LMFDB - Embedded newform 8036.2.a.t.1.18

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:	$0$
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			1.18
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	$-0.195469\pi$
$3$	2.83122	1.63460	0.817302	+	0.576210i	$0.195469\pi$
$4$	0	0
$5$	−2.22134	−0.993412	−0.496706	−	0.867919i	$-0.665457\pi$
$5$	−2.22134	−0.993412	−0.496706	+	0.867919i	$0.665457\pi$
$6$	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	$-0.101159\pi$
$13$	6.85001	1.89985	0.949925	+	0.312477i	$0.101159\pi$
$14$	0	0
$15$	−6.28908	−1.62383
$16$	0	0
$17$	1.59524	0.386902	0.193451	−	0.981110i	$-0.438032\pi$
$17$	1.59524	0.386902	0.193451	+	0.981110i	$0.438032\pi$
$18$	0	0
$19$	−2.36268	−0.542036	−0.271018	−	0.962574i	$-0.587360\pi$
$19$	−2.36268	−0.542036	−0.271018	+	0.962574i	$0.587360\pi$
$20$	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	$-0.439855\pi$
$31$	2.09156	0.375655	0.187827	+	0.982202i	$0.439855\pi$
$32$	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	$-0.668599\pi$
$47$	−6.92760	−1.01049	−0.505247	+	0.862974i	$0.668599\pi$
$48$	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	$-0.565670\pi$
$59$	−3.14694	−0.409697	−0.204848	+	0.978794i	$0.565670\pi$
$60$	0	0
$61$	3.64717	0.466972	0.233486	−	0.972360i	$-0.424987\pi$
$61$	3.64717	0.466972	0.233486	+	0.972360i	$0.424987\pi$
$62$	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	$-0.428722\pi$
$73$	3.79458	0.444122	0.222061	+	0.975033i	$0.428722\pi$
$74$	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	$-0.603642\pi$
$83$	−5.82845	−0.639755	−0.319878	+	0.947459i	$0.603642\pi$
$84$	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	$-0.302720\pi$
$89$	10.9595	1.16170	0.580850	+	0.814011i	$0.302720\pi$
$90$	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	$-0.239368\pi$
$97$	14.3858	1.46065	0.730327	+	0.683098i	$0.239368\pi$
$98$	0	0
$99$	−7.60035	−0.763864
$100$	0	0
$101$	10.9232	1.08690	0.543448	−	0.839443i	$-0.317118\pi$
$101$	10.9232	1.08690	0.543448	+	0.839443i	$0.317118\pi$
$102$	0	0
$103$	−10.3706	−1.02184	−0.510922	−	0.859627i	$-0.670696\pi$
$103$	−10.3706	−1.02184	−0.510922	+	0.859627i	$0.670696\pi$
$104$	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	$-0.524276\pi$
$131$	−1.74407	−0.152380	−0.0761900	+	0.997093i	$0.524276\pi$
$132$	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	$-0.464078\pi$
$139$	2.65539	0.225227	0.112614	+	0.993639i	$0.464078\pi$
$140$	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	$-0.413581\pi$
$157$	6.72031	0.536339	0.268169	+	0.963372i	$0.413581\pi$
$158$	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	$-0.506093\pi$
$167$	−0.494722	−0.0382827	−0.0191414	+	0.999817i	$0.506093\pi$
$168$	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	$-0.515831\pi$
$173$	−1.30778	−0.0994283	−0.0497142	+	0.998763i	$0.515831\pi$
$174$	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	$-0.439720\pi$
$181$	5.06514	0.376489	0.188244	+	0.982122i	$0.439720\pi$
$182$	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	$-0.519109\pi$
$199$	−1.69274	−0.119995	−0.0599975	+	0.998199i	$0.519109\pi$
$200$	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	$-0.403824\pi$
$223$	8.88732	0.595139	0.297569	+	0.954700i	$0.403824\pi$
$224$	0	0
$225$	−0.329366	−0.0219578
$226$	0	0
$227$	21.1796	1.40574	0.702869	−	0.711320i	$-0.251902\pi$
$227$	21.1796	1.40574	0.702869	+	0.711320i	$0.251902\pi$
$228$	0	0
$229$	2.00178	0.132281	0.0661407	−	0.997810i	$-0.478931\pi$
$229$	2.00178	0.132281	0.0661407	+	0.997810i	$0.478931\pi$
$230$	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	$-0.463076\pi$
$241$	3.59352	0.231479	0.115739	+	0.993280i	$0.463076\pi$
$242$	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	$-0.225351\pi$
$251$	24.0715	1.51938	0.759690	+	0.650285i	$0.225351\pi$
$252$	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	$-0.377408\pi$
$257$	12.0453	0.751367	0.375684	+	0.926748i	$0.377408\pi$
$258$	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	$-0.612335\pi$
$269$	−11.3375	−0.691260	−0.345630	+	0.938371i	$0.612335\pi$
$270$	0	0
$271$	−20.4018	−1.23932	−0.619661	−	0.784870i	$-0.712730\pi$
$271$	−20.4018	−1.23932	−0.619661	+	0.784870i	$0.712730\pi$
$272$	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	$-0.279768\pi$
$283$	21.4651	1.27597	0.637985	+	0.770049i	$0.279768\pi$
$284$	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	$-0.649534\pi$
$293$	−15.4975	−0.905372	−0.452686	+	0.891670i	$0.649534\pi$
$294$	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	$-0.494290\pi$
$307$	0.628626	0.0358776	0.0179388	+	0.999839i	$0.494290\pi$
$308$	0	0
$309$	−29.3614	−1.67031
$310$	0	0
$311$	25.0846	1.42242	0.711209	−	0.702981i	$-0.248148\pi$
$311$	25.0846	1.42242	0.711209	+	0.702981i	$0.248148\pi$
$312$	0	0
$313$	6.85414	0.387419	0.193709	−	0.981059i	$-0.437948\pi$
$313$	6.85414	0.387419	0.193709	+	0.981059i	$0.437948\pi$
$314$	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	$-0.217756\pi$
$349$	28.9559	1.54998	0.774988	+	0.631976i	$0.217756\pi$
$350$	0	0
$351$	39.0938	2.08667
$352$	0	0
$353$	12.2613	0.652605	0.326302	−	0.945265i	$-0.394197\pi$
$353$	12.2613	0.652605	0.326302	+	0.945265i	$0.394197\pi$
$354$	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	$-0.637645\pi$
$367$	−16.0566	−0.838147	−0.419074	+	0.907952i	$0.637645\pi$
$368$	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	$-0.441219\pi$
$383$	7.18694	0.367235	0.183618	+	0.982998i	$0.441219\pi$
$384$	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	$-0.671966\pi$
$397$	−20.4967	−1.02870	−0.514349	+	0.857581i	$0.671966\pi$
$398$	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	$-0.559207\pi$
$409$	−7.48011	−0.369867	−0.184934	+	0.982751i	$0.559207\pi$
$410$	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	$-0.599861\pi$
$419$	−12.6338	−0.617202	−0.308601	+	0.951192i	$0.599861\pi$
$420$	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	$-0.635854\pi$
$433$	−17.2278	−0.827916	−0.413958	+	0.910296i	$0.635854\pi$
$434$	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	$-0.673361\pi$
$439$	−21.7108	−1.03620	−0.518100	+	0.855320i	$0.673361\pi$
$440$	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	$-0.297134\pi$
$461$	25.5523	1.19009	0.595045	+	0.803692i	$0.297134\pi$
$462$	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	$-0.300173\pi$
$467$	25.3853	1.17469	0.587346	+	0.809336i	$0.300173\pi$
$468$	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	$-0.622507\pi$
$479$	−16.4336	−0.750871	−0.375436	+	0.926848i	$0.622507\pi$
$480$	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	$-0.574869\pi$
$503$	−10.4533	−0.466090	−0.233045	+	0.972466i	$0.574869\pi$
$504$	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	$-0.372094\pi$
$509$	17.6473	0.782201	0.391101	+	0.920348i	$0.372094\pi$
$510$	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	$-0.623963\pi$
$521$	−17.3323	−0.759342	−0.379671	+	0.925122i	$0.623963\pi$
$522$	0	0
$523$	19.9734	0.873377	0.436688	−	0.899613i	$-0.356151\pi$
$523$	19.9734	0.873377	0.436688	+	0.899613i	$0.356151\pi$
$524$	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	$-0.755643\pi$
$563$	−34.1455	−1.43906	−0.719531	+	0.694461i	$0.755643\pi$
$564$	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	$-0.152363\pi$
$577$	42.6424	1.77522	0.887612	+	0.460591i	$0.152363\pi$
$578$	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	$-0.366317\pi$
$587$	19.7575	0.815478	0.407739	+	0.913098i	$0.366317\pi$
$588$	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	$-0.565770\pi$
$593$	−9.99169	−0.410310	−0.205155	+	0.978730i	$0.565770\pi$
$594$	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	$-0.278130\pi$
$601$	31.4747	1.28388	0.641940	+	0.766755i	$0.278130\pi$
$602$	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	$-0.386163\pi$
$607$	17.2489	0.700110	0.350055	+	0.936729i	$0.386163\pi$
$608$	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	$-0.240556\pi$
$619$	36.2135	1.45554	0.727772	+	0.685819i	$0.240556\pi$
$620$	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	$-0.574481\pi$
$643$	−11.7587	−0.463717	−0.231859	+	0.972749i	$0.574481\pi$
$644$	0	0
$645$	−22.5779	−0.889003
$646$	0	0
$647$	3.88143	0.152595	0.0762974	−	0.997085i	$-0.475690\pi$
$647$	3.88143	0.152595	0.0762974	+	0.997085i	$0.475690\pi$
$648$	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	$-0.260167\pi$
$661$	35.1796	1.36833	0.684164	+	0.729328i	$0.260167\pi$
$662$	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	$-0.619192\pi$
$677$	−19.0337	−0.731525	−0.365762	+	0.930708i	$0.619192\pi$
$678$	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	$-0.515571\pi$
$691$	−2.57078	−0.0977972	−0.0488986	+	0.998804i	$0.515571\pi$
$692$	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	$-0.811185\pi$
$719$	−44.4669	−1.65833	−0.829167	+	0.559001i	$0.811185\pi$
$720$	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	$-0.543985\pi$
$727$	−7.42795	−0.275487	−0.137744	+	0.990468i	$0.543985\pi$
$728$	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	$-0.712285\pi$
$733$	−33.4940	−1.23713	−0.618564	+	0.785734i	$0.712285\pi$
$734$	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	$-0.644811\pi$
$761$	−24.2431	−0.878812	−0.439406	+	0.898289i	$0.644811\pi$
$762$	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	$-0.705348\pi$
$769$	−33.3488	−1.20259	−0.601295	+	0.799027i	$0.705348\pi$
$770$	0	0
$771$	34.1029	1.22819
$772$	0	0
$773$	−35.1822	−1.26542	−0.632709	−	0.774390i	$-0.718057\pi$
$773$	−35.1822	−1.26542	−0.632709	+	0.774390i	$0.718057\pi$
$774$	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	$-0.209669\pi$
$787$	44.3690	1.58158	0.790792	+	0.612086i	$0.209669\pi$
$788$	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	$-0.150581\pi$
$797$	50.2614	1.78035	0.890176	+	0.455617i	$0.150581\pi$
$798$	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	$-0.610973\pi$
$811$	−19.4569	−0.683224	−0.341612	+	0.939841i	$0.610973\pi$
$812$	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	$-0.702149\pi$
$829$	−34.1613	−1.18647	−0.593235	+	0.805030i	$0.702149\pi$
$830$	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	$-0.729335\pi$
$839$	−38.2196	−1.31949	−0.659743	+	0.751491i	$0.729335\pi$
$840$	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	$-0.632622\pi$
$853$	−23.6391	−0.809387	−0.404694	+	0.914452i	$0.632622\pi$
$854$	0	0
$855$	26.3244	0.900274
$856$	0	0
$857$	−35.6249	−1.21692	−0.608461	−	0.793584i	$-0.708213\pi$
$857$	−35.6249	−1.21692	−0.608461	+	0.793584i	$0.708213\pi$
$858$	0	0
$859$	−6.02525	−0.205579	−0.102790	−	0.994703i	$-0.532777\pi$
$859$	−6.02525	−0.205579	−0.102790	+	0.994703i	$0.532777\pi$
$860$	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	$-0.513851\pi$
$881$	−2.58235	−0.0870016	−0.0435008	+	0.999053i	$0.513851\pi$
$882$	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	$-0.323118\pi$
$887$	31.4224	1.05506	0.527530	+	0.849536i	$0.323118\pi$
$888$	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	$-0.417748\pi$
$929$	15.5772	0.511072	0.255536	+	0.966800i	$0.417748\pi$
$930$	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	$-0.446190\pi$
$937$	10.3002	0.336493	0.168246	+	0.985745i	$0.446190\pi$
$938$	0	0
$939$	19.4055	0.633276
$940$	0	0
$941$	2.78075	0.0906499	0.0453250	−	0.998972i	$-0.485568\pi$
$941$	2.78075	0.0906499	0.0453250	+	0.998972i	$0.485568\pi$
$942$	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	$-0.482353\pi$
$971$	3.45335	0.110823	0.0554116	+	0.998464i	$0.482353\pi$
$972$	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	$-0.435370\pi$
$983$	12.6446	0.403299	0.201649	+	0.979458i	$0.435370\pi$
$984$	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	$-0.337853\pi$
$997$	30.7957	0.975308	0.487654	+	0.873037i	$0.337853\pi$
$998$	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.t.1.18	yes	20
7.6	odd	2		8036.2.a.s.1.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.3	✓	20	7.6	odd	2
8036.2.a.t.1.18	yes	20	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
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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