LMFDB - Embedded newform 8036.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.83122$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.83122 q^{3} +2.22134 q^{5} +5.01578 q^{9} +O(q^{10})$	$q-2.83122 q^{3} +2.22134 q^{5} +5.01578 q^{9} -1.51529 q^{11} -6.85001 q^{13} -6.28908 q^{15} -1.59524 q^{17} +2.36268 q^{19} +4.25612 q^{23} -0.0656660 q^{25} -5.70711 q^{27} +1.45086 q^{29} -2.09156 q^{31} +4.29011 q^{33} +7.96332 q^{37} +19.3939 q^{39} +1.00000 q^{41} +3.59001 q^{43} +11.1417 q^{45} +6.92760 q^{47} +4.51646 q^{51} -7.14984 q^{53} -3.36596 q^{55} -6.68926 q^{57} +3.14694 q^{59} -3.64717 q^{61} -15.2162 q^{65} +8.76983 q^{67} -12.0500 q^{69} -0.569520 q^{71} -3.79458 q^{73} +0.185915 q^{75} -11.7557 q^{79} +1.11072 q^{81} +5.82845 q^{83} -3.54356 q^{85} -4.10769 q^{87} -10.9595 q^{89} +5.92165 q^{93} +5.24831 q^{95} -14.3858 q^{97} -7.60035 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.83122	−1.63460	−0.817302	−	0.576210i	$-0.804531\pi$
$3$	−2.83122	−1.63460	−0.817302	+	0.576210i	$0.804531\pi$
$4$	0	0
$5$	2.22134	0.993412	0.496706	−	0.867919i	$-0.334543\pi$
$5$	2.22134	0.993412	0.496706	+	0.867919i	$0.334543\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.01578	1.67193
$10$	0	0
$11$	−1.51529	−0.456876	−0.228438	−	0.973558i	$-0.573362\pi$
$11$	−1.51529	−0.456876	−0.228438	+	0.973558i	$0.573362\pi$
$12$	0	0
$13$	−6.85001	−1.89985	−0.949925	−	0.312477i	$-0.898841\pi$
$13$	−6.85001	−1.89985	−0.949925	+	0.312477i	$0.898841\pi$
$14$	0	0
$15$	−6.28908	−1.62383
$16$	0	0
$17$	−1.59524	−0.386902	−0.193451	−	0.981110i	$-0.561968\pi$
$17$	−1.59524	−0.386902	−0.193451	+	0.981110i	$0.561968\pi$
$18$	0	0
$19$	2.36268	0.542036	0.271018	−	0.962574i	$-0.412640\pi$
$19$	2.36268	0.542036	0.271018	+	0.962574i	$0.412640\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.25612	0.887463	0.443732	−	0.896160i	$-0.353654\pi$
$23$	4.25612	0.887463	0.443732	+	0.896160i	$0.353654\pi$
$24$	0	0
$25$	−0.0656660	−0.0131332
$26$	0	0
$27$	−5.70711	−1.09833
$28$	0	0
$29$	1.45086	0.269417	0.134709	−	0.990885i	$-0.456990\pi$
$29$	1.45086	0.269417	0.134709	+	0.990885i	$0.456990\pi$
$30$	0	0
$31$	−2.09156	−0.375655	−0.187827	−	0.982202i	$-0.560145\pi$
$31$	−2.09156	−0.375655	−0.187827	+	0.982202i	$0.560145\pi$
$32$	0	0
$33$	4.29011	0.746811
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.96332	1.30916	0.654581	−	0.755992i	$-0.272845\pi$
$37$	7.96332	1.30916	0.654581	+	0.755992i	$0.272845\pi$
$38$	0	0
$39$	19.3939	3.10550
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	3.59001	0.547472	0.273736	−	0.961805i	$-0.411741\pi$
$43$	3.59001	0.547472	0.273736	+	0.961805i	$0.411741\pi$
$44$	0	0
$45$	11.1417	1.66091
$46$	0	0
$47$	6.92760	1.01049	0.505247	−	0.862974i	$-0.331401\pi$
$47$	6.92760	1.01049	0.505247	+	0.862974i	$0.331401\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.51646	0.632431
$52$	0	0
$53$	−7.14984	−0.982106	−0.491053	−	0.871130i	$-0.663388\pi$
$53$	−7.14984	−0.982106	−0.491053	+	0.871130i	$0.663388\pi$
$54$	0	0
$55$	−3.36596	−0.453866
$56$	0	0
$57$	−6.68926	−0.886014
$58$	0	0
$59$	3.14694	0.409697	0.204848	−	0.978794i	$-0.434330\pi$
$59$	3.14694	0.409697	0.204848	+	0.978794i	$0.434330\pi$
$60$	0	0
$61$	−3.64717	−0.466972	−0.233486	−	0.972360i	$-0.575013\pi$
$61$	−3.64717	−0.466972	−0.233486	+	0.972360i	$0.575013\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−15.2162	−1.88733
$66$	0	0
$67$	8.76983	1.07141	0.535703	−	0.844407i	$-0.320047\pi$
$67$	8.76983	1.07141	0.535703	+	0.844407i	$0.320047\pi$
$68$	0	0
$69$	−12.0500	−1.45065
$70$	0	0
$71$	−0.569520	−0.0675896	−0.0337948	−	0.999429i	$-0.510759\pi$
$71$	−0.569520	−0.0675896	−0.0337948	+	0.999429i	$0.510759\pi$
$72$	0	0
$73$	−3.79458	−0.444122	−0.222061	−	0.975033i	$-0.571278\pi$
$73$	−3.79458	−0.444122	−0.222061	+	0.975033i	$0.571278\pi$
$74$	0	0
$75$	0.185915	0.0214676
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.7557	−1.32262	−0.661308	−	0.750115i	$-0.729998\pi$
$79$	−11.7557	−1.32262	−0.661308	+	0.750115i	$0.729998\pi$
$80$	0	0
$81$	1.11072	0.123413
$82$	0	0
$83$	5.82845	0.639755	0.319878	−	0.947459i	$-0.396358\pi$
$83$	5.82845	0.639755	0.319878	+	0.947459i	$0.396358\pi$
$84$	0	0
$85$	−3.54356	−0.384353
$86$	0	0
$87$	−4.10769	−0.440390
$88$	0	0
$89$	−10.9595	−1.16170	−0.580850	−	0.814011i	$-0.697280\pi$
$89$	−10.9595	−1.16170	−0.580850	+	0.814011i	$0.697280\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.92165	0.614046
$94$	0	0
$95$	5.24831	0.538465
$96$	0	0
$97$	−14.3858	−1.46065	−0.730327	−	0.683098i	$-0.760632\pi$
$97$	−14.3858	−1.46065	−0.730327	+	0.683098i	$0.760632\pi$
$98$	0	0
$99$	−7.60035	−0.763864
$100$	0	0
$101$	−10.9232	−1.08690	−0.543448	−	0.839443i	$-0.682882\pi$
$101$	−10.9232	−1.08690	−0.543448	+	0.839443i	$0.682882\pi$
$102$	0	0
$103$	10.3706	1.02184	0.510922	−	0.859627i	$-0.329304\pi$
$103$	10.3706	1.02184	0.510922	+	0.859627i	$0.329304\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.1998	−1.08272	−0.541361	−	0.840790i	$-0.682091\pi$
$107$	−11.1998	−1.08272	−0.541361	+	0.840790i	$0.682091\pi$
$108$	0	0
$109$	11.8977	1.13959	0.569797	−	0.821785i	$-0.307022\pi$
$109$	11.8977	1.13959	0.569797	+	0.821785i	$0.307022\pi$
$110$	0	0
$111$	−22.5459	−2.13996
$112$	0	0
$113$	11.9048	1.11991	0.559955	−	0.828523i	$-0.310818\pi$
$113$	11.9048	1.11991	0.559955	+	0.828523i	$0.310818\pi$
$114$	0	0
$115$	9.45428	0.881616
$116$	0	0
$117$	−34.3581	−3.17641
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.70390	−0.791264
$122$	0	0
$123$	−2.83122	−0.255282
$124$	0	0
$125$	−11.2525	−1.00646
$126$	0	0
$127$	11.2729	1.00030	0.500152	−	0.865938i	$-0.333277\pi$
$127$	11.2729	1.00030	0.500152	+	0.865938i	$0.333277\pi$
$128$	0	0
$129$	−10.1641	−0.894899
$130$	0	0
$131$	1.74407	0.152380	0.0761900	−	0.997093i	$-0.475724\pi$
$131$	1.74407	0.152380	0.0761900	+	0.997093i	$0.475724\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−12.6774	−1.09110
$136$	0	0
$137$	19.1965	1.64007	0.820035	−	0.572313i	$-0.193954\pi$
$137$	19.1965	1.64007	0.820035	+	0.572313i	$0.193954\pi$
$138$	0	0
$139$	−2.65539	−0.225227	−0.112614	−	0.993639i	$-0.535922\pi$
$139$	−2.65539	−0.225227	−0.112614	+	0.993639i	$0.535922\pi$
$140$	0	0
$141$	−19.6135	−1.65176
$142$	0	0
$143$	10.3797	0.867997
$144$	0	0
$145$	3.22284	0.267642
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.6148	−1.27922	−0.639608	−	0.768701i	$-0.720903\pi$
$149$	−15.6148	−1.27922	−0.639608	+	0.768701i	$0.720903\pi$
$150$	0	0
$151$	17.8020	1.44871	0.724353	−	0.689429i	$-0.242139\pi$
$151$	17.8020	1.44871	0.724353	+	0.689429i	$0.242139\pi$
$152$	0	0
$153$	−8.00136	−0.646871
$154$	0	0
$155$	−4.64605	−0.373180
$156$	0	0
$157$	−6.72031	−0.536339	−0.268169	−	0.963372i	$-0.586419\pi$
$157$	−6.72031	−0.536339	−0.268169	+	0.963372i	$0.586419\pi$
$158$	0	0
$159$	20.2427	1.60535
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.87978	0.225562	0.112781	−	0.993620i	$-0.464024\pi$
$163$	2.87978	0.225562	0.112781	+	0.993620i	$0.464024\pi$
$164$	0	0
$165$	9.52977	0.741891
$166$	0	0
$167$	0.494722	0.0382827	0.0191414	−	0.999817i	$-0.493907\pi$
$167$	0.494722	0.0382827	0.0191414	+	0.999817i	$0.493907\pi$
$168$	0	0
$169$	33.9226	2.60943
$170$	0	0
$171$	11.8507	0.906245
$172$	0	0
$173$	1.30778	0.0994283	0.0497142	−	0.998763i	$-0.484169\pi$
$173$	1.30778	0.0994283	0.0497142	+	0.998763i	$0.484169\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.90967	−0.669692
$178$	0	0
$179$	9.65003	0.721277	0.360639	−	0.932706i	$-0.382559\pi$
$179$	9.65003	0.721277	0.360639	+	0.932706i	$0.382559\pi$
$180$	0	0
$181$	−5.06514	−0.376489	−0.188244	−	0.982122i	$-0.560280\pi$
$181$	−5.06514	−0.376489	−0.188244	+	0.982122i	$0.560280\pi$
$182$	0	0
$183$	10.3259	0.763315
$184$	0	0
$185$	17.6892	1.30054
$186$	0	0
$187$	2.41724	0.176766
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.83889	−0.277773	−0.138886	−	0.990308i	$-0.544352\pi$
$191$	−3.83889	−0.277773	−0.138886	+	0.990308i	$0.544352\pi$
$192$	0	0
$193$	4.78954	0.344759	0.172379	−	0.985031i	$-0.444854\pi$
$193$	4.78954	0.344759	0.172379	+	0.985031i	$0.444854\pi$
$194$	0	0
$195$	43.0803	3.08504
$196$	0	0
$197$	−3.95862	−0.282040	−0.141020	−	0.990007i	$-0.545038\pi$
$197$	−3.95862	−0.282040	−0.141020	+	0.990007i	$0.545038\pi$
$198$	0	0
$199$	1.69274	0.119995	0.0599975	−	0.998199i	$-0.480891\pi$
$199$	1.69274	0.119995	0.0599975	+	0.998199i	$0.480891\pi$
$200$	0	0
$201$	−24.8293	−1.75132
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.22134	0.155145
$206$	0	0
$207$	21.3478	1.48377
$208$	0	0
$209$	−3.58014	−0.247643
$210$	0	0
$211$	−11.7471	−0.808701	−0.404350	−	0.914604i	$-0.632502\pi$
$211$	−11.7471	−0.808701	−0.404350	+	0.914604i	$0.632502\pi$
$212$	0	0
$213$	1.61243	0.110482
$214$	0	0
$215$	7.97462	0.543865
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.7433	0.725964
$220$	0	0
$221$	10.9274	0.735055
$222$	0	0
$223$	−8.88732	−0.595139	−0.297569	−	0.954700i	$-0.596176\pi$
$223$	−8.88732	−0.595139	−0.297569	+	0.954700i	$0.596176\pi$
$224$	0	0
$225$	−0.329366	−0.0219578
$226$	0	0
$227$	−21.1796	−1.40574	−0.702869	−	0.711320i	$-0.748098\pi$
$227$	−21.1796	−1.40574	−0.702869	+	0.711320i	$0.748098\pi$
$228$	0	0
$229$	−2.00178	−0.132281	−0.0661407	−	0.997810i	$-0.521069\pi$
$229$	−2.00178	−0.132281	−0.0661407	+	0.997810i	$0.521069\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.98056	0.391800	0.195900	−	0.980624i	$-0.437237\pi$
$233$	5.98056	0.391800	0.195900	+	0.980624i	$0.437237\pi$
$234$	0	0
$235$	15.3885	1.00384
$236$	0	0
$237$	33.2828	2.16195
$238$	0	0
$239$	−23.4140	−1.51452	−0.757262	−	0.653111i	$-0.773464\pi$
$239$	−23.4140	−1.51452	−0.757262	+	0.653111i	$0.773464\pi$
$240$	0	0
$241$	−3.59352	−0.231479	−0.115739	−	0.993280i	$-0.536924\pi$
$241$	−3.59352	−0.231479	−0.115739	+	0.993280i	$0.536924\pi$
$242$	0	0
$243$	13.9767	0.896603
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−16.1844	−1.02979
$248$	0	0
$249$	−16.5016	−1.04575
$250$	0	0
$251$	−24.0715	−1.51938	−0.759690	−	0.650285i	$-0.774649\pi$
$251$	−24.0715	−1.51938	−0.759690	+	0.650285i	$0.774649\pi$
$252$	0	0
$253$	−6.44925	−0.405461
$254$	0	0
$255$	10.0326	0.628264
$256$	0	0
$257$	−12.0453	−0.751367	−0.375684	−	0.926748i	$-0.622592\pi$
$257$	−12.0453	−0.751367	−0.375684	+	0.926748i	$0.622592\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	7.27718	0.450446
$262$	0	0
$263$	−22.8321	−1.40789	−0.703944	−	0.710256i	$-0.748579\pi$
$263$	−22.8321	−1.40789	−0.703944	+	0.710256i	$0.748579\pi$
$264$	0	0
$265$	−15.8822	−0.975635
$266$	0	0
$267$	31.0286	1.89892
$268$	0	0
$269$	11.3375	0.691260	0.345630	−	0.938371i	$-0.387665\pi$
$269$	11.3375	0.691260	0.345630	+	0.938371i	$0.387665\pi$
$270$	0	0
$271$	20.4018	1.23932	0.619661	−	0.784870i	$-0.287270\pi$
$271$	20.4018	1.23932	0.619661	+	0.784870i	$0.287270\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.0995029	0.00600025
$276$	0	0
$277$	−31.9214	−1.91797	−0.958987	−	0.283451i	$-0.908521\pi$
$277$	−31.9214	−1.91797	−0.958987	+	0.283451i	$0.908521\pi$
$278$	0	0
$279$	−10.4908	−0.628067
$280$	0	0
$281$	23.4883	1.40120	0.700598	−	0.713556i	$-0.252917\pi$
$281$	23.4883	1.40120	0.700598	+	0.713556i	$0.252917\pi$
$282$	0	0
$283$	−21.4651	−1.27597	−0.637985	−	0.770049i	$-0.720232\pi$
$283$	−21.4651	−1.27597	−0.637985	+	0.770049i	$0.720232\pi$
$284$	0	0
$285$	−14.8591	−0.880176
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.4552	−0.850307
$290$	0	0
$291$	40.7292	2.38759
$292$	0	0
$293$	15.4975	0.905372	0.452686	−	0.891670i	$-0.350466\pi$
$293$	15.4975	0.905372	0.452686	+	0.891670i	$0.350466\pi$
$294$	0	0
$295$	6.99041	0.406998
$296$	0	0
$297$	8.64791	0.501803
$298$	0	0
$299$	−29.1545	−1.68605
$300$	0	0
$301$	0	0
$302$	0	0
$303$	30.9258	1.77664
$304$	0	0
$305$	−8.10159	−0.463896
$306$	0	0
$307$	−0.628626	−0.0358776	−0.0179388	−	0.999839i	$-0.505710\pi$
$307$	−0.628626	−0.0358776	−0.0179388	+	0.999839i	$0.505710\pi$
$308$	0	0
$309$	−29.3614	−1.67031
$310$	0	0
$311$	−25.0846	−1.42242	−0.711209	−	0.702981i	$-0.751852\pi$
$311$	−25.0846	−1.42242	−0.711209	+	0.702981i	$0.751852\pi$
$312$	0	0
$313$	−6.85414	−0.387419	−0.193709	−	0.981059i	$-0.562052\pi$
$313$	−6.85414	−0.387419	−0.193709	+	0.981059i	$0.562052\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.8442	−1.17072	−0.585362	−	0.810772i	$-0.699048\pi$
$317$	−20.8442	−1.17072	−0.585362	+	0.810772i	$0.699048\pi$
$318$	0	0
$319$	−2.19847	−0.123090
$320$	0	0
$321$	31.7089	1.76982
$322$	0	0
$323$	−3.76903	−0.209715
$324$	0	0
$325$	0.449813	0.0249511
$326$	0	0
$327$	−33.6850	−1.86278
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.72356	−0.424525	−0.212263	−	0.977213i	$-0.568083\pi$
$331$	−7.72356	−0.424525	−0.212263	+	0.977213i	$0.568083\pi$
$332$	0	0
$333$	39.9422	2.18882
$334$	0	0
$335$	19.4807	1.06435
$336$	0	0
$337$	19.0522	1.03784	0.518918	−	0.854824i	$-0.326335\pi$
$337$	19.0522	1.03784	0.518918	+	0.854824i	$0.326335\pi$
$338$	0	0
$339$	−33.7051	−1.83061
$340$	0	0
$341$	3.16931	0.171628
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−26.7671	−1.44109
$346$	0	0
$347$	10.6948	0.574128	0.287064	−	0.957911i	$-0.407321\pi$
$347$	10.6948	0.574128	0.287064	+	0.957911i	$0.407321\pi$
$348$	0	0
$349$	−28.9559	−1.54998	−0.774988	−	0.631976i	$-0.782244\pi$
$349$	−28.9559	−1.54998	−0.774988	+	0.631976i	$0.782244\pi$
$350$	0	0
$351$	39.0938	2.08667
$352$	0	0
$353$	−12.2613	−0.652605	−0.326302	−	0.945265i	$-0.605803\pi$
$353$	−12.2613	−0.652605	−0.326302	+	0.945265i	$0.605803\pi$
$354$	0	0
$355$	−1.26510	−0.0671443
$356$	0	0
$357$	0	0
$358$	0	0
$359$	34.9616	1.84520	0.922600	−	0.385757i	$-0.126060\pi$
$359$	34.9616	1.84520	0.922600	+	0.385757i	$0.126060\pi$
$360$	0	0
$361$	−13.4177	−0.706197
$362$	0	0
$363$	24.6426	1.29340
$364$	0	0
$365$	−8.42904	−0.441196
$366$	0	0
$367$	16.0566	0.838147	0.419074	−	0.907952i	$-0.362355\pi$
$367$	16.0566	0.838147	0.419074	+	0.907952i	$0.362355\pi$
$368$	0	0
$369$	5.01578	0.261111
$370$	0	0
$371$	0	0
$372$	0	0
$373$	36.0911	1.86872	0.934362	−	0.356325i	$-0.115971\pi$
$373$	36.0911	1.86872	0.934362	+	0.356325i	$0.115971\pi$
$374$	0	0
$375$	31.8584	1.64516
$376$	0	0
$377$	−9.93838	−0.511853
$378$	0	0
$379$	−24.3348	−1.24999	−0.624996	−	0.780628i	$-0.714900\pi$
$379$	−24.3348	−1.24999	−0.624996	+	0.780628i	$0.714900\pi$
$380$	0	0
$381$	−31.9159	−1.63510
$382$	0	0
$383$	−7.18694	−0.367235	−0.183618	−	0.982998i	$-0.558781\pi$
$383$	−7.18694	−0.367235	−0.183618	+	0.982998i	$0.558781\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.0067	0.915332
$388$	0	0
$389$	−30.8869	−1.56603	−0.783013	−	0.622005i	$-0.786318\pi$
$389$	−30.8869	−1.56603	−0.783013	+	0.622005i	$0.786318\pi$
$390$	0	0
$391$	−6.78952	−0.343361
$392$	0	0
$393$	−4.93784	−0.249081
$394$	0	0
$395$	−26.1133	−1.31390
$396$	0	0
$397$	20.4967	1.02870	0.514349	−	0.857581i	$-0.328034\pi$
$397$	20.4967	1.02870	0.514349	+	0.857581i	$0.328034\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.81685	−0.190605	−0.0953023	−	0.995448i	$-0.530382\pi$
$401$	−3.81685	−0.190605	−0.0953023	+	0.995448i	$0.530382\pi$
$402$	0	0
$403$	14.3272	0.713687
$404$	0	0
$405$	2.46727	0.122600
$406$	0	0
$407$	−12.0667	−0.598125
$408$	0	0
$409$	7.48011	0.369867	0.184934	−	0.982751i	$-0.440793\pi$
$409$	7.48011	0.369867	0.184934	+	0.982751i	$0.440793\pi$
$410$	0	0
$411$	−54.3495	−2.68086
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.9469	0.635540
$416$	0	0
$417$	7.51798	0.368157
$418$	0	0
$419$	12.6338	0.617202	0.308601	−	0.951192i	$-0.400139\pi$
$419$	12.6338	0.617202	0.308601	+	0.951192i	$0.400139\pi$
$420$	0	0
$421$	−22.5003	−1.09660	−0.548299	−	0.836282i	$-0.684724\pi$
$421$	−22.5003	−1.09660	−0.548299	+	0.836282i	$0.684724\pi$
$422$	0	0
$423$	34.7473	1.68947
$424$	0	0
$425$	0.104753	0.00508126
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−29.3873	−1.41883
$430$	0	0
$431$	5.30910	0.255730	0.127865	−	0.991792i	$-0.459188\pi$
$431$	5.30910	0.255730	0.127865	+	0.991792i	$0.459188\pi$
$432$	0	0
$433$	17.2278	0.827916	0.413958	−	0.910296i	$-0.364146\pi$
$433$	17.2278	0.827916	0.413958	+	0.910296i	$0.364146\pi$
$434$	0	0
$435$	−9.12456	−0.437489
$436$	0	0
$437$	10.0559	0.481037
$438$	0	0
$439$	21.7108	1.03620	0.518100	−	0.855320i	$-0.326639\pi$
$439$	21.7108	1.03620	0.518100	+	0.855320i	$0.326639\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.3891	0.921206	0.460603	−	0.887606i	$-0.347633\pi$
$443$	19.3891	0.921206	0.460603	+	0.887606i	$0.347633\pi$
$444$	0	0
$445$	−24.3446	−1.15405
$446$	0	0
$447$	44.2089	2.09101
$448$	0	0
$449$	−27.0936	−1.27863	−0.639314	−	0.768946i	$-0.720782\pi$
$449$	−27.0936	−1.27863	−0.639314	+	0.768946i	$0.720782\pi$
$450$	0	0
$451$	−1.51529	−0.0713521
$452$	0	0
$453$	−50.4013	−2.36806
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.1428	1.45680	0.728400	−	0.685152i	$-0.240264\pi$
$457$	31.1428	1.45680	0.728400	+	0.685152i	$0.240264\pi$
$458$	0	0
$459$	9.10419	0.424947
$460$	0	0
$461$	−25.5523	−1.19009	−0.595045	−	0.803692i	$-0.702866\pi$
$461$	−25.5523	−1.19009	−0.595045	+	0.803692i	$0.702866\pi$
$462$	0	0
$463$	−18.7078	−0.869424	−0.434712	−	0.900570i	$-0.643150\pi$
$463$	−18.7078	−0.869424	−0.434712	+	0.900570i	$0.643150\pi$
$464$	0	0
$465$	13.1540	0.610000
$466$	0	0
$467$	−25.3853	−1.17469	−0.587346	−	0.809336i	$-0.699827\pi$
$467$	−25.3853	−1.17469	−0.587346	+	0.809336i	$0.699827\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	19.0266	0.876701
$472$	0	0
$473$	−5.43990	−0.250127
$474$	0	0
$475$	−0.155148	−0.00711867
$476$	0	0
$477$	−35.8620	−1.64201
$478$	0	0
$479$	16.4336	0.750871	0.375436	−	0.926848i	$-0.377493\pi$
$479$	16.4336	0.750871	0.375436	+	0.926848i	$0.377493\pi$
$480$	0	0
$481$	−54.5488	−2.48721
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−31.9556	−1.45103
$486$	0	0
$487$	10.1082	0.458047	0.229023	−	0.973421i	$-0.426447\pi$
$487$	10.1082	0.458047	0.229023	+	0.973421i	$0.426447\pi$
$488$	0	0
$489$	−8.15328	−0.368704
$490$	0	0
$491$	−24.3172	−1.09742	−0.548710	−	0.836013i	$-0.684881\pi$
$491$	−24.3172	−1.09742	−0.548710	+	0.836013i	$0.684881\pi$
$492$	0	0
$493$	−2.31446	−0.104238
$494$	0	0
$495$	−16.8829	−0.758831
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−23.6831	−1.06020	−0.530101	−	0.847934i	$-0.677846\pi$
$499$	−23.6831	−1.06020	−0.530101	+	0.847934i	$0.677846\pi$
$500$	0	0
$501$	−1.40066	−0.0625770
$502$	0	0
$503$	10.4533	0.466090	0.233045	−	0.972466i	$-0.425131\pi$
$503$	10.4533	0.466090	0.233045	+	0.972466i	$0.425131\pi$
$504$	0	0
$505$	−24.2640	−1.07974
$506$	0	0
$507$	−96.0422	−4.26539
$508$	0	0
$509$	−17.6473	−0.782201	−0.391101	−	0.920348i	$-0.627906\pi$
$509$	−17.6473	−0.782201	−0.391101	+	0.920348i	$0.627906\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−13.4841	−0.595336
$514$	0	0
$515$	23.0366	1.01511
$516$	0	0
$517$	−10.4973	−0.461671
$518$	0	0
$519$	−3.70259	−0.162526
$520$	0	0
$521$	17.3323	0.759342	0.379671	−	0.925122i	$-0.376037\pi$
$521$	17.3323	0.759342	0.379671	+	0.925122i	$0.376037\pi$
$522$	0	0
$523$	−19.9734	−0.873377	−0.436688	−	0.899613i	$-0.643849\pi$
$523$	−19.9734	−0.873377	−0.436688	+	0.899613i	$0.643849\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.33653	0.145341
$528$	0	0
$529$	−4.88541	−0.212409
$530$	0	0
$531$	15.7844	0.684983
$532$	0	0
$533$	−6.85001	−0.296707
$534$	0	0
$535$	−24.8784	−1.07559
$536$	0	0
$537$	−27.3213	−1.17900
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.96600	−0.127518	−0.0637591	−	0.997965i	$-0.520309\pi$
$541$	−2.96600	−0.127518	−0.0637591	+	0.997965i	$0.520309\pi$
$542$	0	0
$543$	14.3405	0.615410
$544$	0	0
$545$	26.4288	1.13209
$546$	0	0
$547$	12.9778	0.554890	0.277445	−	0.960741i	$-0.410512\pi$
$547$	12.9778	0.554890	0.277445	+	0.960741i	$0.410512\pi$
$548$	0	0
$549$	−18.2934	−0.780744
$550$	0	0
$551$	3.42791	0.146034
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−50.0819	−2.12586
$556$	0	0
$557$	−41.4658	−1.75696	−0.878482	−	0.477775i	$-0.841443\pi$
$557$	−41.4658	−1.75696	−0.878482	+	0.477775i	$0.841443\pi$
$558$	0	0
$559$	−24.5916	−1.04011
$560$	0	0
$561$	−6.84373	−0.288943
$562$	0	0
$563$	34.1455	1.43906	0.719531	−	0.694461i	$-0.244357\pi$
$563$	34.1455	1.43906	0.719531	+	0.694461i	$0.244357\pi$
$564$	0	0
$565$	26.4446	1.11253
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.6113	−0.696380	−0.348190	−	0.937424i	$-0.613204\pi$
$569$	−16.6113	−0.696380	−0.348190	+	0.937424i	$0.613204\pi$
$570$	0	0
$571$	−37.7089	−1.57807	−0.789034	−	0.614350i	$-0.789418\pi$
$571$	−37.7089	−1.57807	−0.789034	+	0.614350i	$0.789418\pi$
$572$	0	0
$573$	10.8687	0.454048
$574$	0	0
$575$	−0.279483	−0.0116552
$576$	0	0
$577$	−42.6424	−1.77522	−0.887612	−	0.460591i	$-0.847637\pi$
$577$	−42.6424	−1.77522	−0.887612	+	0.460591i	$0.847637\pi$
$578$	0	0
$579$	−13.5602	−0.563544
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.8341	0.448701
$584$	0	0
$585$	−76.3210	−3.15548
$586$	0	0
$587$	−19.7575	−0.815478	−0.407739	−	0.913098i	$-0.633683\pi$
$587$	−19.7575	−0.815478	−0.407739	+	0.913098i	$0.633683\pi$
$588$	0	0
$589$	−4.94168	−0.203618
$590$	0	0
$591$	11.2077	0.461023
$592$	0	0
$593$	9.99169	0.410310	0.205155	−	0.978730i	$-0.434230\pi$
$593$	9.99169	0.410310	0.205155	+	0.978730i	$0.434230\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.79250	−0.196144
$598$	0	0
$599$	−26.8488	−1.09701	−0.548507	−	0.836146i	$-0.684804\pi$
$599$	−26.8488	−1.09701	−0.548507	+	0.836146i	$0.684804\pi$
$600$	0	0
$601$	−31.4747	−1.28388	−0.641940	−	0.766755i	$-0.721870\pi$
$601$	−31.4747	−1.28388	−0.641940	+	0.766755i	$0.721870\pi$
$602$	0	0
$603$	43.9875	1.79131
$604$	0	0
$605$	−19.3343	−0.786051
$606$	0	0
$607$	−17.2489	−0.700110	−0.350055	−	0.936729i	$-0.613837\pi$
$607$	−17.2489	−0.700110	−0.350055	+	0.936729i	$0.613837\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−47.4542	−1.91979
$612$	0	0
$613$	43.6203	1.76181	0.880904	−	0.473295i	$-0.156935\pi$
$613$	43.6203	1.76181	0.880904	+	0.473295i	$0.156935\pi$
$614$	0	0
$615$	−6.28908	−0.253600
$616$	0	0
$617$	−26.1497	−1.05275	−0.526373	−	0.850254i	$-0.676448\pi$
$617$	−26.1497	−1.05275	−0.526373	+	0.850254i	$0.676448\pi$
$618$	0	0
$619$	−36.2135	−1.45554	−0.727772	−	0.685819i	$-0.759444\pi$
$619$	−36.2135	−1.45554	−0.727772	+	0.685819i	$0.759444\pi$
$620$	0	0
$621$	−24.2902	−0.974731
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−24.6674	−0.986694
$626$	0	0
$627$	10.1361	0.404799
$628$	0	0
$629$	−12.7034	−0.506517
$630$	0	0
$631$	27.3123	1.08728	0.543642	−	0.839317i	$-0.317045\pi$
$631$	27.3123	1.08728	0.543642	+	0.839317i	$0.317045\pi$
$632$	0	0
$633$	33.2584	1.32190
$634$	0	0
$635$	25.0408	0.993714
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.85659	−0.113005
$640$	0	0
$641$	−32.2351	−1.27321	−0.636605	−	0.771190i	$-0.719662\pi$
$641$	−32.2351	−1.27321	−0.636605	+	0.771190i	$0.719662\pi$
$642$	0	0
$643$	11.7587	0.463717	0.231859	−	0.972749i	$-0.425519\pi$
$643$	11.7587	0.463717	0.231859	+	0.972749i	$0.425519\pi$
$644$	0	0
$645$	−22.5779	−0.889003
$646$	0	0
$647$	−3.88143	−0.152595	−0.0762974	−	0.997085i	$-0.524310\pi$
$647$	−3.88143	−0.152595	−0.0762974	+	0.997085i	$0.524310\pi$
$648$	0	0
$649$	−4.76852	−0.187181
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.5009	−1.11533	−0.557664	−	0.830067i	$-0.688302\pi$
$653$	−28.5009	−1.11533	−0.557664	+	0.830067i	$0.688302\pi$
$654$	0	0
$655$	3.87416	0.151376
$656$	0	0
$657$	−19.0328	−0.742540
$658$	0	0
$659$	−15.3075	−0.596295	−0.298148	−	0.954520i	$-0.596369\pi$
$659$	−15.3075	−0.596295	−0.298148	+	0.954520i	$0.596369\pi$
$660$	0	0
$661$	−35.1796	−1.36833	−0.684164	−	0.729328i	$-0.739833\pi$
$661$	−35.1796	−1.36833	−0.684164	+	0.729328i	$0.739833\pi$
$662$	0	0
$663$	−30.9378	−1.20152
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.17503	0.239098
$668$	0	0
$669$	25.1619	0.972816
$670$	0	0
$671$	5.52651	0.213349
$672$	0	0
$673$	19.5437	0.753355	0.376678	−	0.926344i	$-0.377066\pi$
$673$	19.5437	0.753355	0.376678	+	0.926344i	$0.377066\pi$
$674$	0	0
$675$	0.374763	0.0144246
$676$	0	0
$677$	19.0337	0.731525	0.365762	−	0.930708i	$-0.380808\pi$
$677$	19.0337	0.731525	0.365762	+	0.930708i	$0.380808\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	59.9639	2.29782
$682$	0	0
$683$	7.28626	0.278801	0.139401	−	0.990236i	$-0.455482\pi$
$683$	7.28626	0.278801	0.139401	+	0.990236i	$0.455482\pi$
$684$	0	0
$685$	42.6419	1.62927
$686$	0	0
$687$	5.66747	0.216227
$688$	0	0
$689$	48.9765	1.86585
$690$	0	0
$691$	2.57078	0.0977972	0.0488986	−	0.998804i	$-0.484429\pi$
$691$	2.57078	0.0977972	0.0488986	+	0.998804i	$0.484429\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.89852	−0.223743
$696$	0	0
$697$	−1.59524	−0.0604239
$698$	0	0
$699$	−16.9323	−0.640437
$700$	0	0
$701$	−35.9963	−1.35956	−0.679781	−	0.733415i	$-0.737925\pi$
$701$	−35.9963	−1.35956	−0.679781	+	0.733415i	$0.737925\pi$
$702$	0	0
$703$	18.8148	0.709612
$704$	0	0
$705$	−43.5683	−1.64088
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−23.7405	−0.891593	−0.445796	−	0.895134i	$-0.647079\pi$
$709$	−23.7405	−0.891593	−0.445796	+	0.895134i	$0.647079\pi$
$710$	0	0
$711$	−58.9638	−2.21132
$712$	0	0
$713$	−8.90192	−0.333380
$714$	0	0
$715$	23.0569	0.862278
$716$	0	0
$717$	66.2900	2.47565
$718$	0	0
$719$	44.4669	1.65833	0.829167	−	0.559001i	$-0.188815\pi$
$719$	44.4669	1.65833	0.829167	+	0.559001i	$0.188815\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10.1740	0.378376
$724$	0	0
$725$	−0.0952720	−0.00353831
$726$	0	0
$727$	7.42795	0.275487	0.137744	−	0.990468i	$-0.456015\pi$
$727$	7.42795	0.275487	0.137744	+	0.990468i	$0.456015\pi$
$728$	0	0
$729$	−42.9031	−1.58900
$730$	0	0
$731$	−5.72692	−0.211818
$732$	0	0
$733$	33.4940	1.23713	0.618564	−	0.785734i	$-0.287715\pi$
$733$	33.4940	1.23713	0.618564	+	0.785734i	$0.287715\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.2888	−0.489500
$738$	0	0
$739$	−49.0508	−1.80436	−0.902181	−	0.431358i	$-0.858035\pi$
$739$	−49.0508	−1.80436	−0.902181	+	0.431358i	$0.858035\pi$
$740$	0	0
$741$	45.8215	1.68329
$742$	0	0
$743$	21.9291	0.804502	0.402251	−	0.915529i	$-0.368228\pi$
$743$	21.9291	0.804502	0.402251	+	0.915529i	$0.368228\pi$
$744$	0	0
$745$	−34.6858	−1.27079
$746$	0	0
$747$	29.2342	1.06962
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−39.3023	−1.43416	−0.717081	−	0.696990i	$-0.754522\pi$
$751$	−39.3023	−1.43416	−0.717081	+	0.696990i	$0.754522\pi$
$752$	0	0
$753$	68.1516	2.48358
$754$	0	0
$755$	39.5442	1.43916
$756$	0	0
$757$	−2.22997	−0.0810495	−0.0405247	−	0.999179i	$-0.512903\pi$
$757$	−2.22997	−0.0810495	−0.0405247	+	0.999179i	$0.512903\pi$
$758$	0	0
$759$	18.2592	0.662768
$760$	0	0
$761$	24.2431	0.878812	0.439406	−	0.898289i	$-0.355189\pi$
$761$	24.2431	0.878812	0.439406	+	0.898289i	$0.355189\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−17.7737	−0.642609
$766$	0	0
$767$	−21.5566	−0.778363
$768$	0	0
$769$	33.3488	1.20259	0.601295	−	0.799027i	$-0.294652\pi$
$769$	33.3488	1.20259	0.601295	+	0.799027i	$0.294652\pi$
$770$	0	0
$771$	34.1029	1.22819
$772$	0	0
$773$	35.1822	1.26542	0.632709	−	0.774390i	$-0.281943\pi$
$773$	35.1822	1.26542	0.632709	+	0.774390i	$0.281943\pi$
$774$	0	0
$775$	0.137344	0.00493355
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.36268	0.0846518
$780$	0	0
$781$	0.862987	0.0308801
$782$	0	0
$783$	−8.28020	−0.295910
$784$	0	0
$785$	−14.9281	−0.532805
$786$	0	0
$787$	−44.3690	−1.58158	−0.790792	−	0.612086i	$-0.790331\pi$
$787$	−44.3690	−1.58158	−0.790792	+	0.612086i	$0.790331\pi$
$788$	0	0
$789$	64.6426	2.30134
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.9832	0.887178
$794$	0	0
$795$	44.9659	1.59478
$796$	0	0
$797$	−50.2614	−1.78035	−0.890176	−	0.455617i	$-0.849419\pi$
$797$	−50.2614	−1.78035	−0.890176	+	0.455617i	$0.849419\pi$
$798$	0	0
$799$	−11.0512	−0.390962
$800$	0	0
$801$	−54.9702	−1.94228
$802$	0	0
$803$	5.74988	0.202909
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−32.0989	−1.12994
$808$	0	0
$809$	5.78059	0.203235	0.101617	−	0.994824i	$-0.467598\pi$
$809$	5.78059	0.203235	0.101617	+	0.994824i	$0.467598\pi$
$810$	0	0
$811$	19.4569	0.683224	0.341612	−	0.939841i	$-0.389027\pi$
$811$	19.4569	0.683224	0.341612	+	0.939841i	$0.389027\pi$
$812$	0	0
$813$	−57.7619	−2.02580
$814$	0	0
$815$	6.39696	0.224076
$816$	0	0
$817$	8.48205	0.296749
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.99034	0.174164	0.0870821	−	0.996201i	$-0.472246\pi$
$821$	4.99034	0.174164	0.0870821	+	0.996201i	$0.472246\pi$
$822$	0	0
$823$	−4.25875	−0.148451	−0.0742254	−	0.997241i	$-0.523648\pi$
$823$	−4.25875	−0.148451	−0.0742254	+	0.997241i	$0.523648\pi$
$824$	0	0
$825$	−0.281714	−0.00980803
$826$	0	0
$827$	−5.79439	−0.201491	−0.100745	−	0.994912i	$-0.532123\pi$
$827$	−5.79439	−0.201491	−0.100745	+	0.994912i	$0.532123\pi$
$828$	0	0
$829$	34.1613	1.18647	0.593235	−	0.805030i	$-0.297851\pi$
$829$	34.1613	1.18647	0.593235	+	0.805030i	$0.297851\pi$
$830$	0	0
$831$	90.3765	3.13512
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.09894	0.0380305
$836$	0	0
$837$	11.9367	0.412594
$838$	0	0
$839$	38.2196	1.31949	0.659743	−	0.751491i	$-0.270665\pi$
$839$	38.2196	1.31949	0.659743	+	0.751491i	$0.270665\pi$
$840$	0	0
$841$	−26.8950	−0.927414
$842$	0	0
$843$	−66.5005	−2.29040
$844$	0	0
$845$	75.3535	2.59224
$846$	0	0
$847$	0	0
$848$	0	0
$849$	60.7724	2.08570
$850$	0	0
$851$	33.8929	1.16183
$852$	0	0
$853$	23.6391	0.809387	0.404694	−	0.914452i	$-0.367378\pi$
$853$	23.6391	0.809387	0.404694	+	0.914452i	$0.367378\pi$
$854$	0	0
$855$	26.3244	0.900274
$856$	0	0
$857$	35.6249	1.21692	0.608461	−	0.793584i	$-0.291787\pi$
$857$	35.6249	1.21692	0.608461	+	0.793584i	$0.291787\pi$
$858$	0	0
$859$	6.02525	0.205579	0.102790	−	0.994703i	$-0.467223\pi$
$859$	6.02525	0.205579	0.102790	+	0.994703i	$0.467223\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.8161	−0.640509	−0.320254	−	0.947332i	$-0.603768\pi$
$863$	−18.8161	−0.640509	−0.320254	+	0.947332i	$0.603768\pi$
$864$	0	0
$865$	2.90501	0.0987733
$866$	0	0
$867$	40.9258	1.38991
$868$	0	0
$869$	17.8132	0.604272
$870$	0	0
$871$	−60.0734	−2.03551
$872$	0	0
$873$	−72.1559	−2.44211
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.20990	0.175926	0.0879630	−	0.996124i	$-0.471964\pi$
$877$	5.20990	0.175926	0.0879630	+	0.996124i	$0.471964\pi$
$878$	0	0
$879$	−43.8767	−1.47992
$880$	0	0
$881$	2.58235	0.0870016	0.0435008	−	0.999053i	$-0.486149\pi$
$881$	2.58235	0.0870016	0.0435008	+	0.999053i	$0.486149\pi$
$882$	0	0
$883$	−37.6520	−1.26709	−0.633546	−	0.773705i	$-0.718401\pi$
$883$	−37.6520	−1.26709	−0.633546	+	0.773705i	$0.718401\pi$
$884$	0	0
$885$	−19.7914	−0.665279
$886$	0	0
$887$	−31.4224	−1.05506	−0.527530	−	0.849536i	$-0.676882\pi$
$887$	−31.4224	−1.05506	−0.527530	+	0.849536i	$0.676882\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.68305	−0.0563844
$892$	0	0
$893$	16.3677	0.547725
$894$	0	0
$895$	21.4360	0.716525
$896$	0	0
$897$	82.5426	2.75602
$898$	0	0
$899$	−3.03455	−0.101208
$900$	0	0
$901$	11.4057	0.379978
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.2514	−0.374008
$906$	0	0
$907$	25.3863	0.842939	0.421469	−	0.906843i	$-0.361514\pi$
$907$	25.3863	0.842939	0.421469	+	0.906843i	$0.361514\pi$
$908$	0	0
$909$	−54.7882	−1.81721
$910$	0	0
$911$	43.5586	1.44316	0.721580	−	0.692331i	$-0.243416\pi$
$911$	43.5586	1.44316	0.721580	+	0.692331i	$0.243416\pi$
$912$	0	0
$913$	−8.83177	−0.292289
$914$	0	0
$915$	22.9374	0.758286
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−33.4412	−1.10312	−0.551562	−	0.834134i	$-0.685968\pi$
$919$	−33.4412	−1.10312	−0.551562	+	0.834134i	$0.685968\pi$
$920$	0	0
$921$	1.77978	0.0586456
$922$	0	0
$923$	3.90122	0.128410
$924$	0	0
$925$	−0.522919	−0.0171935
$926$	0	0
$927$	52.0166	1.70845
$928$	0	0
$929$	−15.5772	−0.511072	−0.255536	−	0.966800i	$-0.582252\pi$
$929$	−15.5772	−0.511072	−0.255536	+	0.966800i	$0.582252\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	71.0200	2.32509
$934$	0	0
$935$	5.36951	0.175602
$936$	0	0
$937$	−10.3002	−0.336493	−0.168246	−	0.985745i	$-0.553810\pi$
$937$	−10.3002	−0.336493	−0.168246	+	0.985745i	$0.553810\pi$
$938$	0	0
$939$	19.4055	0.633276
$940$	0	0
$941$	−2.78075	−0.0906499	−0.0453250	−	0.998972i	$-0.514432\pi$
$941$	−2.78075	−0.0906499	−0.0453250	+	0.998972i	$0.514432\pi$
$942$	0	0
$943$	4.25612	0.138598
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22.1990	−0.721372	−0.360686	−	0.932687i	$-0.617457\pi$
$947$	−22.1990	−0.721372	−0.360686	+	0.932687i	$0.617457\pi$
$948$	0	0
$949$	25.9929	0.843766
$950$	0	0
$951$	59.0143	1.91367
$952$	0	0
$953$	−35.3033	−1.14359	−0.571793	−	0.820398i	$-0.693752\pi$
$953$	−35.3033	−1.14359	−0.571793	+	0.820398i	$0.693752\pi$
$954$	0	0
$955$	−8.52747	−0.275943
$956$	0	0
$957$	6.22433	0.201204
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−26.6254	−0.858884
$962$	0	0
$963$	−56.1755	−1.81023
$964$	0	0
$965$	10.6392	0.342488
$966$	0	0
$967$	20.0664	0.645292	0.322646	−	0.946520i	$-0.395428\pi$
$967$	20.0664	0.645292	0.322646	+	0.946520i	$0.395428\pi$
$968$	0	0
$969$	10.6709	0.342800
$970$	0	0
$971$	−3.45335	−0.110823	−0.0554116	−	0.998464i	$-0.517647\pi$
$971$	−3.45335	−0.110823	−0.0554116	+	0.998464i	$0.517647\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.27352	−0.0407852
$976$	0	0
$977$	18.2468	0.583768	0.291884	−	0.956454i	$-0.405718\pi$
$977$	18.2468	0.583768	0.291884	+	0.956454i	$0.405718\pi$
$978$	0	0
$979$	16.6067	0.530753
$980$	0	0
$981$	59.6763	1.90532
$982$	0	0
$983$	−12.6446	−0.403299	−0.201649	−	0.979458i	$-0.564630\pi$
$983$	−12.6446	−0.403299	−0.201649	+	0.979458i	$0.564630\pi$
$984$	0	0
$985$	−8.79342	−0.280182
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.2795	0.485861
$990$	0	0
$991$	−38.9308	−1.23668	−0.618339	−	0.785911i	$-0.712194\pi$
$991$	−38.9308	−1.23668	−0.618339	+	0.785911i	$0.712194\pi$
$992$	0	0
$993$	21.8671	0.693930
$994$	0	0
$995$	3.76014	0.119204
$996$	0	0
$997$	−30.7957	−0.975308	−0.487654	−	0.873037i	$-0.662147\pi$
$997$	−30.7957	−0.975308	−0.487654	+	0.873037i	$0.662147\pi$
$998$	0	0
$999$	−45.4475	−1.43790

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.s.1.3	✓	20
7.6	odd	2		8036.2.a.t.1.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.3	✓	20	1.1	even	1	trivial
8036.2.a.t.1.18	yes	20	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-2.83122 q^{3} +2.22134 q^{5} +5.01578 q^{9} +O(q^{10})\)	\(q-2.83122 q^{3} +2.22134 q^{5} +5.01578 q^{9} -1.51529 q^{11} -6.85001 q^{13} -6.28908 q^{15} -1.59524 q^{17} +2.36268 q^{19} +4.25612 q^{23} -0.0656660 q^{25} -5.70711 q^{27} +1.45086 q^{29} -2.09156 q^{31} +4.29011 q^{33} +7.96332 q^{37} +19.3939 q^{39} +1.00000 q^{41} +3.59001 q^{43} +11.1417 q^{45} +6.92760 q^{47} +4.51646 q^{51} -7.14984 q^{53} -3.36596 q^{55} -6.68926 q^{57} +3.14694 q^{59} -3.64717 q^{61} -15.2162 q^{65} +8.76983 q^{67} -12.0500 q^{69} -0.569520 q^{71} -3.79458 q^{73} +0.185915 q^{75} -11.7557 q^{79} +1.11072 q^{81} +5.82845 q^{83} -3.54356 q^{85} -4.10769 q^{87} -10.9595 q^{89} +5.92165 q^{93} +5.24831 q^{95} -14.3858 q^{97} -7.60035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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