1.53993	−0.0791008	−0.0395504	−	0.999218i	$-0.512593\pi$
$379$	−1.53993	−0.0791008	−0.0395504	+	0.999218i	$0.512593\pi$
$380$	0	0
$381$	31.4081	1.60909
$382$	0	0
$383$	9.90066	0.505900	0.252950	−	0.967479i	$-0.418599\pi$
$383$	9.90066	0.505900	0.252950	+	0.967479i	$0.418599\pi$
$384$	0	0
$385$	19.3599	0.986671
$386$	0	0
$387$	3.11333	0.158259
$388$	0	0
$389$	3.66812	0.185981	0.0929906	−	0.995667i	$-0.470357\pi$
$389$	3.66812	0.185981	0.0929906	+	0.995667i	$0.470357\pi$
$390$	0	0
$391$	3.19235	0.161444
$392$	0	0
$393$	−23.5703	−1.18897
$394$	0	0
$395$	41.9856	2.11252
$396$	0	0
$397$	−27.6079	−1.38560	−0.692801	−	0.721129i	$-0.743623\pi$
$397$	−27.6079	−1.38560	−0.692801	+	0.721129i	$0.743623\pi$
$398$	0	0
$399$	−35.8219	−1.79334
$400$	0	0
$401$	22.3927	1.11824	0.559119	−	0.829088i	$-0.311140\pi$
$401$	22.3927	1.11824	0.559119	+	0.829088i	$0.311140\pi$
$402$	0	0
$403$	46.8791	2.33521
$404$	0	0
$405$	24.4186	1.21337
$406$	0	0
$407$	−0.374042	−0.0185406
$408$	0	0
$409$	21.8110	1.07848	0.539242	−	0.842151i	$-0.318711\pi$
$409$	21.8110	1.07848	0.539242	+	0.842151i	$0.318711\pi$
$410$	0	0
$411$	7.12792	0.351595
$412$	0	0
$413$	−2.74257	−0.134953
$414$	0	0
$415$	10.1750	0.499469
$416$	0	0
$417$	17.5992	0.861835
$418$	0	0
$419$	35.9337	1.75548	0.877738	−	0.479141i	$-0.159052\pi$
$419$	35.9337	1.75548	0.877738	+	0.479141i	$0.159052\pi$
$420$	0	0
$421$	36.7511	1.79114	0.895570	−	0.444921i	$-0.146768\pi$
$421$	36.7511	1.79114	0.895570	+	0.444921i	$0.146768\pi$
$422$	0	0
$423$	18.9087	0.919370
$424$	0	0
$425$	−2.75684	−0.133726
$426$	0	0
$427$	−21.3126	−1.03139
$428$	0	0
$429$	−20.5930	−0.994238
$430$	0	0
$431$	−33.8345	−1.62975	−0.814875	−	0.579637i	$-0.803194\pi$
$431$	−33.8345	−1.62975	−0.814875	+	0.579637i	$0.803194\pi$
$432$	0	0
$433$	1.16595	0.0560320	0.0280160	−	0.999607i	$-0.491081\pi$
$433$	1.16595	0.0560320	0.0280160	+	0.999607i	$0.491081\pi$
$434$	0	0
$435$	−12.3116	−0.590298
$436$	0	0
$437$	11.9565	0.571955
$438$	0	0
$439$	23.0230	1.09883	0.549414	−	0.835550i	$-0.314851\pi$
$439$	23.0230	1.09883	0.549414	+	0.835550i	$0.314851\pi$
$440$	0	0
$441$	50.1494	2.38807
$442$	0	0
$443$	−11.9529	−0.567897	−0.283949	−	0.958839i	$-0.591645\pi$
$443$	−11.9529	−0.567897	−0.283949	+	0.958839i	$0.591645\pi$
$444$	0	0
$445$	−48.8896	−2.31759
$446$	0	0
$447$	51.4817	2.43500
$448$	0	0
$449$	14.7853	0.697759	0.348880	−	0.937168i	$-0.386562\pi$
$449$	14.7853	0.697759	0.348880	+	0.937168i	$0.386562\pi$
$450$	0	0
$451$	−13.6498	−0.642746
$452$	0	0
$453$	−35.3786	−1.66223
$454$	0	0
$455$	−81.9388	−3.84135
$456$	0	0
$457$	16.2850	0.761780	0.380890	−	0.924620i	$-0.375618\pi$
$457$	16.2850	0.761780	0.380890	+	0.924620i	$0.375618\pi$
$458$	0	0
$459$	−0.368001	−0.0171768
$460$	0	0
$461$	−6.89191	−0.320988	−0.160494	−	0.987037i	$-0.551309\pi$
$461$	−6.89191	−0.320988	−0.160494	+	0.987037i	$0.551309\pi$
$462$	0	0
$463$	32.9900	1.53318	0.766589	−	0.642138i	$-0.221953\pi$
$463$	32.9900	1.53318	0.766589	+	0.642138i	$0.221953\pi$
$464$	0	0
$465$	−57.1324	−2.64945
$466$	0	0
$467$	−4.34048	−0.200853	−0.100427	−	0.994944i	$-0.532021\pi$
$467$	−4.34048	−0.200853	−0.100427	+	0.994944i	$0.532021\pi$
$468$	0	0
$469$	73.5303	3.39531
$470$	0	0
$471$	50.6406	2.33340
$472$	0	0
$473$	−1.36796	−0.0628987
$474$	0	0
$475$	−10.3253	−0.473759
$476$	0	0
$477$	−34.5626	−1.58251
$478$	0	0
$479$	16.5262	0.755100	0.377550	−	0.925989i	$-0.376767\pi$
$479$	16.5262	0.755100	0.377550	+	0.925989i	$0.376767\pi$
$480$	0	0
$481$	1.58310	0.0721830
$482$	0	0
$483$	−46.9597	−2.13674
$484$	0	0
$485$	−6.03573	−0.274069
$486$	0	0
$487$	31.1925	1.41347	0.706734	−	0.707479i	$-0.250168\pi$
$487$	31.1925	1.41347	0.706734	+	0.707479i	$0.250168\pi$
$488$	0	0
$489$	41.3402	1.86947
$490$	0	0
$491$	−34.8121	−1.57105	−0.785525	−	0.618830i	$-0.787607\pi$
$491$	−34.8121	−1.57105	−0.785525	+	0.618830i	$0.787607\pi$
$492$	0	0
$493$	−1.37591	−0.0619678
$494$	0	0
$495$	12.9210	0.580754
$496$	0	0
$497$	41.9973	1.88383
$498$	0	0
$499$	−17.1690	−0.768590	−0.384295	−	0.923210i	$-0.625555\pi$
$499$	−17.1690	−0.768590	−0.384295	+	0.923210i	$0.625555\pi$
$500$	0	0
$501$	36.5456	1.63274
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−0.851462	−0.0378896
$506$	0	0
$507$	54.8310	2.43513
$508$	0	0
$509$	18.0265	0.799012	0.399506	−	0.916730i	$-0.369182\pi$
$509$	18.0265	0.799012	0.399506	+	0.916730i	$0.369182\pi$
$510$	0	0
$511$	41.7174	1.84547
$512$	0	0
$513$	−1.37829	−0.0608531
$514$	0	0
$515$	3.38949	0.149359
$516$	0	0
$517$	−8.30823	−0.365396
$518$	0	0
$519$	−48.9144	−2.14710
$520$	0	0
$521$	32.3064	1.41537	0.707684	−	0.706529i	$-0.249740\pi$
$521$	32.3064	1.41537	0.707684	+	0.706529i	$0.249740\pi$
$522$	0	0
$523$	14.1858	0.620301	0.310151	−	0.950687i	$-0.399621\pi$
$523$	14.1858	0.620301	0.310151	+	0.950687i	$0.399621\pi$
$524$	0	0
$525$	40.5533	1.76989
$526$	0	0
$527$	−6.38493	−0.278132
$528$	0	0
$529$	−7.32602	−0.318523
$530$	0	0
$531$	−1.83041	−0.0794332
$532$	0	0
$533$	57.7716	2.50237
$534$	0	0
$535$	24.1341	1.04341
$536$	0	0
$537$	26.5238	1.14458
$538$	0	0
$539$	−22.0350	−0.949115
$540$	0	0
$541$	11.6803	0.502174	0.251087	−	0.967965i	$-0.419212\pi$
$541$	11.6803	0.502174	0.251087	+	0.967965i	$0.419212\pi$
$542$	0	0
$543$	−35.4802	−1.52260
$544$	0	0
$545$	22.2973	0.955110
$546$	0	0
$547$	26.2683	1.12315	0.561575	−	0.827426i	$-0.310196\pi$
$547$	26.2683	1.12315	0.561575	+	0.827426i	$0.310196\pi$
$548$	0	0
$549$	−14.2242	−0.607075
$550$	0	0
$551$	−5.15326	−0.219536
$552$	0	0
$553$	−69.0222	−2.93512
$554$	0	0
$555$	−1.92935	−0.0818962
$556$	0	0
$557$	−31.4214	−1.33137	−0.665683	−	0.746235i	$-0.731860\pi$
$557$	−31.4214	−1.33137	−0.665683	+	0.746235i	$0.731860\pi$
$558$	0	0
$559$	5.78974	0.244880
$560$	0	0
$561$	2.80476	0.118417
$562$	0	0
$563$	27.0863	1.14155	0.570777	−	0.821105i	$-0.306642\pi$
$563$	27.0863	1.14155	0.570777	+	0.821105i	$0.306642\pi$
$564$	0	0
$565$	13.4094	0.564138
$566$	0	0
$567$	−40.1429	−1.68584
$568$	0	0
$569$	−42.7959	−1.79410	−0.897048	−	0.441933i	$-0.854293\pi$
$569$	−42.7959	−1.79410	−0.897048	+	0.441933i	$0.854293\pi$
$570$	0	0
$571$	30.1437	1.26148	0.630738	−	0.775996i	$-0.282752\pi$
$571$	30.1437	1.26148	0.630738	+	0.775996i	$0.282752\pi$
$572$	0	0
$573$	−52.2904	−2.18446
$574$	0	0
$575$	−13.5357	−0.564478
$576$	0	0
$577$	43.0021	1.79020	0.895101	−	0.445863i	$-0.147103\pi$
$577$	43.0021	1.79020	0.895101	+	0.445863i	$0.147103\pi$
$578$	0	0
$579$	−30.6730	−1.27472
$580$	0	0
$581$	−16.7271	−0.693957
$582$	0	0
$583$	15.1864	0.628955
$584$	0	0
$585$	−54.6867	−2.26102
$586$	0	0
$587$	−20.2319	−0.835062	−0.417531	−	0.908663i	$-0.637104\pi$
$587$	−20.2319	−0.835062	−0.417531	+	0.908663i	$0.637104\pi$
$588$	0	0
$589$	−23.9138	−0.985350
$590$	0	0
$591$	−43.8280	−1.80284
$592$	0	0
$593$	−26.9710	−1.10757	−0.553783	−	0.832661i	$-0.686816\pi$
$593$	−26.9710	−1.10757	−0.553783	+	0.832661i	$0.686816\pi$
$594$	0	0
$595$	11.1601	0.457518
$596$	0	0
$597$	30.9634	1.26725
$598$	0	0
$599$	−34.8870	−1.42544	−0.712722	−	0.701447i	$-0.752538\pi$
$599$	−34.8870	−1.42544	−0.712722	+	0.701447i	$0.752538\pi$
$600$	0	0
$601$	−2.60284	−0.106172	−0.0530861	−	0.998590i	$-0.516906\pi$
$601$	−2.60284	−0.106172	−0.0530861	+	0.998590i	$0.516906\pi$
$602$	0	0
$603$	49.0748	1.99848
$604$	0	0
$605$	26.2396	1.06679
$606$	0	0
$607$	10.3915	0.421779	0.210889	−	0.977510i	$-0.432364\pi$
$607$	10.3915	0.421779	0.210889	+	0.977510i	$0.432364\pi$
$608$	0	0
$609$	20.2397	0.820155
$610$	0	0
$611$	35.1638	1.42257
$612$	0	0
$613$	−39.3138	−1.58787	−0.793935	−	0.608003i	$-0.791971\pi$
$613$	−39.3138	−1.58787	−0.793935	+	0.608003i	$0.791971\pi$
$614$	0	0
$615$	−70.4073	−2.83910
$616$	0	0
$617$	−1.60270	−0.0645224	−0.0322612	−	0.999479i	$-0.510271\pi$
$617$	−1.60270	−0.0645224	−0.0322612	+	0.999479i	$0.510271\pi$
$618$	0	0
$619$	16.6467	0.669086	0.334543	−	0.942380i	$-0.391418\pi$
$619$	16.6467	0.669086	0.334543	+	0.942380i	$0.391418\pi$
$620$	0	0
$621$	−1.80683	−0.0725057
$622$	0	0
$623$	80.3720	3.22004
$624$	0	0
$625$	−30.4056	−1.21622
$626$	0	0
$627$	10.5048	0.419522
$628$	0	0
$629$	−0.215618	−0.00859724
$630$	0	0
$631$	−18.3441	−0.730268	−0.365134	−	0.930955i	$-0.618977\pi$
$631$	−18.3441	−0.730268	−0.365134	+	0.930955i	$0.618977\pi$
$632$	0	0
$633$	46.7007	1.85619
$634$	0	0
$635$	−36.6481	−1.45434
$636$	0	0
$637$	93.2610	3.69514
$638$	0	0
$639$	28.0294	1.10882
$640$	0	0
$641$	−38.1419	−1.50651	−0.753257	−	0.657726i	$-0.771519\pi$
$641$	−38.1419	−1.50651	−0.753257	+	0.657726i	$0.771519\pi$
$642$	0	0
$643$	28.4317	1.12124	0.560619	−	0.828074i	$-0.310563\pi$
$643$	28.4317	1.12124	0.560619	+	0.828074i	$0.310563\pi$
$644$	0	0
$645$	−7.05606	−0.277832
$646$	0	0
$647$	−32.5533	−1.27980	−0.639901	−	0.768457i	$-0.721025\pi$
$647$	−32.5533	−1.27980	−0.639901	+	0.768457i	$0.721025\pi$
$648$	0	0
$649$	0.804261	0.0315700
$650$	0	0
$651$	93.9227	3.68112
$652$	0	0
$653$	−23.3809	−0.914965	−0.457482	−	0.889219i	$-0.651249\pi$
$653$	−23.3809	−0.914965	−0.457482	+	0.889219i	$0.651249\pi$
$654$	0	0
$655$	27.5028	1.07462
$656$	0	0
$657$	27.8426	1.08624
$658$	0	0
$659$	15.1928	0.591827	0.295914	−	0.955215i	$-0.404376\pi$
$659$	15.1928	0.591827	0.295914	+	0.955215i	$0.404376\pi$
$660$	0	0
$661$	−6.22635	−0.242177	−0.121089	−	0.992642i	$-0.538638\pi$
$661$	−6.22635	−0.242177	−0.121089	+	0.992642i	$0.538638\pi$
$662$	0	0
$663$	−11.8709	−0.461027
$664$	0	0
$665$	41.7983	1.62087
$666$	0	0
$667$	−6.75552	−0.261575
$668$	0	0
$669$	4.79760	0.185486
$670$	0	0
$671$	6.24995	0.241277
$672$	0	0
$673$	−7.39377	−0.285009	−0.142504	−	0.989794i	$-0.545516\pi$
$673$	−7.39377	−0.285009	−0.142504	+	0.989794i	$0.545516\pi$
$674$	0	0
$675$	1.56034	0.0600575
$676$	0	0
$677$	−25.3068	−0.972620	−0.486310	−	0.873786i	$-0.661657\pi$
$677$	−25.3068	−0.972620	−0.486310	+	0.873786i	$0.661657\pi$
$678$	0	0
$679$	9.92244	0.380788
$680$	0	0
$681$	16.2785	0.623795
$682$	0	0
$683$	16.3461	0.625467	0.312733	−	0.949841i	$-0.398755\pi$
$683$	16.3461	0.625467	0.312733	+	0.949841i	$0.398755\pi$
$684$	0	0
$685$	−8.31713	−0.317781
$686$	0	0
$687$	18.3919	0.701695
$688$	0	0
$689$	−64.2748	−2.44867
$690$	0	0
$691$	−23.8789	−0.908398	−0.454199	−	0.890900i	$-0.650074\pi$
$691$	−23.8789	−0.908398	−0.454199	+	0.890900i	$0.650074\pi$
$692$	0	0
$693$	−21.2414	−0.806894
$694$	0	0
$695$	−20.5354	−0.778951
$696$	0	0
$697$	−7.86849	−0.298040
$698$	0	0
$699$	−3.58001	−0.135408
$700$	0	0
$701$	−15.1203	−0.571086	−0.285543	−	0.958366i	$-0.592174\pi$
$701$	−15.1203	−0.571086	−0.285543	+	0.958366i	$0.592174\pi$
$702$	0	0
$703$	−0.807563	−0.0304578
$704$	0	0
$705$	−42.8547	−1.61400
$706$	0	0
$707$	1.39976	0.0526434
$708$	0	0
$709$	2.22944	0.0837284	0.0418642	−	0.999123i	$-0.486670\pi$
$709$	2.22944	0.0837284	0.0418642	+	0.999123i	$0.486670\pi$
$710$	0	0
$711$	−46.0660	−1.72761
$712$	0	0
$713$	−31.3491	−1.17403
$714$	0	0
$715$	24.0287	0.898621
$716$	0	0
$717$	−2.40101	−0.0896673
$718$	0	0
$719$	6.42954	0.239782	0.119891	−	0.992787i	$-0.461746\pi$
$719$	6.42954	0.239782	0.119891	+	0.992787i	$0.461746\pi$
$720$	0	0
$721$	−5.57214	−0.207517
$722$	0	0
$723$	−17.9176	−0.666361
$724$	0	0
$725$	5.83392	0.216666
$726$	0	0
$727$	6.94655	0.257633	0.128817	−	0.991668i	$-0.458882\pi$
$727$	6.94655	0.257633	0.128817	+	0.991668i	$0.458882\pi$
$728$	0	0
$729$	−30.1963	−1.11838
$730$	0	0
$731$	−0.788563	−0.0291660
$732$	0	0
$733$	−39.8352	−1.47135	−0.735674	−	0.677336i	$-0.763134\pi$
$733$	−39.8352	−1.47135	−0.735674	+	0.677336i	$0.763134\pi$
$734$	0	0
$735$	−113.659	−4.19237
$736$	0	0
$737$	−21.5629	−0.794278
$738$	0	0
$739$	−35.6853	−1.31270	−0.656352	−	0.754454i	$-0.727902\pi$
$739$	−35.6853	−1.31270	−0.656352	+	0.754454i	$0.727902\pi$
$740$	0	0
$741$	−44.4606	−1.63330
$742$	0	0
$743$	−49.5322	−1.81716	−0.908581	−	0.417710i	$-0.862833\pi$
$743$	−49.5322	−1.81716	−0.908581	+	0.417710i	$0.862833\pi$
$744$	0	0
$745$	−60.0708	−2.20082
$746$	0	0
$747$	−11.1638	−0.408463
$748$	0	0
$749$	−39.6753	−1.44970
$750$	0	0
$751$	2.73350	0.0997468	0.0498734	−	0.998756i	$-0.484118\pi$
$751$	2.73350	0.0997468	0.0498734	+	0.998756i	$0.484118\pi$
$752$	0	0
$753$	29.0185	1.05749
$754$	0	0
$755$	41.2811	1.50237
$756$	0	0
$757$	−20.3479	−0.739556	−0.369778	−	0.929120i	$-0.620566\pi$
$757$	−20.3479	−0.739556	−0.369778	+	0.929120i	$0.620566\pi$
$758$	0	0
$759$	13.7710	0.499855
$760$	0	0
$761$	−28.0875	−1.01817	−0.509086	−	0.860716i	$-0.670016\pi$
$761$	−28.0875	−1.01817	−0.509086	+	0.860716i	$0.670016\pi$
$762$	0	0
$763$	−36.6556	−1.32702
$764$	0	0
$765$	7.44832	0.269295
$766$	0	0
$767$	−3.40396	−0.122910
$768$	0	0
$769$	−45.2058	−1.63016	−0.815082	−	0.579346i	$-0.803308\pi$
$769$	−45.2058	−1.63016	−0.815082	+	0.579346i	$0.803308\pi$
$770$	0	0
$771$	37.2483	1.34147
$772$	0	0
$773$	−4.59755	−0.165362	−0.0826811	−	0.996576i	$-0.526348\pi$
$773$	−4.59755	−0.165362	−0.0826811	+	0.996576i	$0.526348\pi$
$774$	0	0
$775$	27.0724	0.972469
$776$	0	0
$777$	3.17175	0.113786
$778$	0	0
$779$	−29.4703	−1.05588
$780$	0	0
$781$	−12.3158	−0.440692
$782$	0	0
$783$	0.778748	0.0278302
$784$	0	0
$785$	−59.0894	−2.10899
$786$	0	0
$787$	−30.4493	−1.08540	−0.542700	−	0.839927i	$-0.682598\pi$
$787$	−30.4493	−1.08540	−0.542700	+	0.839927i	$0.682598\pi$
$788$	0	0
$789$	25.9756	0.924754
$790$	0	0
$791$	−22.0444	−0.783808
$792$	0	0
$793$	−26.4523	−0.939349
$794$	0	0
$795$	78.3328	2.77818
$796$	0	0
$797$	−7.05039	−0.249738	−0.124869	−	0.992173i	$-0.539851\pi$
$797$	−7.05039	−0.249738	−0.124869	+	0.992173i	$0.539851\pi$
$798$	0	0
$799$	−4.78930	−0.169433
$800$	0	0
$801$	53.6410	1.89531
$802$	0	0
$803$	−12.2337	−0.431717
$804$	0	0
$805$	54.7943	1.93124
$806$	0	0
$807$	−58.0245	−2.04256
$808$	0	0
$809$	29.7457	1.04580	0.522902	−	0.852393i	$-0.324849\pi$
$809$	29.7457	1.04580	0.522902	+	0.852393i	$0.324849\pi$
$810$	0	0
$811$	15.0108	0.527102	0.263551	−	0.964645i	$-0.415106\pi$
$811$	15.0108	0.527102	0.263551	+	0.964645i	$0.415106\pi$
$812$	0	0
$813$	−5.48162	−0.192249
$814$	0	0
$815$	−48.2373	−1.68968
$816$	0	0
$817$	−2.95344	−0.103328
$818$	0	0
$819$	89.9021	3.14143
$820$	0	0
$821$	48.4753	1.69180	0.845900	−	0.533341i	$-0.179064\pi$
$821$	48.4753	1.69180	0.845900	+	0.533341i	$0.179064\pi$
$822$	0	0
$823$	15.0152	0.523399	0.261699	−	0.965149i	$-0.415717\pi$
$823$	15.0152	0.523399	0.261699	+	0.965149i	$0.415717\pi$
$824$	0	0
$825$	−11.8923	−0.414037
$826$	0	0
$827$	−23.5517	−0.818974	−0.409487	−	0.912316i	$-0.634292\pi$
$827$	−23.5517	−0.818974	−0.409487	+	0.912316i	$0.634292\pi$
$828$	0	0
$829$	−14.2507	−0.494948	−0.247474	−	0.968895i	$-0.579600\pi$
$829$	−14.2507	−0.494948	−0.247474	+	0.968895i	$0.579600\pi$
$830$	0	0
$831$	−53.1744	−1.84460
$832$	0	0
$833$	−12.7021	−0.440103
$834$	0	0
$835$	−42.6428	−1.47572
$836$	0	0
$837$	3.61379	0.124911
$838$	0	0
$839$	−29.6199	−1.02259	−0.511296	−	0.859405i	$-0.670834\pi$
$839$	−29.6199	−1.02259	−0.511296	+	0.859405i	$0.670834\pi$
$840$	0	0
$841$	−26.0884	−0.899598
$842$	0	0
$843$	−62.1003	−2.13885
$844$	0	0
$845$	−63.9789	−2.20094
$846$	0	0
$847$	−43.1366	−1.48219
$848$	0	0
$849$	46.2782	1.58826
$850$	0	0
$851$	−1.05865	−0.0362901
$852$	0	0
$853$	23.2492	0.796038	0.398019	−	0.917377i	$-0.369698\pi$
$853$	23.2492	0.796038	0.398019	+	0.917377i	$0.369698\pi$
$854$	0	0
$855$	27.8966	0.954042
$856$	0	0
$857$	30.9537	1.05736	0.528680	−	0.848821i	$-0.322687\pi$
$857$	30.9537	1.05736	0.528680	+	0.848821i	$0.322687\pi$
$858$	0	0
$859$	29.8245	1.01760	0.508800	−	0.860885i	$-0.330089\pi$
$859$	29.8245	1.01760	0.508800	+	0.860885i	$0.330089\pi$
$860$	0	0
$861$	115.746	3.94461
$862$	0	0
$863$	30.5564	1.04015	0.520077	−	0.854120i	$-0.325903\pi$
$863$	30.5564	1.04015	0.520077	+	0.854120i	$0.325903\pi$
$864$	0	0
$865$	57.0752	1.94061
$866$	0	0
$867$	−40.6566	−1.38077
$868$	0	0
$869$	20.2408	0.686624
$870$	0	0
$871$	91.2627	3.09232
$872$	0	0
$873$	6.62233	0.224132
$874$	0	0
$875$	21.8824	0.739760
$876$	0	0
$877$	−43.4689	−1.46784	−0.733921	−	0.679235i	$-0.762312\pi$
$877$	−43.4689	−1.46784	−0.733921	+	0.679235i	$0.762312\pi$
$878$	0	0
$879$	52.0391	1.75523
$880$	0	0
$881$	−47.5342	−1.60147	−0.800733	−	0.599021i	$-0.795557\pi$
$881$	−47.5342	−1.60147	−0.800733	+	0.599021i	$0.795557\pi$
$882$	0	0
$883$	−27.2791	−0.918016	−0.459008	−	0.888432i	$-0.651795\pi$
$883$	−27.2791	−0.918016	−0.459008	+	0.888432i	$0.651795\pi$
$884$	0	0
$885$	4.14846	0.139449
$886$	0	0
$887$	−10.7462	−0.360824	−0.180412	−	0.983591i	$-0.557743\pi$
$887$	−10.7462	−0.360824	−0.180412	+	0.983591i	$0.557743\pi$
$888$	0	0
$889$	60.2477	2.02064
$890$	0	0
$891$	11.7719	0.394375
$892$	0	0
$893$	−17.9376	−0.600259
$894$	0	0
$895$	−30.9489	−1.03451
$896$	0	0
$897$	−58.2843	−1.94606
$898$	0	0
$899$	13.5115	0.450635
$900$	0	0
$901$	8.75422	0.291645
$902$	0	0
$903$	11.5998	0.386017
$904$	0	0
$905$	41.3996	1.37617
$906$	0	0
$907$	−41.1145	−1.36518	−0.682592	−	0.730800i	$-0.739147\pi$
$907$	−41.1145	−1.36518	−0.682592	+	0.730800i	$0.739147\pi$
$908$	0	0
$909$	0.934213	0.0309859
$910$	0	0
$911$	−5.62224	−0.186273	−0.0931366	−	0.995653i	$-0.529689\pi$
$911$	−5.62224	−0.186273	−0.0931366	+	0.995653i	$0.529689\pi$
$912$	0	0
$913$	4.90525	0.162340
$914$	0	0
$915$	32.2379	1.06575
$916$	0	0
$917$	−45.2131	−1.49307
$918$	0	0
$919$	−24.6883	−0.814393	−0.407197	−	0.913341i	$-0.633494\pi$
$919$	−24.6883	−0.814393	−0.407197	+	0.913341i	$0.633494\pi$
$920$	0	0
$921$	−41.2082	−1.35786
$922$	0	0
$923$	52.1252	1.71572
$924$	0	0
$925$	0.914228	0.0300596
$926$	0	0
$927$	−3.71890	−0.122145
$928$	0	0
$929$	25.0134	0.820663	0.410332	−	0.911936i	$-0.365413\pi$
$929$	25.0134	0.820663	0.410332	+	0.911936i	$0.365413\pi$
$930$	0	0
$931$	−47.5740	−1.55917
$932$	0	0
$933$	−32.9067	−1.07732
$934$	0	0
$935$	−3.27270	−0.107029
$936$	0	0
$937$	−10.0787	−0.329256	−0.164628	−	0.986356i	$-0.552642\pi$
$937$	−10.0787	−0.329256	−0.164628	+	0.986356i	$0.552642\pi$
$938$	0	0
$939$	19.1746	0.625740
$940$	0	0
$941$	28.1793	0.918618	0.459309	−	0.888277i	$-0.348097\pi$
$941$	28.1793	0.918618	0.459309	+	0.888277i	$0.348097\pi$
$942$	0	0
$943$	−38.6332	−1.25807
$944$	0	0
$945$	−6.31646	−0.205474
$946$	0	0
$947$	19.8688	0.645649	0.322824	−	0.946459i	$-0.395368\pi$
$947$	19.8688	0.645649	0.322824	+	0.946459i	$0.395368\pi$
$948$	0	0
$949$	51.7779	1.68078
$950$	0	0
$951$	−26.4527	−0.857787
$952$	0	0
$953$	−1.98984	−0.0644574	−0.0322287	−	0.999481i	$-0.510260\pi$
$953$	−1.98984	−0.0644574	−0.0322287	+	0.999481i	$0.510260\pi$
$954$	0	0
$955$	61.0144	1.97438
$956$	0	0
$957$	−5.93532	−0.191862
$958$	0	0
$959$	13.6729	0.441522
$960$	0	0
$961$	31.7005	1.02260
$962$	0	0
$963$	−26.4797	−0.853295
$964$	0	0
$965$	35.7904	1.15213
$966$	0	0
$967$	9.39714	0.302192	0.151096	−	0.988519i	$-0.451720\pi$
$967$	9.39714	0.302192	0.151096	+	0.988519i	$0.451720\pi$
$968$	0	0
$969$	6.05553	0.194532
$970$	0	0
$971$	24.8946	0.798904	0.399452	−	0.916754i	$-0.369200\pi$
$971$	24.8946	0.798904	0.399452	+	0.916754i	$0.369200\pi$
$972$	0	0
$973$	33.7591	1.08227
$974$	0	0
$975$	50.3330	1.61195
$976$	0	0
$977$	−9.10258	−0.291217	−0.145609	−	0.989342i	$-0.546514\pi$
$977$	−9.10258	−0.291217	−0.145609	+	0.989342i	$0.546514\pi$
$978$	0	0
$979$	−23.5692	−0.753275
$980$	0	0
$981$	−24.4642	−0.781083
$982$	0	0
$983$	−61.0516	−1.94724	−0.973621	−	0.228170i	$-0.926726\pi$
$983$	−61.0516	−1.94724	−0.973621	+	0.228170i	$0.926726\pi$
$984$	0	0
$985$	51.1402	1.62946
$986$	0	0
$987$	70.4509	2.24248
$988$	0	0
$989$	−3.87173	−0.123114
$990$	0	0
$991$	36.7089	1.16610	0.583048	−	0.812438i	$-0.301860\pi$
$991$	36.7089	1.16610	0.583048	+	0.812438i	$0.301860\pi$
$992$	0	0
$993$	−7.41902	−0.235436
$994$	0	0
$995$	−36.1292	−1.14537
$996$	0	0
$997$	−49.0109	−1.55219	−0.776095	−	0.630616i	$-0.782802\pi$
$997$	−49.0109	−1.55219	−0.776095	+	0.630616i	$0.782802\pi$
$998$	0	0
$999$	0.122037	0.00386108

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.t.1.17		21
4.3	odd	2		2012.2.a.a.1.5	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.a.1.5	✓	21	4.3	odd	2
8048.2.a.t.1.17		21	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)

\(f(q)\)	\(=\)	\(q+2.48667 q^{3} -2.90154 q^{5} +4.76998 q^{7} +3.18353 q^{9} +O(q^{10})\)	\(q+2.48667 q^{3} -2.90154 q^{5} +4.76998 q^{7} +3.18353 q^{9} -1.39880 q^{11} +5.92030 q^{13} -7.21518 q^{15} -0.806345 q^{17} -3.02004 q^{19} +11.8614 q^{21} -3.95904 q^{23} +3.41894 q^{25} +0.456381 q^{27} +1.70635 q^{29} +7.91836 q^{31} -3.47837 q^{33} -13.8403 q^{35} +0.267401 q^{37} +14.7218 q^{39} +9.75823 q^{41} +0.977947 q^{43} -9.23714 q^{45} +5.93952 q^{47} +15.7528 q^{49} -2.00511 q^{51} -10.8567 q^{53} +4.05869 q^{55} -7.50985 q^{57} -0.574963 q^{59} -4.46807 q^{61} +15.1854 q^{63} -17.1780 q^{65} +15.4152 q^{67} -9.84482 q^{69} +8.80449 q^{71} +8.74581 q^{73} +8.50177 q^{75} -6.67227 q^{77} -14.4701 q^{79} -8.41572 q^{81} -3.50674 q^{83} +2.33964 q^{85} +4.24314 q^{87} +16.8495 q^{89} +28.2397 q^{91} +19.6904 q^{93} +8.76278 q^{95} +2.08018 q^{97} -4.45314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

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