LMFDB - Embedded newform 8048.2.a.x.1.20

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:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+0.105343 q^{3} -1.39521 q^{5} +1.93153 q^{7} -2.98890 q^{9} +O(q^{10})$	$q+0.105343 q^{3} -1.39521 q^{5} +1.93153 q^{7} -2.98890 q^{9} +4.24653 q^{11} -3.77608 q^{13} -0.146976 q^{15} -3.39357 q^{17} +3.84418 q^{19} +0.203473 q^{21} -0.348338 q^{23} -3.05338 q^{25} -0.630890 q^{27} +4.79864 q^{29} +5.47956 q^{31} +0.447344 q^{33} -2.69489 q^{35} +7.77720 q^{37} -0.397785 q^{39} -7.43679 q^{41} -10.1664 q^{43} +4.17015 q^{45} -6.22680 q^{47} -3.26919 q^{49} -0.357490 q^{51} +5.11233 q^{53} -5.92481 q^{55} +0.404958 q^{57} +7.70880 q^{59} -3.79413 q^{61} -5.77315 q^{63} +5.26844 q^{65} -1.85379 q^{67} -0.0366951 q^{69} -4.40403 q^{71} +11.8458 q^{73} -0.321653 q^{75} +8.20231 q^{77} -2.36090 q^{79} +8.90025 q^{81} +1.20958 q^{83} +4.73475 q^{85} +0.505504 q^{87} -0.581037 q^{89} -7.29362 q^{91} +0.577234 q^{93} -5.36344 q^{95} -5.14557 q^{97} -12.6925 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * 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
$3$	0.105343	0.0608199	0.0304100	−	0.999538i	$-0.490319\pi$
$3$	0.105343	0.0608199	0.0304100	+	0.999538i	$0.490319\pi$
$4$	0	0
$5$	−1.39521	−0.623958	−0.311979	−	0.950089i	$-0.600992\pi$
$5$	−1.39521	−0.623958	−0.311979	+	0.950089i	$0.600992\pi$
$6$	0	0
$7$	1.93153	0.730049	0.365025	−	0.930998i	$-0.381061\pi$
$7$	1.93153	0.730049	0.365025	+	0.930998i	$0.381061\pi$
$8$	0	0
$9$	−2.98890	−0.996301
$10$	0	0
$11$	4.24653	1.28038	0.640189	−	0.768217i	$-0.278856\pi$
$11$	4.24653	1.28038	0.640189	+	0.768217i	$0.278856\pi$
$12$	0	0
$13$	−3.77608	−1.04730	−0.523649	−	0.851934i	$-0.675430\pi$
$13$	−3.77608	−1.04730	−0.523649	+	0.851934i	$0.675430\pi$
$14$	0	0
$15$	−0.146976	−0.0379491
$16$	0	0
$17$	−3.39357	−0.823062	−0.411531	−	0.911396i	$-0.635006\pi$
$17$	−3.39357	−0.823062	−0.411531	+	0.911396i	$0.635006\pi$
$18$	0	0
$19$	3.84418	0.881915	0.440958	−	0.897528i	$-0.354639\pi$
$19$	3.84418	0.881915	0.440958	+	0.897528i	$0.354639\pi$
$20$	0	0
$21$	0.203473	0.0444016
$22$	0	0
$23$	−0.348338	−0.0726336	−0.0363168	−	0.999340i	$-0.511563\pi$
$23$	−0.348338	−0.0726336	−0.0363168	+	0.999340i	$0.511563\pi$
$24$	0	0
$25$	−3.05338	−0.610677
$26$	0	0
$27$	−0.630890	−0.121415
$28$	0	0
$29$	4.79864	0.891085	0.445542	−	0.895261i	$-0.353011\pi$
$29$	4.79864	0.891085	0.445542	+	0.895261i	$0.353011\pi$
$30$	0	0
$31$	5.47956	0.984158	0.492079	−	0.870551i	$-0.336237\pi$
$31$	5.47956	0.984158	0.492079	+	0.870551i	$0.336237\pi$
$32$	0	0
$33$	0.447344	0.0778725
$34$	0	0
$35$	−2.69489	−0.455520
$36$	0	0
$37$	7.77720	1.27856	0.639282	−	0.768972i	$-0.279232\pi$
$37$	7.77720	1.27856	0.639282	+	0.768972i	$0.279232\pi$
$38$	0	0
$39$	−0.397785	−0.0636966
$40$	0	0
$41$	−7.43679	−1.16143	−0.580716	−	0.814106i	$-0.697227\pi$
$41$	−7.43679	−1.16143	−0.580716	+	0.814106i	$0.697227\pi$
$42$	0	0
$43$	−10.1664	−1.55036	−0.775180	−	0.631741i	$-0.782341\pi$
$43$	−10.1664	−1.55036	−0.775180	+	0.631741i	$0.782341\pi$
$44$	0	0
$45$	4.17015	0.621649
$46$	0	0
$47$	−6.22680	−0.908272	−0.454136	−	0.890932i	$-0.650052\pi$
$47$	−6.22680	−0.908272	−0.454136	+	0.890932i	$0.650052\pi$
$48$	0	0
$49$	−3.26919	−0.467028
$50$	0	0
$51$	−0.357490	−0.0500586
$52$	0	0
$53$	5.11233	0.702233	0.351116	−	0.936332i	$-0.385802\pi$
$53$	5.11233	0.702233	0.351116	+	0.936332i	$0.385802\pi$
$54$	0	0
$55$	−5.92481	−0.798902
$56$	0	0
$57$	0.404958	0.0536380
$58$	0	0
$59$	7.70880	1.00360	0.501800	−	0.864984i	$-0.332671\pi$
$59$	7.70880	1.00360	0.501800	+	0.864984i	$0.332671\pi$
$60$	0	0
$61$	−3.79413	−0.485788	−0.242894	−	0.970053i	$-0.578097\pi$
$61$	−3.79413	−0.485788	−0.242894	+	0.970053i	$0.578097\pi$
$62$	0	0
$63$	−5.77315	−0.727349
$64$	0	0
$65$	5.26844	0.653469
$66$	0	0
$67$	−1.85379	−0.226476	−0.113238	−	0.993568i	$-0.536122\pi$
$67$	−1.85379	−0.226476	−0.113238	+	0.993568i	$0.536122\pi$
$68$	0	0
$69$	−0.0366951	−0.00441757
$70$	0	0
$71$	−4.40403	−0.522663	−0.261331	−	0.965249i	$-0.584161\pi$
$71$	−4.40403	−0.522663	−0.261331	+	0.965249i	$0.584161\pi$
$72$	0	0
$73$	11.8458	1.38645	0.693224	−	0.720722i	$-0.256190\pi$
$73$	11.8458	1.38645	0.693224	+	0.720722i	$0.256190\pi$
$74$	0	0
$75$	−0.321653	−0.0371413
$76$	0	0
$77$	8.20231	0.934739
$78$	0	0
$79$	−2.36090	−0.265622	−0.132811	−	0.991141i	$-0.542400\pi$
$79$	−2.36090	−0.265622	−0.132811	+	0.991141i	$0.542400\pi$
$80$	0	0
$81$	8.90025	0.988916
$82$	0	0
$83$	1.20958	0.132769	0.0663845	−	0.997794i	$-0.478854\pi$
$83$	1.20958	0.132769	0.0663845	+	0.997794i	$0.478854\pi$
$84$	0	0
$85$	4.73475	0.513556
$86$	0	0
$87$	0.505504	0.0541957
$88$	0	0
$89$	−0.581037	−0.0615898	−0.0307949	−	0.999526i	$-0.509804\pi$
$89$	−0.581037	−0.0615898	−0.0307949	+	0.999526i	$0.509804\pi$
$90$	0	0
$91$	−7.29362	−0.764579
$92$	0	0
$93$	0.577234	0.0598564
$94$	0	0
$95$	−5.36344	−0.550278
$96$	0	0
$97$	−5.14557	−0.522454	−0.261227	−	0.965277i	$-0.584127\pi$
$97$	−5.14557	−0.522454	−0.261227	+	0.965277i	$0.584127\pi$
$98$	0	0
$99$	−12.6925	−1.27564
$100$	0	0
$101$	−15.0851	−1.50102	−0.750512	−	0.660857i	$-0.770193\pi$
$101$	−15.0851	−1.50102	−0.750512	+	0.660857i	$0.770193\pi$
$102$	0	0
$103$	7.81578	0.770112	0.385056	−	0.922893i	$-0.374182\pi$
$103$	7.81578	0.770112	0.385056	+	0.922893i	$0.374182\pi$
$104$	0	0
$105$	−0.283889	−0.0277047
$106$	0	0
$107$	−6.61709	−0.639698	−0.319849	−	0.947468i	$-0.603632\pi$
$107$	−6.61709	−0.639698	−0.319849	+	0.947468i	$0.603632\pi$
$108$	0	0
$109$	−13.8587	−1.32742	−0.663710	−	0.747990i	$-0.731019\pi$
$109$	−13.8587	−1.32742	−0.663710	+	0.747990i	$0.731019\pi$
$110$	0	0
$111$	0.819275	0.0777622
$112$	0	0
$113$	−2.80765	−0.264122	−0.132061	−	0.991242i	$-0.542159\pi$
$113$	−2.80765	−0.264122	−0.132061	+	0.991242i	$0.542159\pi$
$114$	0	0
$115$	0.486006	0.0453203
$116$	0	0
$117$	11.2863	1.04342
$118$	0	0
$119$	−6.55478	−0.600876
$120$	0	0
$121$	7.03306	0.639369
$122$	0	0
$123$	−0.783415	−0.0706382
$124$	0	0
$125$	11.2362	1.00499
$126$	0	0
$127$	−0.217830	−0.0193293	−0.00966463	−	0.999953i	$-0.503076\pi$
$127$	−0.217830	−0.0193293	−0.00966463	+	0.999953i	$0.503076\pi$
$128$	0	0
$129$	−1.07096	−0.0942927
$130$	0	0
$131$	17.1971	1.50252	0.751260	−	0.660007i	$-0.229447\pi$
$131$	17.1971	1.50252	0.751260	+	0.660007i	$0.229447\pi$
$132$	0	0
$133$	7.42515	0.643842
$134$	0	0
$135$	0.880225	0.0757577
$136$	0	0
$137$	9.20685	0.786594	0.393297	−	0.919411i	$-0.371334\pi$
$137$	9.20685	0.786594	0.393297	+	0.919411i	$0.371334\pi$
$138$	0	0
$139$	−7.40364	−0.627968	−0.313984	−	0.949428i	$-0.601664\pi$
$139$	−7.40364	−0.627968	−0.313984	+	0.949428i	$0.601664\pi$
$140$	0	0
$141$	−0.655951	−0.0552411
$142$	0	0
$143$	−16.0353	−1.34094
$144$	0	0
$145$	−6.69512	−0.555999
$146$	0	0
$147$	−0.344387	−0.0284046
$148$	0	0
$149$	0.331497	0.0271573	0.0135786	−	0.999908i	$-0.495678\pi$
$149$	0.331497	0.0271573	0.0135786	+	0.999908i	$0.495678\pi$
$150$	0	0
$151$	−8.17266	−0.665081	−0.332541	−	0.943089i	$-0.607906\pi$
$151$	−8.17266	−0.665081	−0.332541	+	0.943089i	$0.607906\pi$
$152$	0	0
$153$	10.1431	0.820017
$154$	0	0
$155$	−7.64514	−0.614073
$156$	0	0
$157$	−18.1777	−1.45073	−0.725367	−	0.688362i	$-0.758330\pi$
$157$	−18.1777	−1.45073	−0.725367	+	0.688362i	$0.758330\pi$
$158$	0	0
$159$	0.538549	0.0427097
$160$	0	0
$161$	−0.672826	−0.0530261
$162$	0	0
$163$	−11.5739	−0.906541	−0.453270	−	0.891373i	$-0.649743\pi$
$163$	−11.5739	−0.906541	−0.453270	+	0.891373i	$0.649743\pi$
$164$	0	0
$165$	−0.624139	−0.0485891
$166$	0	0
$167$	−15.9588	−1.23493	−0.617466	−	0.786598i	$-0.711841\pi$
$167$	−15.9588	−1.23493	−0.617466	+	0.786598i	$0.711841\pi$
$168$	0	0
$169$	1.25881	0.0968318
$170$	0	0
$171$	−11.4899	−0.878653
$172$	0	0
$173$	−16.8521	−1.28124	−0.640619	−	0.767859i	$-0.721322\pi$
$173$	−16.8521	−1.28124	−0.640619	+	0.767859i	$0.721322\pi$
$174$	0	0
$175$	−5.89770	−0.445824
$176$	0	0
$177$	0.812070	0.0610389
$178$	0	0
$179$	9.83612	0.735186	0.367593	−	0.929987i	$-0.380182\pi$
$179$	9.83612	0.735186	0.367593	+	0.929987i	$0.380182\pi$
$180$	0	0
$181$	−17.9963	−1.33766	−0.668828	−	0.743417i	$-0.733204\pi$
$181$	−17.9963	−1.33766	−0.668828	+	0.743417i	$0.733204\pi$
$182$	0	0
$183$	−0.399686	−0.0295456
$184$	0	0
$185$	−10.8508	−0.797770
$186$	0	0
$187$	−14.4109	−1.05383
$188$	0	0
$189$	−1.21858	−0.0886389
$190$	0	0
$191$	−2.10213	−0.152105	−0.0760523	−	0.997104i	$-0.524232\pi$
$191$	−2.10213	−0.152105	−0.0760523	+	0.997104i	$0.524232\pi$
$192$	0	0
$193$	2.40192	0.172894	0.0864470	−	0.996256i	$-0.472449\pi$
$193$	2.40192	0.172894	0.0864470	+	0.996256i	$0.472449\pi$
$194$	0	0
$195$	0.554994	0.0397439
$196$	0	0
$197$	−27.0876	−1.92991	−0.964957	−	0.262409i	$-0.915483\pi$
$197$	−27.0876	−1.92991	−0.964957	+	0.262409i	$0.915483\pi$
$198$	0	0
$199$	−7.84680	−0.556245	−0.278123	−	0.960546i	$-0.589712\pi$
$199$	−7.84680	−0.556245	−0.278123	+	0.960546i	$0.589712\pi$
$200$	0	0
$201$	−0.195284	−0.0137743
$202$	0	0
$203$	9.26871	0.650536
$204$	0	0
$205$	10.3759	0.724684
$206$	0	0
$207$	1.04115	0.0723649
$208$	0	0
$209$	16.3244	1.12919
$210$	0	0
$211$	−5.22198	−0.359496	−0.179748	−	0.983713i	$-0.557528\pi$
$211$	−5.22198	−0.359496	−0.179748	+	0.983713i	$0.557528\pi$
$212$	0	0
$213$	−0.463935	−0.0317883
$214$	0	0
$215$	14.1843	0.967358
$216$	0	0
$217$	10.5839	0.718484
$218$	0	0
$219$	1.24788	0.0843237
$220$	0	0
$221$	12.8144	0.861990
$222$	0	0
$223$	27.7492	1.85823	0.929113	−	0.369796i	$-0.120572\pi$
$223$	27.7492	1.85823	0.929113	+	0.369796i	$0.120572\pi$
$224$	0	0
$225$	9.12627	0.608418
$226$	0	0
$227$	19.2160	1.27541	0.637705	−	0.770280i	$-0.279884\pi$
$227$	19.2160	1.27541	0.637705	+	0.770280i	$0.279884\pi$
$228$	0	0
$229$	−4.97424	−0.328707	−0.164353	−	0.986402i	$-0.552554\pi$
$229$	−4.97424	−0.328707	−0.164353	+	0.986402i	$0.552554\pi$
$230$	0	0
$231$	0.864057	0.0568508
$232$	0	0
$233$	22.8946	1.49987	0.749937	−	0.661509i	$-0.230084\pi$
$233$	22.8946	1.49987	0.749937	+	0.661509i	$0.230084\pi$
$234$	0	0
$235$	8.68770	0.566723
$236$	0	0
$237$	−0.248705	−0.0161551
$238$	0	0
$239$	−4.25733	−0.275384	−0.137692	−	0.990475i	$-0.543968\pi$
$239$	−4.25733	−0.275384	−0.137692	+	0.990475i	$0.543968\pi$
$240$	0	0
$241$	19.9772	1.28684	0.643421	−	0.765512i	$-0.277514\pi$
$241$	19.9772	1.28684	0.643421	+	0.765512i	$0.277514\pi$
$242$	0	0
$243$	2.83025	0.181561
$244$	0	0
$245$	4.56122	0.291406
$246$	0	0
$247$	−14.5159	−0.923628
$248$	0	0
$249$	0.127421	0.00807500
$250$	0	0
$251$	−25.0312	−1.57996	−0.789979	−	0.613134i	$-0.789909\pi$
$251$	−25.0312	−1.57996	−0.789979	+	0.613134i	$0.789909\pi$
$252$	0	0
$253$	−1.47923	−0.0929985
$254$	0	0
$255$	0.498774	0.0312344
$256$	0	0
$257$	−3.16575	−0.197474	−0.0987372	−	0.995114i	$-0.531480\pi$
$257$	−3.16575	−0.197474	−0.0987372	+	0.995114i	$0.531480\pi$
$258$	0	0
$259$	15.0219	0.933415
$260$	0	0
$261$	−14.3427	−0.887789
$262$	0	0
$263$	−27.1038	−1.67129	−0.835646	−	0.549269i	$-0.814906\pi$
$263$	−27.1038	−1.67129	−0.835646	+	0.549269i	$0.814906\pi$
$264$	0	0
$265$	−7.13278	−0.438163
$266$	0	0
$267$	−0.0612083	−0.00374589
$268$	0	0
$269$	23.0596	1.40597	0.702984	−	0.711206i	$-0.251851\pi$
$269$	23.0596	1.40597	0.702984	+	0.711206i	$0.251851\pi$
$270$	0	0
$271$	12.7769	0.776144	0.388072	−	0.921629i	$-0.373141\pi$
$271$	12.7769	0.776144	0.388072	+	0.921629i	$0.373141\pi$
$272$	0	0
$273$	−0.768333	−0.0465016
$274$	0	0
$275$	−12.9663	−0.781898
$276$	0	0
$277$	−30.0530	−1.80571	−0.902854	−	0.429947i	$-0.858532\pi$
$277$	−30.0530	−1.80571	−0.902854	+	0.429947i	$0.858532\pi$
$278$	0	0
$279$	−16.3779	−0.980518
$280$	0	0
$281$	11.3618	0.677786	0.338893	−	0.940825i	$-0.389948\pi$
$281$	11.3618	0.677786	0.338893	+	0.940825i	$0.389948\pi$
$282$	0	0
$283$	25.7708	1.53192	0.765958	−	0.642891i	$-0.222265\pi$
$283$	25.7708	1.53192	0.765958	+	0.642891i	$0.222265\pi$
$284$	0	0
$285$	−0.565002	−0.0334679
$286$	0	0
$287$	−14.3644	−0.847903
$288$	0	0
$289$	−5.48368	−0.322569
$290$	0	0
$291$	−0.542051	−0.0317756
$292$	0	0
$293$	2.95127	0.172415	0.0862076	−	0.996277i	$-0.472525\pi$
$293$	2.95127	0.172415	0.0862076	+	0.996277i	$0.472525\pi$
$294$	0	0
$295$	−10.7554	−0.626204
$296$	0	0
$297$	−2.67910	−0.155457
$298$	0	0
$299$	1.31536	0.0760690
$300$	0	0
$301$	−19.6367	−1.13184
$302$	0	0
$303$	−1.58911	−0.0912921
$304$	0	0
$305$	5.29361	0.303111
$306$	0	0
$307$	−31.3727	−1.79053	−0.895266	−	0.445532i	$-0.853015\pi$
$307$	−31.3727	−1.79053	−0.895266	+	0.445532i	$0.853015\pi$
$308$	0	0
$309$	0.823340	0.0468382
$310$	0	0
$311$	6.60266	0.374402	0.187201	−	0.982322i	$-0.440058\pi$
$311$	6.60266	0.374402	0.187201	+	0.982322i	$0.440058\pi$
$312$	0	0
$313$	−18.3434	−1.03683	−0.518415	−	0.855129i	$-0.673478\pi$
$313$	−18.3434	−1.03683	−0.518415	+	0.855129i	$0.673478\pi$
$314$	0	0
$315$	8.05477	0.453835
$316$	0	0
$317$	−7.13627	−0.400813	−0.200407	−	0.979713i	$-0.564226\pi$
$317$	−7.13627	−0.400813	−0.200407	+	0.979713i	$0.564226\pi$
$318$	0	0
$319$	20.3776	1.14093
$320$	0	0
$321$	−0.697065	−0.0389064
$322$	0	0
$323$	−13.0455	−0.725871
$324$	0	0
$325$	11.5298	0.639560
$326$	0	0
$327$	−1.45992	−0.0807336
$328$	0	0
$329$	−12.0272	−0.663084
$330$	0	0
$331$	−0.207540	−0.0114074	−0.00570372	−	0.999984i	$-0.501816\pi$
$331$	−0.207540	−0.0114074	−0.00570372	+	0.999984i	$0.501816\pi$
$332$	0	0
$333$	−23.2453	−1.27383
$334$	0	0
$335$	2.58642	0.141311
$336$	0	0
$337$	2.33324	0.127100	0.0635498	−	0.997979i	$-0.479758\pi$
$337$	2.33324	0.127100	0.0635498	+	0.997979i	$0.479758\pi$
$338$	0	0
$339$	−0.295767	−0.0160639
$340$	0	0
$341$	23.2691	1.26009
$342$	0	0
$343$	−19.8352	−1.07100
$344$	0	0
$345$	0.0511974	0.00275638
$346$	0	0
$347$	−24.0588	−1.29154	−0.645771	−	0.763531i	$-0.723464\pi$
$347$	−24.0588	−1.29154	−0.645771	+	0.763531i	$0.723464\pi$
$348$	0	0
$349$	21.1154	1.13028	0.565142	−	0.824994i	$-0.308821\pi$
$349$	21.1154	1.13028	0.565142	+	0.824994i	$0.308821\pi$
$350$	0	0
$351$	2.38229	0.127157
$352$	0	0
$353$	−22.6654	−1.20636	−0.603180	−	0.797605i	$-0.706100\pi$
$353$	−22.6654	−1.20636	−0.603180	+	0.797605i	$0.706100\pi$
$354$	0	0
$355$	6.14456	0.326119
$356$	0	0
$357$	−0.690502	−0.0365452
$358$	0	0
$359$	6.57959	0.347257	0.173629	−	0.984811i	$-0.444451\pi$
$359$	6.57959	0.347257	0.173629	+	0.984811i	$0.444451\pi$
$360$	0	0
$361$	−4.22228	−0.222225
$362$	0	0
$363$	0.740885	0.0388864
$364$	0	0
$365$	−16.5274	−0.865085
$366$	0	0
$367$	−22.4510	−1.17193	−0.585966	−	0.810335i	$-0.699285\pi$
$367$	−22.4510	−1.17193	−0.585966	+	0.810335i	$0.699285\pi$
$368$	0	0
$369$	22.2278	1.15714
$370$	0	0
$371$	9.87462	0.512665
$372$	0	0
$373$	−9.44340	−0.488961	−0.244480	−	0.969654i	$-0.578617\pi$
$373$	−9.44340	−0.488961	−0.244480	+	0.969654i	$0.578617\pi$
$374$	0	0
$375$	1.18365	0.0611237
$376$	0	0
$377$	−18.1201	−0.933231
$378$	0	0
$379$	−22.3470	−1.14789	−0.573944	−	0.818894i	$-0.694587\pi$
$379$	−22.3470	−1.14789	−0.573944	+	0.818894i	$0.694587\pi$
$380$	0	0
$381$	−0.0229469	−0.00117560
$382$	0	0
$383$	−20.3045	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$383$	−20.3045	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$384$	0	0
$385$	−11.4440	−0.583238
$386$	0	0
$387$	30.3863	1.54462
$388$	0	0
$389$	20.2559	1.02701	0.513506	−	0.858086i	$-0.328346\pi$
$389$	20.2559	1.02701	0.513506	+	0.858086i	$0.328346\pi$
$390$	0	0
$391$	1.18211	0.0597819
$392$	0	0
$393$	1.81160	0.0913831
$394$	0	0
$395$	3.29396	0.165737
$396$	0	0
$397$	−17.1209	−0.859273	−0.429637	−	0.903002i	$-0.641358\pi$
$397$	−17.1209	−0.859273	−0.429637	+	0.903002i	$0.641358\pi$
$398$	0	0
$399$	0.782189	0.0391584
$400$	0	0
$401$	18.4253	0.920114	0.460057	−	0.887889i	$-0.347829\pi$
$401$	18.4253	0.920114	0.460057	+	0.887889i	$0.347829\pi$
$402$	0	0
$403$	−20.6913	−1.03071
$404$	0	0
$405$	−12.4177	−0.617042
$406$	0	0
$407$	33.0262	1.63705
$408$	0	0
$409$	−13.1804	−0.651730	−0.325865	−	0.945416i	$-0.605655\pi$
$409$	−13.1804	−0.651730	−0.325865	+	0.945416i	$0.605655\pi$
$410$	0	0
$411$	0.969879	0.0478406
$412$	0	0
$413$	14.8898	0.732678
$414$	0	0
$415$	−1.68762	−0.0828422
$416$	0	0
$417$	−0.779923	−0.0381930
$418$	0	0
$419$	−18.6959	−0.913353	−0.456676	−	0.889633i	$-0.650960\pi$
$419$	−18.6959	−0.913353	−0.456676	+	0.889633i	$0.650960\pi$
$420$	0	0
$421$	2.14768	0.104672	0.0523358	−	0.998630i	$-0.483333\pi$
$421$	2.14768	0.104672	0.0523358	+	0.998630i	$0.483333\pi$
$422$	0	0
$423$	18.6113	0.904913
$424$	0	0
$425$	10.3619	0.502625
$426$	0	0
$427$	−7.32847	−0.354650
$428$	0	0
$429$	−1.68921	−0.0815557
$430$	0	0
$431$	2.58413	0.124473	0.0622367	−	0.998061i	$-0.480177\pi$
$431$	2.58413	0.124473	0.0622367	+	0.998061i	$0.480177\pi$
$432$	0	0
$433$	−16.7818	−0.806480	−0.403240	−	0.915094i	$-0.632116\pi$
$433$	−16.7818	−0.806480	−0.403240	+	0.915094i	$0.632116\pi$
$434$	0	0
$435$	−0.705285	−0.0338158
$436$	0	0
$437$	−1.33908	−0.0640567
$438$	0	0
$439$	4.90192	0.233956	0.116978	−	0.993135i	$-0.462679\pi$
$439$	4.90192	0.233956	0.116978	+	0.993135i	$0.462679\pi$
$440$	0	0
$441$	9.77131	0.465300
$442$	0	0
$443$	−40.4351	−1.92113	−0.960566	−	0.278052i	$-0.910311\pi$
$443$	−40.4351	−1.92113	−0.960566	+	0.278052i	$0.910311\pi$
$444$	0	0
$445$	0.810669	0.0384294
$446$	0	0
$447$	0.0349209	0.00165170
$448$	0	0
$449$	−21.6894	−1.02358	−0.511792	−	0.859109i	$-0.671018\pi$
$449$	−21.6894	−1.02358	−0.511792	+	0.859109i	$0.671018\pi$
$450$	0	0
$451$	−31.5806	−1.48707
$452$	0	0
$453$	−0.860934	−0.0404502
$454$	0	0
$455$	10.1761	0.477065
$456$	0	0
$457$	32.7455	1.53177	0.765884	−	0.642979i	$-0.222302\pi$
$457$	32.7455	1.53177	0.765884	+	0.642979i	$0.222302\pi$
$458$	0	0
$459$	2.14097	0.0999319
$460$	0	0
$461$	10.2732	0.478473	0.239236	−	0.970961i	$-0.423103\pi$
$461$	10.2732	0.478473	0.239236	+	0.970961i	$0.423103\pi$
$462$	0	0
$463$	10.6626	0.495531	0.247765	−	0.968820i	$-0.420304\pi$
$463$	10.6626	0.495531	0.247765	+	0.968820i	$0.420304\pi$
$464$	0	0
$465$	−0.805364	−0.0373479
$466$	0	0
$467$	−2.78490	−0.128870	−0.0644349	−	0.997922i	$-0.520524\pi$
$467$	−2.78490	−0.128870	−0.0644349	+	0.997922i	$0.520524\pi$
$468$	0	0
$469$	−3.58064	−0.165339
$470$	0	0
$471$	−1.91489	−0.0882336
$472$	0	0
$473$	−43.1719	−1.98505
$474$	0	0
$475$	−11.7378	−0.538565
$476$	0	0
$477$	−15.2803	−0.699635
$478$	0	0
$479$	0.333967	0.0152593	0.00762967	−	0.999971i	$-0.497571\pi$
$479$	0.333967	0.0152593	0.00762967	+	0.999971i	$0.497571\pi$
$480$	0	0
$481$	−29.3674	−1.33904
$482$	0	0
$483$	−0.0708776	−0.00322504
$484$	0	0
$485$	7.17916	0.325989
$486$	0	0
$487$	6.23216	0.282406	0.141203	−	0.989981i	$-0.454903\pi$
$487$	6.23216	0.282406	0.141203	+	0.989981i	$0.454903\pi$
$488$	0	0
$489$	−1.21924	−0.0551357
$490$	0	0
$491$	−24.6952	−1.11448	−0.557240	−	0.830352i	$-0.688140\pi$
$491$	−24.6952	−1.11448	−0.557240	+	0.830352i	$0.688140\pi$
$492$	0	0
$493$	−16.2845	−0.733418
$494$	0	0
$495$	17.7087	0.795947
$496$	0	0
$497$	−8.50652	−0.381569
$498$	0	0
$499$	3.46385	0.155063	0.0775316	−	0.996990i	$-0.475296\pi$
$499$	3.46385	0.155063	0.0775316	+	0.996990i	$0.475296\pi$
$500$	0	0
$501$	−1.68115	−0.0751084
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	21.0469	0.936575
$506$	0	0
$507$	0.132607	0.00588931
$508$	0	0
$509$	−30.9293	−1.37092	−0.685459	−	0.728112i	$-0.740398\pi$
$509$	−30.9293	−1.37092	−0.685459	+	0.728112i	$0.740398\pi$
$510$	0	0
$511$	22.8805	1.01218
$512$	0	0
$513$	−2.42526	−0.107078
$514$	0	0
$515$	−10.9047	−0.480517
$516$	0	0
$517$	−26.4423	−1.16293
$518$	0	0
$519$	−1.77525	−0.0779248
$520$	0	0
$521$	31.0796	1.36162	0.680812	−	0.732459i	$-0.261627\pi$
$521$	31.0796	1.36162	0.680812	+	0.732459i	$0.261627\pi$
$522$	0	0
$523$	23.0224	1.00670	0.503350	−	0.864082i	$-0.332101\pi$
$523$	23.0224	1.00670	0.503350	+	0.864082i	$0.332101\pi$
$524$	0	0
$525$	−0.621283	−0.0271150
$526$	0	0
$527$	−18.5953	−0.810023
$528$	0	0
$529$	−22.8787	−0.994724
$530$	0	0
$531$	−23.0409	−0.999888
$532$	0	0
$533$	28.0820	1.21636
$534$	0	0
$535$	9.23224	0.399145
$536$	0	0
$537$	1.03617	0.0447140
$538$	0	0
$539$	−13.8827	−0.597972
$540$	0	0
$541$	−10.9406	−0.470373	−0.235187	−	0.971950i	$-0.575570\pi$
$541$	−10.9406	−0.470373	−0.235187	+	0.971950i	$0.575570\pi$
$542$	0	0
$543$	−1.89579	−0.0813561
$544$	0	0
$545$	19.3358	0.828254
$546$	0	0
$547$	−1.43113	−0.0611905	−0.0305953	−	0.999532i	$-0.509740\pi$
$547$	−1.43113	−0.0611905	−0.0305953	+	0.999532i	$0.509740\pi$
$548$	0	0
$549$	11.3403	0.483992
$550$	0	0
$551$	18.4468	0.785862
$552$	0	0
$553$	−4.56015	−0.193917
$554$	0	0
$555$	−1.14306	−0.0485203
$556$	0	0
$557$	−21.2625	−0.900921	−0.450460	−	0.892796i	$-0.648740\pi$
$557$	−21.2625	−0.900921	−0.450460	+	0.892796i	$0.648740\pi$
$558$	0	0
$559$	38.3891	1.62369
$560$	0	0
$561$	−1.51809	−0.0640939
$562$	0	0
$563$	25.2190	1.06285	0.531427	−	0.847104i	$-0.321656\pi$
$563$	25.2190	1.06285	0.531427	+	0.847104i	$0.321656\pi$
$564$	0	0
$565$	3.91727	0.164801
$566$	0	0
$567$	17.1911	0.721958
$568$	0	0
$569$	9.28589	0.389285	0.194642	−	0.980874i	$-0.437645\pi$
$569$	9.28589	0.389285	0.194642	+	0.980874i	$0.437645\pi$
$570$	0	0
$571$	17.4534	0.730400	0.365200	−	0.930929i	$-0.381001\pi$
$571$	17.4534	0.730400	0.365200	+	0.930929i	$0.381001\pi$
$572$	0	0
$573$	−0.221445	−0.00925098
$574$	0	0
$575$	1.06361	0.0443557
$576$	0	0
$577$	32.5989	1.35711	0.678554	−	0.734550i	$-0.262607\pi$
$577$	32.5989	1.35711	0.678554	+	0.734550i	$0.262607\pi$
$578$	0	0
$579$	0.253026	0.0105154
$580$	0	0
$581$	2.33635	0.0969279
$582$	0	0
$583$	21.7097	0.899124
$584$	0	0
$585$	−15.7468	−0.651052
$586$	0	0
$587$	−12.1843	−0.502901	−0.251451	−	0.967870i	$-0.580908\pi$
$587$	−12.1843	−0.502901	−0.251451	+	0.967870i	$0.580908\pi$
$588$	0	0
$589$	21.0644	0.867944
$590$	0	0
$591$	−2.85350	−0.117377
$592$	0	0
$593$	26.0989	1.07175	0.535877	−	0.844296i	$-0.319981\pi$
$593$	26.0989	1.07175	0.535877	+	0.844296i	$0.319981\pi$
$594$	0	0
$595$	9.14531	0.374921
$596$	0	0
$597$	−0.826607	−0.0338308
$598$	0	0
$599$	−6.97364	−0.284935	−0.142468	−	0.989799i	$-0.545504\pi$
$599$	−6.97364	−0.284935	−0.142468	+	0.989799i	$0.545504\pi$
$600$	0	0
$601$	9.81815	0.400491	0.200245	−	0.979746i	$-0.435826\pi$
$601$	9.81815	0.400491	0.200245	+	0.979746i	$0.435826\pi$
$602$	0	0
$603$	5.54079	0.225638
$604$	0	0
$605$	−9.81260	−0.398939
$606$	0	0
$607$	−16.7486	−0.679804	−0.339902	−	0.940461i	$-0.610394\pi$
$607$	−16.7486	−0.679804	−0.339902	+	0.940461i	$0.610394\pi$
$608$	0	0
$609$	0.976396	0.0395656
$610$	0	0
$611$	23.5129	0.951231
$612$	0	0
$613$	−39.2177	−1.58399	−0.791993	−	0.610530i	$-0.790956\pi$
$613$	−39.2177	−1.58399	−0.791993	+	0.610530i	$0.790956\pi$
$614$	0	0
$615$	1.09303	0.0440752
$616$	0	0
$617$	0.477601	0.0192275	0.00961376	−	0.999954i	$-0.496940\pi$
$617$	0.477601	0.0192275	0.00961376	+	0.999954i	$0.496940\pi$
$618$	0	0
$619$	−24.2851	−0.976099	−0.488049	−	0.872816i	$-0.662291\pi$
$619$	−24.2851	−0.976099	−0.488049	+	0.872816i	$0.662291\pi$
$620$	0	0
$621$	0.219763	0.00881880
$622$	0	0
$623$	−1.12229	−0.0449636
$624$	0	0
$625$	−0.409916	−0.0163966
$626$	0	0
$627$	1.71967	0.0686770
$628$	0	0
$629$	−26.3925	−1.05234
$630$	0	0
$631$	−48.3314	−1.92404	−0.962021	−	0.272975i	$-0.911992\pi$
$631$	−48.3314	−1.92404	−0.962021	+	0.272975i	$0.911992\pi$
$632$	0	0
$633$	−0.550101	−0.0218645
$634$	0	0
$635$	0.303919	0.0120606
$636$	0	0
$637$	12.3448	0.489117
$638$	0	0
$639$	13.1632	0.520729
$640$	0	0
$641$	1.12605	0.0444765	0.0222382	−	0.999753i	$-0.492921\pi$
$641$	1.12605	0.0444765	0.0222382	+	0.999753i	$0.492921\pi$
$642$	0	0
$643$	35.4046	1.39622	0.698111	−	0.715990i	$-0.254024\pi$
$643$	35.4046	1.39622	0.698111	+	0.715990i	$0.254024\pi$
$644$	0	0
$645$	1.49422	0.0588347
$646$	0	0
$647$	−8.20554	−0.322593	−0.161297	−	0.986906i	$-0.551568\pi$
$647$	−8.20554	−0.322593	−0.161297	+	0.986906i	$0.551568\pi$
$648$	0	0
$649$	32.7357	1.28499
$650$	0	0
$651$	1.11495	0.0436981
$652$	0	0
$653$	−24.6156	−0.963285	−0.481642	−	0.876368i	$-0.659960\pi$
$653$	−24.6156	−0.963285	−0.481642	+	0.876368i	$0.659960\pi$
$654$	0	0
$655$	−23.9936	−0.937508
$656$	0	0
$657$	−35.4060	−1.38132
$658$	0	0
$659$	5.93739	0.231288	0.115644	−	0.993291i	$-0.463107\pi$
$659$	5.93739	0.231288	0.115644	+	0.993291i	$0.463107\pi$
$660$	0	0
$661$	−17.1573	−0.667343	−0.333671	−	0.942689i	$-0.608288\pi$
$661$	−17.1573	−0.667343	−0.333671	+	0.942689i	$0.608288\pi$
$662$	0	0
$663$	1.34991	0.0524262
$664$	0	0
$665$	−10.3596	−0.401730
$666$	0	0
$667$	−1.67155	−0.0647227
$668$	0	0
$669$	2.92319	0.113017
$670$	0	0
$671$	−16.1119	−0.621993
$672$	0	0
$673$	−40.7106	−1.56928	−0.784640	−	0.619952i	$-0.787152\pi$
$673$	−40.7106	−1.56928	−0.784640	+	0.619952i	$0.787152\pi$
$674$	0	0
$675$	1.92635	0.0741453
$676$	0	0
$677$	−36.7166	−1.41113	−0.705567	−	0.708643i	$-0.749308\pi$
$677$	−36.7166	−1.41113	−0.705567	+	0.708643i	$0.749308\pi$
$678$	0	0
$679$	−9.93882	−0.381417
$680$	0	0
$681$	2.02428	0.0775704
$682$	0	0
$683$	45.2956	1.73319	0.866594	−	0.499014i	$-0.166304\pi$
$683$	45.2956	1.73319	0.866594	+	0.499014i	$0.166304\pi$
$684$	0	0
$685$	−12.8455	−0.490801
$686$	0	0
$687$	−0.524002	−0.0199919
$688$	0	0
$689$	−19.3046	−0.735447
$690$	0	0
$691$	−9.35254	−0.355787	−0.177894	−	0.984050i	$-0.556928\pi$
$691$	−9.35254	−0.355787	−0.177894	+	0.984050i	$0.556928\pi$
$692$	0	0
$693$	−24.5159	−0.931282
$694$	0	0
$695$	10.3296	0.391826
$696$	0	0
$697$	25.2373	0.955930
$698$	0	0
$699$	2.41179	0.0912222
$700$	0	0
$701$	29.7290	1.12285	0.561425	−	0.827528i	$-0.310253\pi$
$701$	29.7290	1.12285	0.561425	+	0.827528i	$0.310253\pi$
$702$	0	0
$703$	29.8970	1.12759
$704$	0	0
$705$	0.915191	0.0344681
$706$	0	0
$707$	−29.1373	−1.09582
$708$	0	0
$709$	17.3149	0.650276	0.325138	−	0.945667i	$-0.394589\pi$
$709$	17.3149	0.650276	0.325138	+	0.945667i	$0.394589\pi$
$710$	0	0
$711$	7.05650	0.264640
$712$	0	0
$713$	−1.90874	−0.0714829
$714$	0	0
$715$	22.3726	0.836688
$716$	0	0
$717$	−0.448481	−0.0167488
$718$	0	0
$719$	−18.7331	−0.698626	−0.349313	−	0.937006i	$-0.613585\pi$
$719$	−18.7331	−0.698626	−0.349313	+	0.937006i	$0.613585\pi$
$720$	0	0
$721$	15.0964	0.562220
$722$	0	0
$723$	2.10446	0.0782656
$724$	0	0
$725$	−14.6521	−0.544165
$726$	0	0
$727$	−35.6360	−1.32167	−0.660833	−	0.750533i	$-0.729797\pi$
$727$	−35.6360	−1.32167	−0.660833	+	0.750533i	$0.729797\pi$
$728$	0	0
$729$	−26.4026	−0.977874
$730$	0	0
$731$	34.5004	1.27604
$732$	0	0
$733$	26.0049	0.960512	0.480256	−	0.877128i	$-0.340544\pi$
$733$	26.0049	0.960512	0.480256	+	0.877128i	$0.340544\pi$
$734$	0	0
$735$	0.480493	0.0177233
$736$	0	0
$737$	−7.87217	−0.289975
$738$	0	0
$739$	−32.1103	−1.18120	−0.590599	−	0.806965i	$-0.701108\pi$
$739$	−32.1103	−1.18120	−0.590599	+	0.806965i	$0.701108\pi$
$740$	0	0
$741$	−1.52916	−0.0561750
$742$	0	0
$743$	5.93157	0.217608	0.108804	−	0.994063i	$-0.465298\pi$
$743$	5.93157	0.217608	0.108804	+	0.994063i	$0.465298\pi$
$744$	0	0
$745$	−0.462508	−0.0169450
$746$	0	0
$747$	−3.61533	−0.132278
$748$	0	0
$749$	−12.7811	−0.467011
$750$	0	0
$751$	30.3584	1.10780	0.553898	−	0.832585i	$-0.313140\pi$
$751$	30.3584	1.10780	0.553898	+	0.832585i	$0.313140\pi$
$752$	0	0
$753$	−2.63687	−0.0960929
$754$	0	0
$755$	11.4026	0.414983
$756$	0	0
$757$	−35.5220	−1.29107	−0.645534	−	0.763731i	$-0.723365\pi$
$757$	−35.5220	−1.29107	−0.645534	+	0.763731i	$0.723365\pi$
$758$	0	0
$759$	−0.155827	−0.00565616
$760$	0	0
$761$	52.4976	1.90303	0.951517	−	0.307595i	$-0.0995241\pi$
$761$	52.4976	1.90303	0.951517	+	0.307595i	$0.0995241\pi$
$762$	0	0
$763$	−26.7684	−0.969083
$764$	0	0
$765$	−14.1517	−0.511656
$766$	0	0
$767$	−29.1091	−1.05107
$768$	0	0
$769$	13.0388	0.470192	0.235096	−	0.971972i	$-0.424460\pi$
$769$	13.0388	0.470192	0.235096	+	0.971972i	$0.424460\pi$
$770$	0	0
$771$	−0.333491	−0.0120104
$772$	0	0
$773$	15.7034	0.564814	0.282407	−	0.959295i	$-0.408867\pi$
$773$	15.7034	0.564814	0.282407	+	0.959295i	$0.408867\pi$
$774$	0	0
$775$	−16.7312	−0.601003
$776$	0	0
$777$	1.58245	0.0567702
$778$	0	0
$779$	−28.5884	−1.02428
$780$	0	0
$781$	−18.7019	−0.669206
$782$	0	0
$783$	−3.02741	−0.108191
$784$	0	0
$785$	25.3617	0.905197
$786$	0	0
$787$	−17.8454	−0.636118	−0.318059	−	0.948071i	$-0.603031\pi$
$787$	−17.8454	−0.636118	−0.318059	+	0.948071i	$0.603031\pi$
$788$	0	0
$789$	−2.85520	−0.101648
$790$	0	0
$791$	−5.42306	−0.192822
$792$	0	0
$793$	14.3270	0.508765
$794$	0	0
$795$	−0.751390	−0.0266491
$796$	0	0
$797$	−5.83409	−0.206654	−0.103327	−	0.994647i	$-0.532949\pi$
$797$	−5.83409	−0.206654	−0.103327	+	0.994647i	$0.532949\pi$
$798$	0	0
$799$	21.1311	0.747564
$800$	0	0
$801$	1.73666	0.0613619
$802$	0	0
$803$	50.3037	1.77518
$804$	0	0
$805$	0.938734	0.0330860
$806$	0	0
$807$	2.42917	0.0855109
$808$	0	0
$809$	0.109724	0.00385770	0.00192885	−	0.999998i	$-0.499386\pi$
$809$	0.109724	0.00385770	0.00192885	+	0.999998i	$0.499386\pi$
$810$	0	0
$811$	22.5883	0.793181	0.396590	−	0.917996i	$-0.370193\pi$
$811$	22.5883	0.793181	0.396590	+	0.917996i	$0.370193\pi$
$812$	0	0
$813$	1.34596	0.0472050
$814$	0	0
$815$	16.1481	0.565643
$816$	0	0
$817$	−39.0814	−1.36729
$818$	0	0
$819$	21.7999	0.761751
$820$	0	0
$821$	22.5253	0.786140	0.393070	−	0.919509i	$-0.371413\pi$
$821$	22.5253	0.786140	0.393070	+	0.919509i	$0.371413\pi$
$822$	0	0
$823$	−21.8390	−0.761259	−0.380630	−	0.924728i	$-0.624293\pi$
$823$	−21.8390	−0.761259	−0.380630	+	0.924728i	$0.624293\pi$
$824$	0	0
$825$	−1.36591	−0.0475550
$826$	0	0
$827$	3.88293	0.135023	0.0675113	−	0.997719i	$-0.478494\pi$
$827$	3.88293	0.135023	0.0675113	+	0.997719i	$0.478494\pi$
$828$	0	0
$829$	−24.7313	−0.858954	−0.429477	−	0.903078i	$-0.641302\pi$
$829$	−24.7313	−0.858954	−0.429477	+	0.903078i	$0.641302\pi$
$830$	0	0
$831$	−3.16588	−0.109823
$832$	0	0
$833$	11.0942	0.384393
$834$	0	0
$835$	22.2659	0.770545
$836$	0	0
$837$	−3.45700	−0.119491
$838$	0	0
$839$	−43.9848	−1.51852	−0.759261	−	0.650786i	$-0.774440\pi$
$839$	−43.9848	−1.51852	−0.759261	+	0.650786i	$0.774440\pi$
$840$	0	0
$841$	−5.97306	−0.205968
$842$	0	0
$843$	1.19688	0.0412229
$844$	0	0
$845$	−1.75631	−0.0604190
$846$	0	0
$847$	13.5846	0.466771
$848$	0	0
$849$	2.71478	0.0931710
$850$	0	0
$851$	−2.70910	−0.0928667
$852$	0	0
$853$	0.776266	0.0265788	0.0132894	−	0.999912i	$-0.495770\pi$
$853$	0.776266	0.0265788	0.0132894	+	0.999912i	$0.495770\pi$
$854$	0	0
$855$	16.0308	0.548242
$856$	0	0
$857$	9.02787	0.308386	0.154193	−	0.988041i	$-0.450722\pi$
$857$	9.02787	0.308386	0.154193	+	0.988041i	$0.450722\pi$
$858$	0	0
$859$	38.7938	1.32363	0.661813	−	0.749669i	$-0.269787\pi$
$859$	38.7938	1.32363	0.661813	+	0.749669i	$0.269787\pi$
$860$	0	0
$861$	−1.51319	−0.0515694
$862$	0	0
$863$	11.2720	0.383705	0.191852	−	0.981424i	$-0.438551\pi$
$863$	11.2720	0.383705	0.191852	+	0.981424i	$0.438551\pi$
$864$	0	0
$865$	23.5122	0.799438
$866$	0	0
$867$	−0.577668	−0.0196186
$868$	0	0
$869$	−10.0256	−0.340097
$870$	0	0
$871$	7.00005	0.237188
$872$	0	0
$873$	15.3796	0.520521
$874$	0	0
$875$	21.7030	0.733695
$876$	0	0
$877$	25.9803	0.877293	0.438647	−	0.898660i	$-0.355458\pi$
$877$	25.9803	0.877293	0.438647	+	0.898660i	$0.355458\pi$
$878$	0	0
$879$	0.310896	0.0104863
$880$	0	0
$881$	20.6579	0.695983	0.347991	−	0.937498i	$-0.386864\pi$
$881$	20.6579	0.695983	0.347991	+	0.937498i	$0.386864\pi$
$882$	0	0
$883$	−5.05581	−0.170142	−0.0850708	−	0.996375i	$-0.527112\pi$
$883$	−5.05581	−0.170142	−0.0850708	+	0.996375i	$0.527112\pi$
$884$	0	0
$885$	−1.13301	−0.0380857
$886$	0	0
$887$	−20.9576	−0.703687	−0.351844	−	0.936059i	$-0.614445\pi$
$887$	−20.9576	−0.703687	−0.351844	+	0.936059i	$0.614445\pi$
$888$	0	0
$889$	−0.420744	−0.0141113
$890$	0	0
$891$	37.7952	1.26619
$892$	0	0
$893$	−23.9369	−0.801019
$894$	0	0
$895$	−13.7235	−0.458725
$896$	0	0
$897$	0.138564	0.00462651
$898$	0	0
$899$	26.2944	0.876968
$900$	0	0
$901$	−17.3491	−0.577981
$902$	0	0
$903$	−2.06859	−0.0688384
$904$	0	0
$905$	25.1087	0.834640
$906$	0	0
$907$	13.5634	0.450367	0.225183	−	0.974316i	$-0.427702\pi$
$907$	13.5634	0.450367	0.225183	+	0.974316i	$0.427702\pi$
$908$	0	0
$909$	45.0879	1.49547
$910$	0	0
$911$	39.9227	1.32270	0.661349	−	0.750078i	$-0.269984\pi$
$911$	39.9227	1.32270	0.661349	+	0.750078i	$0.269984\pi$
$912$	0	0
$913$	5.13654	0.169995
$914$	0	0
$915$	0.557646	0.0184352
$916$	0	0
$917$	33.2167	1.09691
$918$	0	0
$919$	−18.3550	−0.605477	−0.302739	−	0.953074i	$-0.597901\pi$
$919$	−18.3550	−0.605477	−0.302739	+	0.953074i	$0.597901\pi$
$920$	0	0
$921$	−3.30490	−0.108900
$922$	0	0
$923$	16.6300	0.547383
$924$	0	0
$925$	−23.7468	−0.780790
$926$	0	0
$927$	−23.3606	−0.767263
$928$	0	0
$929$	−15.7751	−0.517563	−0.258781	−	0.965936i	$-0.583321\pi$
$929$	−15.7751	−0.517563	−0.258781	+	0.965936i	$0.583321\pi$
$930$	0	0
$931$	−12.5674	−0.411879
$932$	0	0
$933$	0.695545	0.0227711
$934$	0	0
$935$	20.1063	0.657546
$936$	0	0
$937$	51.4629	1.68122	0.840609	−	0.541642i	$-0.182197\pi$
$937$	51.4629	1.68122	0.840609	+	0.541642i	$0.182197\pi$
$938$	0	0
$939$	−1.93235	−0.0630600
$940$	0	0
$941$	−38.7105	−1.26193	−0.630963	−	0.775813i	$-0.717340\pi$
$941$	−38.7105	−1.26193	−0.630963	+	0.775813i	$0.717340\pi$
$942$	0	0
$943$	2.59052	0.0843590
$944$	0	0
$945$	1.70018	0.0553069
$946$	0	0
$947$	−28.7467	−0.934141	−0.467071	−	0.884220i	$-0.654691\pi$
$947$	−28.7467	−0.934141	−0.467071	+	0.884220i	$0.654691\pi$
$948$	0	0
$949$	−44.7308	−1.45202
$950$	0	0
$951$	−0.751758	−0.0243774
$952$	0	0
$953$	5.37207	0.174018	0.0870092	−	0.996208i	$-0.472269\pi$
$953$	5.37207	0.174018	0.0870092	+	0.996208i	$0.472269\pi$
$954$	0	0
$955$	2.93291	0.0949068
$956$	0	0
$957$	2.14664	0.0693910
$958$	0	0
$959$	17.7833	0.574253
$960$	0	0
$961$	−0.974425	−0.0314331
$962$	0	0
$963$	19.7778	0.637332
$964$	0	0
$965$	−3.35119	−0.107879
$966$	0	0
$967$	−5.41547	−0.174150	−0.0870749	−	0.996202i	$-0.527752\pi$
$967$	−5.41547	−0.174150	−0.0870749	+	0.996202i	$0.527752\pi$
$968$	0	0
$969$	−1.37425	−0.0441474
$970$	0	0
$971$	22.1723	0.711544	0.355772	−	0.934573i	$-0.384218\pi$
$971$	22.1723	0.711544	0.355772	+	0.934573i	$0.384218\pi$
$972$	0	0
$973$	−14.3003	−0.458448
$974$	0	0
$975$	1.21459	0.0388980
$976$	0	0
$977$	−26.8074	−0.857644	−0.428822	−	0.903389i	$-0.641071\pi$
$977$	−26.8074	−0.857644	−0.428822	+	0.903389i	$0.641071\pi$
$978$	0	0
$979$	−2.46739	−0.0788582
$980$	0	0
$981$	41.4222	1.32251
$982$	0	0
$983$	2.54309	0.0811119	0.0405560	−	0.999177i	$-0.487087\pi$
$983$	2.54309	0.0811119	0.0405560	+	0.999177i	$0.487087\pi$
$984$	0	0
$985$	37.7930	1.20418
$986$	0	0
$987$	−1.26699	−0.0403287
$988$	0	0
$989$	3.54134	0.112608
$990$	0	0
$991$	58.9346	1.87212	0.936059	−	0.351842i	$-0.114445\pi$
$991$	58.9346	1.87212	0.936059	+	0.351842i	$0.114445\pi$
$992$	0	0
$993$	−0.0218630	−0.000693800 0
$994$	0	0
$995$	10.9479	0.347073
$996$	0	0
$997$	−21.1321	−0.669262	−0.334631	−	0.942349i	$-0.608612\pi$
$997$	−21.1321	−0.669262	−0.334631	+	0.942349i	$0.608612\pi$
$998$	0	0
$999$	−4.90656	−0.155237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.x.1.20		33
4.3	odd	2		4024.2.a.g.1.14	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.14	✓	33	4.3	odd	2
8048.2.a.x.1.20		33	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)

\(f(q)\)	\(=\)	\(q+0.105343 q^{3} -1.39521 q^{5} +1.93153 q^{7} -2.98890 q^{9} +O(q^{10})\)	\(q+0.105343 q^{3} -1.39521 q^{5} +1.93153 q^{7} -2.98890 q^{9} +4.24653 q^{11} -3.77608 q^{13} -0.146976 q^{15} -3.39357 q^{17} +3.84418 q^{19} +0.203473 q^{21} -0.348338 q^{23} -3.05338 q^{25} -0.630890 q^{27} +4.79864 q^{29} +5.47956 q^{31} +0.447344 q^{33} -2.69489 q^{35} +7.77720 q^{37} -0.397785 q^{39} -7.43679 q^{41} -10.1664 q^{43} +4.17015 q^{45} -6.22680 q^{47} -3.26919 q^{49} -0.357490 q^{51} +5.11233 q^{53} -5.92481 q^{55} +0.404958 q^{57} +7.70880 q^{59} -3.79413 q^{61} -5.77315 q^{63} +5.26844 q^{65} -1.85379 q^{67} -0.0366951 q^{69} -4.40403 q^{71} +11.8458 q^{73} -0.321653 q^{75} +8.20231 q^{77} -2.36090 q^{79} +8.90025 q^{81} +1.20958 q^{83} +4.73475 q^{85} +0.505504 q^{87} -0.581037 q^{89} -7.29362 q^{91} +0.577234 q^{93} -5.36344 q^{95} -5.14557 q^{97} -12.6925 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * q^99

Introduction

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