LMFDB - Embedded newform 8036.2.a.m.1.7

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 $ x^8 - 14x^6 - 4x^5 + 60x^4 + 31x^3 - 75x^2 - 60x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.24904$ 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)

$f(q)$	$=$	$q+2.24904 q^{3} -1.47097 q^{5} +2.05817 q^{9} +O(q^{10})$	$q+2.24904 q^{3} -1.47097 q^{5} +2.05817 q^{9} -0.494253 q^{11} +5.52254 q^{13} -3.30827 q^{15} -4.25624 q^{17} -0.960935 q^{19} -2.30001 q^{23} -2.83624 q^{25} -2.11820 q^{27} +1.25851 q^{29} -8.26477 q^{31} -1.11159 q^{33} -8.22617 q^{37} +12.4204 q^{39} +1.00000 q^{41} +5.33579 q^{43} -3.02752 q^{45} +3.66765 q^{47} -9.57245 q^{51} +8.54635 q^{53} +0.727033 q^{55} -2.16118 q^{57} +1.47245 q^{59} -6.23371 q^{61} -8.12350 q^{65} +3.91517 q^{67} -5.17281 q^{69} -10.1617 q^{71} +0.775836 q^{73} -6.37881 q^{75} -9.98543 q^{79} -10.9384 q^{81} -7.68368 q^{83} +6.26081 q^{85} +2.83043 q^{87} -6.83079 q^{89} -18.5878 q^{93} +1.41351 q^{95} -6.83506 q^{97} -1.01726 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.24904	1.29848	0.649241	−	0.760582i	$-0.275086\pi$
$3$	2.24904	1.29848	0.649241	+	0.760582i	$0.275086\pi$
$4$	0	0
$5$	−1.47097	−0.657839	−0.328919	−	0.944358i	$-0.606684\pi$
$5$	−1.47097	−0.657839	−0.328919	+	0.944358i	$0.606684\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.05817	0.686058
$10$	0	0
$11$	−0.494253	−0.149023	−0.0745114	−	0.997220i	$-0.523740\pi$
$11$	−0.494253	−0.149023	−0.0745114	+	0.997220i	$0.523740\pi$
$12$	0	0
$13$	5.52254	1.53168	0.765838	−	0.643033i	$-0.222324\pi$
$13$	5.52254	1.53168	0.765838	+	0.643033i	$0.222324\pi$
$14$	0	0
$15$	−3.30827	−0.854193
$16$	0	0
$17$	−4.25624	−1.03229	−0.516145	−	0.856501i	$-0.672633\pi$
$17$	−4.25624	−1.03229	−0.516145	+	0.856501i	$0.672633\pi$
$18$	0	0
$19$	−0.960935	−0.220454	−0.110227	−	0.993906i	$-0.535158\pi$
$19$	−0.960935	−0.220454	−0.110227	+	0.993906i	$0.535158\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.30001	−0.479585	−0.239793	−	0.970824i	$-0.577079\pi$
$23$	−2.30001	−0.479585	−0.239793	+	0.970824i	$0.577079\pi$
$24$	0	0
$25$	−2.83624	−0.567248
$26$	0	0
$27$	−2.11820	−0.407648
$28$	0	0
$29$	1.25851	0.233699	0.116850	−	0.993150i	$-0.462720\pi$
$29$	1.25851	0.233699	0.116850	+	0.993150i	$0.462720\pi$
$30$	0	0
$31$	−8.26477	−1.48440	−0.742198	−	0.670181i	$-0.766217\pi$
$31$	−8.26477	−1.48440	−0.742198	+	0.670181i	$0.766217\pi$
$32$	0	0
$33$	−1.11159	−0.193504
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.22617	−1.35237	−0.676187	−	0.736730i	$-0.736369\pi$
$37$	−8.22617	−1.35237	−0.676187	+	0.736730i	$0.736369\pi$
$38$	0	0
$39$	12.4204	1.98886
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	5.33579	0.813700	0.406850	−	0.913495i	$-0.366627\pi$
$43$	5.33579	0.813700	0.406850	+	0.913495i	$0.366627\pi$
$44$	0	0
$45$	−3.02752	−0.451316
$46$	0	0
$47$	3.66765	0.534982	0.267491	−	0.963560i	$-0.413805\pi$
$47$	3.66765	0.534982	0.267491	+	0.963560i	$0.413805\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−9.57245	−1.34041
$52$	0	0
$53$	8.54635	1.17393	0.586966	−	0.809612i	$-0.300322\pi$
$53$	8.54635	1.17393	0.586966	+	0.809612i	$0.300322\pi$
$54$	0	0
$55$	0.727033	0.0980331
$56$	0	0
$57$	−2.16118	−0.286255
$58$	0	0
$59$	1.47245	0.191696	0.0958481	−	0.995396i	$-0.469444\pi$
$59$	1.47245	0.191696	0.0958481	+	0.995396i	$0.469444\pi$
$60$	0	0
$61$	−6.23371	−0.798145	−0.399072	−	0.916919i	$-0.630668\pi$
$61$	−6.23371	−0.798145	−0.399072	+	0.916919i	$0.630668\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.12350	−1.00760
$66$	0	0
$67$	3.91517	0.478314	0.239157	−	0.970981i	$-0.423129\pi$
$67$	3.91517	0.478314	0.239157	+	0.970981i	$0.423129\pi$
$68$	0	0
$69$	−5.17281	−0.622733
$70$	0	0
$71$	−10.1617	−1.20597	−0.602987	−	0.797751i	$-0.706023\pi$
$71$	−10.1617	−1.20597	−0.602987	+	0.797751i	$0.706023\pi$
$72$	0	0
$73$	0.775836	0.0908048	0.0454024	−	0.998969i	$-0.485543\pi$
$73$	0.775836	0.0908048	0.0454024	+	0.998969i	$0.485543\pi$
$74$	0	0
$75$	−6.37881	−0.736562
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.98543	−1.12345	−0.561724	−	0.827325i	$-0.689862\pi$
$79$	−9.98543	−1.12345	−0.561724	+	0.827325i	$0.689862\pi$
$80$	0	0
$81$	−10.9384	−1.21538
$82$	0	0
$83$	−7.68368	−0.843393	−0.421697	−	0.906737i	$-0.638565\pi$
$83$	−7.68368	−0.843393	−0.421697	+	0.906737i	$0.638565\pi$
$84$	0	0
$85$	6.26081	0.679080
$86$	0	0
$87$	2.83043	0.303454
$88$	0	0
$89$	−6.83079	−0.724063	−0.362031	−	0.932166i	$-0.617917\pi$
$89$	−6.83079	−0.724063	−0.362031	+	0.932166i	$0.617917\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−18.5878	−1.92746
$94$	0	0
$95$	1.41351	0.145023
$96$	0	0
$97$	−6.83506	−0.693995	−0.346998	−	0.937866i	$-0.612799\pi$
$97$	−6.83506	−0.693995	−0.346998	+	0.937866i	$0.612799\pi$
$98$	0	0
$99$	−1.01726	−0.102238
$100$	0	0
$101$	2.49543	0.248305	0.124152	−	0.992263i	$-0.460379\pi$
$101$	2.49543	0.248305	0.124152	+	0.992263i	$0.460379\pi$
$102$	0	0
$103$	−2.27911	−0.224567	−0.112284	−	0.993676i	$-0.535817\pi$
$103$	−2.27911	−0.224567	−0.112284	+	0.993676i	$0.535817\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.05987	−0.585830	−0.292915	−	0.956138i	$-0.594625\pi$
$107$	−6.05987	−0.585830	−0.292915	+	0.956138i	$0.594625\pi$
$108$	0	0
$109$	−5.55318	−0.531898	−0.265949	−	0.963987i	$-0.585685\pi$
$109$	−5.55318	−0.531898	−0.265949	+	0.963987i	$0.585685\pi$
$110$	0	0
$111$	−18.5010	−1.75603
$112$	0	0
$113$	9.60020	0.903111	0.451555	−	0.892243i	$-0.350869\pi$
$113$	9.60020	0.903111	0.451555	+	0.892243i	$0.350869\pi$
$114$	0	0
$115$	3.38325	0.315490
$116$	0	0
$117$	11.3663	1.05082
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.7557	−0.977792
$122$	0	0
$123$	2.24904	0.202789
$124$	0	0
$125$	11.5269	1.03100
$126$	0	0
$127$	4.29583	0.381193	0.190596	−	0.981668i	$-0.438958\pi$
$127$	4.29583	0.381193	0.190596	+	0.981668i	$0.438958\pi$
$128$	0	0
$129$	12.0004	1.05658
$130$	0	0
$131$	12.8643	1.12396	0.561981	−	0.827150i	$-0.310039\pi$
$131$	12.8643	1.12396	0.561981	+	0.827150i	$0.310039\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.11582	0.268167
$136$	0	0
$137$	−19.8399	−1.69504	−0.847520	−	0.530764i	$-0.821905\pi$
$137$	−19.8399	−1.69504	−0.847520	+	0.530764i	$0.821905\pi$
$138$	0	0
$139$	10.3831	0.880685	0.440342	−	0.897830i	$-0.354857\pi$
$139$	10.3831	0.880685	0.440342	+	0.897830i	$0.354857\pi$
$140$	0	0
$141$	8.24869	0.694665
$142$	0	0
$143$	−2.72953	−0.228255
$144$	0	0
$145$	−1.85123	−0.153736
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.3985	−1.34342	−0.671710	−	0.740815i	$-0.734440\pi$
$149$	−16.3985	−1.34342	−0.671710	+	0.740815i	$0.734440\pi$
$150$	0	0
$151$	−2.89266	−0.235402	−0.117701	−	0.993049i	$-0.537552\pi$
$151$	−2.89266	−0.235402	−0.117701	+	0.993049i	$0.537552\pi$
$152$	0	0
$153$	−8.76008	−0.708211
$154$	0	0
$155$	12.1572	0.976494
$156$	0	0
$157$	−21.5181	−1.71733	−0.858665	−	0.512537i	$-0.828706\pi$
$157$	−21.5181	−1.71733	−0.858665	+	0.512537i	$0.828706\pi$
$158$	0	0
$159$	19.2211	1.52433
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.03732	0.316228	0.158114	−	0.987421i	$-0.449459\pi$
$163$	4.03732	0.316228	0.158114	+	0.987421i	$0.449459\pi$
$164$	0	0
$165$	1.63512	0.127294
$166$	0	0
$167$	−15.8080	−1.22326	−0.611630	−	0.791144i	$-0.709486\pi$
$167$	−15.8080	−1.22326	−0.611630	+	0.791144i	$0.709486\pi$
$168$	0	0
$169$	17.4984	1.34603
$170$	0	0
$171$	−1.97777	−0.151244
$172$	0	0
$173$	13.9471	1.06038	0.530190	−	0.847879i	$-0.322121\pi$
$173$	13.9471	1.06038	0.530190	+	0.847879i	$0.322121\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.31159	0.248914
$178$	0	0
$179$	16.4518	1.22967	0.614833	−	0.788657i	$-0.289223\pi$
$179$	16.4518	1.22967	0.614833	+	0.788657i	$0.289223\pi$
$180$	0	0
$181$	1.78054	0.132347	0.0661733	−	0.997808i	$-0.478921\pi$
$181$	1.78054	0.132347	0.0661733	+	0.997808i	$0.478921\pi$
$182$	0	0
$183$	−14.0198	−1.03638
$184$	0	0
$185$	12.1005	0.889644
$186$	0	0
$187$	2.10366	0.153835
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.2536	1.61021	0.805107	−	0.593130i	$-0.202108\pi$
$191$	22.2536	1.61021	0.805107	+	0.593130i	$0.202108\pi$
$192$	0	0
$193$	−4.34825	−0.312994	−0.156497	−	0.987678i	$-0.550020\pi$
$193$	−4.34825	−0.312994	−0.156497	+	0.987678i	$0.550020\pi$
$194$	0	0
$195$	−18.2701	−1.30835
$196$	0	0
$197$	−18.5511	−1.32171	−0.660854	−	0.750514i	$-0.729806\pi$
$197$	−18.5511	−1.32171	−0.660854	+	0.750514i	$0.729806\pi$
$198$	0	0
$199$	−6.02988	−0.427447	−0.213724	−	0.976894i	$-0.568559\pi$
$199$	−6.02988	−0.427447	−0.213724	+	0.976894i	$0.568559\pi$
$200$	0	0
$201$	8.80536	0.621082
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.47097	−0.102737
$206$	0	0
$207$	−4.73382	−0.329023
$208$	0	0
$209$	0.474945	0.0328526
$210$	0	0
$211$	−7.97038	−0.548704	−0.274352	−	0.961629i	$-0.588463\pi$
$211$	−7.97038	−0.548704	−0.274352	+	0.961629i	$0.588463\pi$
$212$	0	0
$213$	−22.8541	−1.56594
$214$	0	0
$215$	−7.84880	−0.535284
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.74489	0.117908
$220$	0	0
$221$	−23.5052	−1.58113
$222$	0	0
$223$	−18.1994	−1.21872	−0.609360	−	0.792894i	$-0.708573\pi$
$223$	−18.1994	−1.21872	−0.609360	+	0.792894i	$0.708573\pi$
$224$	0	0
$225$	−5.83748	−0.389165
$226$	0	0
$227$	−2.57940	−0.171201	−0.0856004	−	0.996330i	$-0.527281\pi$
$227$	−2.57940	−0.171201	−0.0856004	+	0.996330i	$0.527281\pi$
$228$	0	0
$229$	22.5532	1.49036	0.745180	−	0.666863i	$-0.232363\pi$
$229$	22.5532	1.49036	0.745180	+	0.666863i	$0.232363\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.54988	−0.101536	−0.0507682	−	0.998710i	$-0.516167\pi$
$233$	−1.54988	−0.101536	−0.0507682	+	0.998710i	$0.516167\pi$
$234$	0	0
$235$	−5.39502	−0.351932
$236$	0	0
$237$	−22.4576	−1.45878
$238$	0	0
$239$	16.3710	1.05895	0.529477	−	0.848324i	$-0.322388\pi$
$239$	16.3710	1.05895	0.529477	+	0.848324i	$0.322388\pi$
$240$	0	0
$241$	11.7603	0.757546	0.378773	−	0.925490i	$-0.376346\pi$
$241$	11.7603	0.757546	0.378773	+	0.925490i	$0.376346\pi$
$242$	0	0
$243$	−18.2464	−1.17051
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.30680	−0.337663
$248$	0	0
$249$	−17.2809	−1.09513
$250$	0	0
$251$	15.1858	0.958517	0.479259	−	0.877674i	$-0.340906\pi$
$251$	15.1858	0.958517	0.479259	+	0.877674i	$0.340906\pi$
$252$	0	0
$253$	1.13679	0.0714692
$254$	0	0
$255$	14.0808	0.881774
$256$	0	0
$257$	0.0946877	0.00590645	0.00295323	−	0.999996i	$-0.499060\pi$
$257$	0.0946877	0.00590645	0.00295323	+	0.999996i	$0.499060\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.59023	0.160331
$262$	0	0
$263$	17.9175	1.10484	0.552420	−	0.833566i	$-0.313705\pi$
$263$	17.9175	1.10484	0.552420	+	0.833566i	$0.313705\pi$
$264$	0	0
$265$	−12.5714	−0.772258
$266$	0	0
$267$	−15.3627	−0.940183
$268$	0	0
$269$	32.3561	1.97279	0.986393	−	0.164406i	$-0.0525706\pi$
$269$	32.3561	1.97279	0.986393	+	0.164406i	$0.0525706\pi$
$270$	0	0
$271$	−16.9810	−1.03152	−0.515762	−	0.856732i	$-0.672491\pi$
$271$	−16.9810	−1.03152	−0.515762	+	0.856732i	$0.672491\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.40182	0.0845329
$276$	0	0
$277$	−0.422608	−0.0253921	−0.0126960	−	0.999919i	$-0.504041\pi$
$277$	−0.422608	−0.0253921	−0.0126960	+	0.999919i	$0.504041\pi$
$278$	0	0
$279$	−17.0103	−1.01838
$280$	0	0
$281$	−10.4772	−0.625016	−0.312508	−	0.949915i	$-0.601169\pi$
$281$	−10.4772	−0.625016	−0.312508	+	0.949915i	$0.601169\pi$
$282$	0	0
$283$	−27.5343	−1.63674	−0.818372	−	0.574689i	$-0.805123\pi$
$283$	−27.5343	−1.63674	−0.818372	+	0.574689i	$0.805123\pi$
$284$	0	0
$285$	3.17904	0.188310
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.11558	0.0656225
$290$	0	0
$291$	−15.3723	−0.901141
$292$	0	0
$293$	22.6387	1.32257	0.661283	−	0.750137i	$-0.270012\pi$
$293$	22.6387	1.32257	0.661283	+	0.750137i	$0.270012\pi$
$294$	0	0
$295$	−2.16593	−0.126105
$296$	0	0
$297$	1.04693	0.0607489
$298$	0	0
$299$	−12.7019	−0.734570
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.61233	0.322420
$304$	0	0
$305$	9.16961	0.525051
$306$	0	0
$307$	−9.00706	−0.514060	−0.257030	−	0.966403i	$-0.582744\pi$
$307$	−9.00706	−0.514060	−0.257030	+	0.966403i	$0.582744\pi$
$308$	0	0
$309$	−5.12580	−0.291597
$310$	0	0
$311$	−25.8029	−1.46315	−0.731573	−	0.681763i	$-0.761213\pi$
$311$	−25.8029	−1.46315	−0.731573	+	0.681763i	$0.761213\pi$
$312$	0	0
$313$	−5.30641	−0.299936	−0.149968	−	0.988691i	$-0.547917\pi$
$313$	−5.30641	−0.299936	−0.149968	+	0.988691i	$0.547917\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.73715	0.153734	0.0768668	−	0.997041i	$-0.475508\pi$
$317$	2.73715	0.153734	0.0768668	+	0.997041i	$0.475508\pi$
$318$	0	0
$319$	−0.622022	−0.0348265
$320$	0	0
$321$	−13.6289	−0.760690
$322$	0	0
$323$	4.08997	0.227572
$324$	0	0
$325$	−15.6632	−0.868840
$326$	0	0
$327$	−12.4893	−0.690660
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11.5489	−0.634787	−0.317394	−	0.948294i	$-0.602808\pi$
$331$	−11.5489	−0.634787	−0.317394	+	0.948294i	$0.602808\pi$
$332$	0	0
$333$	−16.9309	−0.927807
$334$	0	0
$335$	−5.75910	−0.314653
$336$	0	0
$337$	−31.6744	−1.72541	−0.862706	−	0.505705i	$-0.831232\pi$
$337$	−31.6744	−1.72541	−0.862706	+	0.505705i	$0.831232\pi$
$338$	0	0
$339$	21.5912	1.17267
$340$	0	0
$341$	4.08489	0.221209
$342$	0	0
$343$	0	0
$344$	0	0
$345$	7.60906	0.409658
$346$	0	0
$347$	−8.14098	−0.437031	−0.218515	−	0.975833i	$-0.570121\pi$
$347$	−8.14098	−0.437031	−0.218515	+	0.975833i	$0.570121\pi$
$348$	0	0
$349$	4.07115	0.217924	0.108962	−	0.994046i	$-0.465247\pi$
$349$	4.07115	0.217924	0.108962	+	0.994046i	$0.465247\pi$
$350$	0	0
$351$	−11.6979	−0.624385
$352$	0	0
$353$	30.5592	1.62650	0.813252	−	0.581912i	$-0.197695\pi$
$353$	30.5592	1.62650	0.813252	+	0.581912i	$0.197695\pi$
$354$	0	0
$355$	14.9476	0.793337
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.0102	−1.53110	−0.765550	−	0.643376i	$-0.777533\pi$
$359$	−29.0102	−1.53110	−0.765550	+	0.643376i	$0.777533\pi$
$360$	0	0
$361$	−18.0766	−0.951400
$362$	0	0
$363$	−24.1900	−1.26965
$364$	0	0
$365$	−1.14123	−0.0597349
$366$	0	0
$367$	16.4440	0.858369	0.429185	−	0.903217i	$-0.358801\pi$
$367$	16.4440	0.858369	0.429185	+	0.903217i	$0.358801\pi$
$368$	0	0
$369$	2.05817	0.107144
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.4364	1.36883	0.684413	−	0.729095i	$-0.260059\pi$
$373$	26.4364	1.36883	0.684413	+	0.729095i	$0.260059\pi$
$374$	0	0
$375$	25.9244	1.33873
$376$	0	0
$377$	6.95016	0.357951
$378$	0	0
$379$	−15.6085	−0.801756	−0.400878	−	0.916132i	$-0.631295\pi$
$379$	−15.6085	−0.801756	−0.400878	+	0.916132i	$0.631295\pi$
$380$	0	0
$381$	9.66148	0.494973
$382$	0	0
$383$	−3.36472	−0.171929	−0.0859645	−	0.996298i	$-0.527397\pi$
$383$	−3.36472	−0.171929	−0.0859645	+	0.996298i	$0.527397\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.9820	0.558246
$388$	0	0
$389$	−1.22512	−0.0621159	−0.0310579	−	0.999518i	$-0.509888\pi$
$389$	−1.22512	−0.0621159	−0.0310579	+	0.999518i	$0.509888\pi$
$390$	0	0
$391$	9.78940	0.495071
$392$	0	0
$393$	28.9324	1.45945
$394$	0	0
$395$	14.6883	0.739048
$396$	0	0
$397$	36.5319	1.83348	0.916742	−	0.399481i	$-0.130810\pi$
$397$	36.5319	1.83348	0.916742	+	0.399481i	$0.130810\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−29.9865	−1.49746	−0.748728	−	0.662877i	$-0.769335\pi$
$401$	−29.9865	−1.49746	−0.748728	+	0.662877i	$0.769335\pi$
$402$	0	0
$403$	−45.6425	−2.27361
$404$	0	0
$405$	16.0901	0.799526
$406$	0	0
$407$	4.06581	0.201535
$408$	0	0
$409$	−37.0251	−1.83077	−0.915386	−	0.402578i	$-0.868114\pi$
$409$	−37.0251	−1.83077	−0.915386	+	0.402578i	$0.868114\pi$
$410$	0	0
$411$	−44.6208	−2.20098
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11.3025	0.554817
$416$	0	0
$417$	23.3520	1.14355
$418$	0	0
$419$	24.4042	1.19222	0.596112	−	0.802901i	$-0.296711\pi$
$419$	24.4042	1.19222	0.596112	+	0.802901i	$0.296711\pi$
$420$	0	0
$421$	−2.51451	−0.122550	−0.0612750	−	0.998121i	$-0.519517\pi$
$421$	−2.51451	−0.122550	−0.0612750	+	0.998121i	$0.519517\pi$
$422$	0	0
$423$	7.54867	0.367029
$424$	0	0
$425$	12.0717	0.585564
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6.13882	−0.296385
$430$	0	0
$431$	17.8325	0.858961	0.429481	−	0.903076i	$-0.358697\pi$
$431$	17.8325	0.858961	0.429481	+	0.903076i	$0.358697\pi$
$432$	0	0
$433$	−26.2172	−1.25992	−0.629960	−	0.776628i	$-0.716929\pi$
$433$	−26.2172	−1.25992	−0.629960	+	0.776628i	$0.716929\pi$
$434$	0	0
$435$	−4.16349	−0.199624
$436$	0	0
$437$	2.21016	0.105726
$438$	0	0
$439$	12.0661	0.575881	0.287941	−	0.957648i	$-0.407029\pi$
$439$	12.0661	0.575881	0.287941	+	0.957648i	$0.407029\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.1899	−1.48187	−0.740937	−	0.671574i	$-0.765619\pi$
$443$	−31.1899	−1.48187	−0.740937	+	0.671574i	$0.765619\pi$
$444$	0	0
$445$	10.0479	0.476316
$446$	0	0
$447$	−36.8809	−1.74441
$448$	0	0
$449$	−12.5606	−0.592770	−0.296385	−	0.955069i	$-0.595781\pi$
$449$	−12.5606	−0.592770	−0.296385	+	0.955069i	$0.595781\pi$
$450$	0	0
$451$	−0.494253	−0.0232735
$452$	0	0
$453$	−6.50571	−0.305665
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.209051	−0.00977900	−0.00488950	−	0.999988i	$-0.501556\pi$
$457$	−0.209051	−0.00977900	−0.00488950	+	0.999988i	$0.501556\pi$
$458$	0	0
$459$	9.01558	0.420811
$460$	0	0
$461$	−10.3033	−0.479873	−0.239936	−	0.970789i	$-0.577127\pi$
$461$	−10.3033	−0.479873	−0.239936	+	0.970789i	$0.577127\pi$
$462$	0	0
$463$	−8.65140	−0.402065	−0.201032	−	0.979585i	$-0.564430\pi$
$463$	−8.65140	−0.402065	−0.201032	+	0.979585i	$0.564430\pi$
$464$	0	0
$465$	27.3421	1.26796
$466$	0	0
$467$	22.7584	1.05313	0.526566	−	0.850134i	$-0.323479\pi$
$467$	22.7584	1.05313	0.526566	+	0.850134i	$0.323479\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−48.3950	−2.22992
$472$	0	0
$473$	−2.63723	−0.121260
$474$	0	0
$475$	2.72544	0.125052
$476$	0	0
$477$	17.5899	0.805385
$478$	0	0
$479$	1.76991	0.0808691	0.0404345	−	0.999182i	$-0.487126\pi$
$479$	1.76991	0.0808691	0.0404345	+	0.999182i	$0.487126\pi$
$480$	0	0
$481$	−45.4293	−2.07140
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.0542	0.456537
$486$	0	0
$487$	−27.1322	−1.22948	−0.614738	−	0.788731i	$-0.710738\pi$
$487$	−27.1322	−1.22948	−0.614738	+	0.788731i	$0.710738\pi$
$488$	0	0
$489$	9.08010	0.410616
$490$	0	0
$491$	−0.410698	−0.0185345	−0.00926727	−	0.999957i	$-0.502950\pi$
$491$	−0.410698	−0.0185345	−0.00926727	+	0.999957i	$0.502950\pi$
$492$	0	0
$493$	−5.35651	−0.241245
$494$	0	0
$495$	1.49636	0.0672564
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.2052	−0.904509	−0.452255	−	0.891889i	$-0.649380\pi$
$499$	−20.2052	−0.904509	−0.452255	+	0.891889i	$0.649380\pi$
$500$	0	0
$501$	−35.5528	−1.58838
$502$	0	0
$503$	−38.0193	−1.69520	−0.847599	−	0.530638i	$-0.821952\pi$
$503$	−38.0193	−1.69520	−0.847599	+	0.530638i	$0.821952\pi$
$504$	0	0
$505$	−3.67071	−0.163345
$506$	0	0
$507$	39.3546	1.74780
$508$	0	0
$509$	43.2599	1.91746	0.958730	−	0.284317i	$-0.0917669\pi$
$509$	43.2599	1.91746	0.958730	+	0.284317i	$0.0917669\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.03545	0.0898675
$514$	0	0
$515$	3.35251	0.147729
$516$	0	0
$517$	−1.81275	−0.0797246
$518$	0	0
$519$	31.3676	1.37689
$520$	0	0
$521$	4.70419	0.206094	0.103047	−	0.994676i	$-0.467141\pi$
$521$	4.70419	0.206094	0.103047	+	0.994676i	$0.467141\pi$
$522$	0	0
$523$	17.1882	0.751588	0.375794	−	0.926703i	$-0.377370\pi$
$523$	17.1882	0.751588	0.375794	+	0.926703i	$0.377370\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	35.1768	1.53233
$528$	0	0
$529$	−17.7100	−0.769998
$530$	0	0
$531$	3.03055	0.131515
$532$	0	0
$533$	5.52254	0.239208
$534$	0	0
$535$	8.91391	0.385382
$536$	0	0
$537$	37.0008	1.59670
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−19.4343	−0.835545	−0.417772	−	0.908552i	$-0.637189\pi$
$541$	−19.4343	−0.835545	−0.417772	+	0.908552i	$0.637189\pi$
$542$	0	0
$543$	4.00451	0.171850
$544$	0	0
$545$	8.16857	0.349903
$546$	0	0
$547$	−24.8699	−1.06336	−0.531681	−	0.846945i	$-0.678439\pi$
$547$	−24.8699	−1.06336	−0.531681	+	0.846945i	$0.678439\pi$
$548$	0	0
$549$	−12.8301	−0.547573
$550$	0	0
$551$	−1.20934	−0.0515198
$552$	0	0
$553$	0	0
$554$	0	0
$555$	27.2144	1.15519
$556$	0	0
$557$	21.7680	0.922338	0.461169	−	0.887312i	$-0.347430\pi$
$557$	21.7680	0.922338	0.461169	+	0.887312i	$0.347430\pi$
$558$	0	0
$559$	29.4671	1.24633
$560$	0	0
$561$	4.73121	0.199752
$562$	0	0
$563$	34.7390	1.46407	0.732036	−	0.681265i	$-0.238570\pi$
$563$	34.7390	1.46407	0.732036	+	0.681265i	$0.238570\pi$
$564$	0	0
$565$	−14.1216	−0.594101
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−38.2133	−1.60199	−0.800993	−	0.598674i	$-0.795695\pi$
$569$	−38.2133	−1.60199	−0.800993	+	0.598674i	$0.795695\pi$
$570$	0	0
$571$	−45.6458	−1.91022	−0.955109	−	0.296256i	$-0.904262\pi$
$571$	−45.6458	−1.91022	−0.955109	+	0.296256i	$0.904262\pi$
$572$	0	0
$573$	50.0492	2.09083
$574$	0	0
$575$	6.52338	0.272044
$576$	0	0
$577$	−5.19932	−0.216451	−0.108225	−	0.994126i	$-0.534517\pi$
$577$	−5.19932	−0.216451	−0.108225	+	0.994126i	$0.534517\pi$
$578$	0	0
$579$	−9.77938	−0.406417
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.22406	−0.174943
$584$	0	0
$585$	−16.7196	−0.691269
$586$	0	0
$587$	−12.2599	−0.506020	−0.253010	−	0.967464i	$-0.581421\pi$
$587$	−12.2599	−0.506020	−0.253010	+	0.967464i	$0.581421\pi$
$588$	0	0
$589$	7.94190	0.327240
$590$	0	0
$591$	−41.7220	−1.71622
$592$	0	0
$593$	12.4659	0.511914	0.255957	−	0.966688i	$-0.417609\pi$
$593$	12.4659	0.511914	0.255957	+	0.966688i	$0.417609\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.5614	−0.555033
$598$	0	0
$599$	−42.8908	−1.75247	−0.876235	−	0.481884i	$-0.839953\pi$
$599$	−42.8908	−1.75247	−0.876235	+	0.481884i	$0.839953\pi$
$600$	0	0
$601$	2.40756	0.0982066	0.0491033	−	0.998794i	$-0.484364\pi$
$601$	2.40756	0.0982066	0.0491033	+	0.998794i	$0.484364\pi$
$602$	0	0
$603$	8.05809	0.328151
$604$	0	0
$605$	15.8214	0.643230
$606$	0	0
$607$	−7.03651	−0.285603	−0.142802	−	0.989751i	$-0.545611\pi$
$607$	−7.03651	−0.285603	−0.142802	+	0.989751i	$0.545611\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20.2547	0.819419
$612$	0	0
$613$	30.5473	1.23379	0.616897	−	0.787044i	$-0.288390\pi$
$613$	30.5473	1.23379	0.616897	+	0.787044i	$0.288390\pi$
$614$	0	0
$615$	−3.30827	−0.133402
$616$	0	0
$617$	−16.2455	−0.654018	−0.327009	−	0.945021i	$-0.606041\pi$
$617$	−16.2455	−0.654018	−0.327009	+	0.945021i	$0.606041\pi$
$618$	0	0
$619$	26.3140	1.05765	0.528825	−	0.848731i	$-0.322633\pi$
$619$	26.3140	1.05765	0.528825	+	0.848731i	$0.322633\pi$
$620$	0	0
$621$	4.87189	0.195502
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−2.77454	−0.110982
$626$	0	0
$627$	1.06817	0.0426586
$628$	0	0
$629$	35.0126	1.39604
$630$	0	0
$631$	33.3206	1.32647	0.663237	−	0.748410i	$-0.269182\pi$
$631$	33.3206	1.32647	0.663237	+	0.748410i	$0.269182\pi$
$632$	0	0
$633$	−17.9257	−0.712483
$634$	0	0
$635$	−6.31904	−0.250764
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−20.9146	−0.827368
$640$	0	0
$641$	34.2462	1.35264	0.676322	−	0.736606i	$-0.263573\pi$
$641$	34.2462	1.35264	0.676322	+	0.736606i	$0.263573\pi$
$642$	0	0
$643$	17.7446	0.699777	0.349889	−	0.936791i	$-0.386219\pi$
$643$	17.7446	0.699777	0.349889	+	0.936791i	$0.386219\pi$
$644$	0	0
$645$	−17.6523	−0.695057
$646$	0	0
$647$	14.3513	0.564209	0.282104	−	0.959384i	$-0.408968\pi$
$647$	14.3513	0.564209	0.282104	+	0.959384i	$0.408968\pi$
$648$	0	0
$649$	−0.727761	−0.0285671
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.0966	1.21690	0.608452	−	0.793591i	$-0.291791\pi$
$653$	31.0966	1.21690	0.608452	+	0.793591i	$0.291791\pi$
$654$	0	0
$655$	−18.9231	−0.739386
$656$	0	0
$657$	1.59681	0.0622973
$658$	0	0
$659$	−24.5409	−0.955980	−0.477990	−	0.878365i	$-0.658634\pi$
$659$	−24.5409	−0.955980	−0.477990	+	0.878365i	$0.658634\pi$
$660$	0	0
$661$	26.5985	1.03456	0.517281	−	0.855816i	$-0.326944\pi$
$661$	26.5985	1.03456	0.517281	+	0.855816i	$0.326944\pi$
$662$	0	0
$663$	−52.8642	−2.05308
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.89458	−0.112079
$668$	0	0
$669$	−40.9311	−1.58249
$670$	0	0
$671$	3.08103	0.118942
$672$	0	0
$673$	−20.9351	−0.806988	−0.403494	−	0.914982i	$-0.632204\pi$
$673$	−20.9351	−0.806988	−0.403494	+	0.914982i	$0.632204\pi$
$674$	0	0
$675$	6.00773	0.231238
$676$	0	0
$677$	−2.82687	−0.108645	−0.0543227	−	0.998523i	$-0.517300\pi$
$677$	−2.82687	−0.108645	−0.0543227	+	0.998523i	$0.517300\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5.80117	−0.222301
$682$	0	0
$683$	14.7116	0.562922	0.281461	−	0.959573i	$-0.409181\pi$
$683$	14.7116	0.562922	0.281461	+	0.959573i	$0.409181\pi$
$684$	0	0
$685$	29.1840	1.11506
$686$	0	0
$687$	50.7231	1.93521
$688$	0	0
$689$	47.1975	1.79808
$690$	0	0
$691$	9.48370	0.360777	0.180388	−	0.983595i	$-0.442265\pi$
$691$	9.48370	0.360777	0.180388	+	0.983595i	$0.442265\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.2733	−0.579349
$696$	0	0
$697$	−4.25624	−0.161217
$698$	0	0
$699$	−3.48575	−0.131843
$700$	0	0
$701$	6.62883	0.250367	0.125184	−	0.992134i	$-0.460048\pi$
$701$	6.62883	0.250367	0.125184	+	0.992134i	$0.460048\pi$
$702$	0	0
$703$	7.90481	0.298136
$704$	0	0
$705$	−12.1336	−0.456978
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−29.4597	−1.10638	−0.553191	−	0.833054i	$-0.686590\pi$
$709$	−29.4597	−1.10638	−0.553191	+	0.833054i	$0.686590\pi$
$710$	0	0
$711$	−20.5517	−0.770751
$712$	0	0
$713$	19.0091	0.711895
$714$	0	0
$715$	4.01506	0.150155
$716$	0	0
$717$	36.8191	1.37503
$718$	0	0
$719$	16.0532	0.598682	0.299341	−	0.954146i	$-0.403233\pi$
$719$	16.0532	0.598682	0.299341	+	0.954146i	$0.403233\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	26.4493	0.983661
$724$	0	0
$725$	−3.56943	−0.132565
$726$	0	0
$727$	41.7494	1.54840	0.774199	−	0.632942i	$-0.218153\pi$
$727$	41.7494	1.54840	0.774199	+	0.632942i	$0.218153\pi$
$728$	0	0
$729$	−8.22146	−0.304498
$730$	0	0
$731$	−22.7104	−0.839975
$732$	0	0
$733$	15.2698	0.564003	0.282002	−	0.959414i	$-0.409002\pi$
$733$	15.2698	0.564003	0.282002	+	0.959414i	$0.409002\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.93508	−0.0712797
$738$	0	0
$739$	51.4608	1.89302	0.946508	−	0.322680i	$-0.104584\pi$
$739$	51.4608	1.89302	0.946508	+	0.322680i	$0.104584\pi$
$740$	0	0
$741$	−11.9352	−0.438450
$742$	0	0
$743$	−14.2055	−0.521148	−0.260574	−	0.965454i	$-0.583912\pi$
$743$	−14.2055	−0.521148	−0.260574	+	0.965454i	$0.583912\pi$
$744$	0	0
$745$	24.1218	0.883753
$746$	0	0
$747$	−15.8143	−0.578617
$748$	0	0
$749$	0	0
$750$	0	0
$751$	48.1537	1.75715	0.878576	−	0.477602i	$-0.158494\pi$
$751$	48.1537	1.75715	0.878576	+	0.477602i	$0.158494\pi$
$752$	0	0
$753$	34.1534	1.24462
$754$	0	0
$755$	4.25503	0.154856
$756$	0	0
$757$	−20.7677	−0.754816	−0.377408	−	0.926047i	$-0.623185\pi$
$757$	−20.7677	−0.754816	−0.377408	+	0.926047i	$0.623185\pi$
$758$	0	0
$759$	2.55668	0.0928015
$760$	0	0
$761$	−7.80893	−0.283073	−0.141537	−	0.989933i	$-0.545204\pi$
$761$	−7.80893	−0.283073	−0.141537	+	0.989933i	$0.545204\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	12.8858	0.465889
$766$	0	0
$767$	8.13164	0.293617
$768$	0	0
$769$	−6.74705	−0.243305	−0.121652	−	0.992573i	$-0.538819\pi$
$769$	−6.74705	−0.243305	−0.121652	+	0.992573i	$0.538819\pi$
$770$	0	0
$771$	0.212956	0.00766943
$772$	0	0
$773$	−24.0422	−0.864738	−0.432369	−	0.901697i	$-0.642322\pi$
$773$	−24.0422	−0.864738	−0.432369	+	0.901697i	$0.642322\pi$
$774$	0	0
$775$	23.4409	0.842021
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.960935	−0.0344291
$780$	0	0
$781$	5.02246	0.179718
$782$	0	0
$783$	−2.66578	−0.0952671
$784$	0	0
$785$	31.6525	1.12973
$786$	0	0
$787$	32.7427	1.16715	0.583576	−	0.812059i	$-0.301653\pi$
$787$	32.7427	1.16715	0.583576	+	0.812059i	$0.301653\pi$
$788$	0	0
$789$	40.2971	1.43462
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−34.4259	−1.22250
$794$	0	0
$795$	−28.2737	−1.00276
$796$	0	0
$797$	−50.3514	−1.78354	−0.891769	−	0.452491i	$-0.850535\pi$
$797$	−50.3514	−1.78354	−0.891769	+	0.452491i	$0.850535\pi$
$798$	0	0
$799$	−15.6104	−0.552257
$800$	0	0
$801$	−14.0590	−0.496749
$802$	0	0
$803$	−0.383459	−0.0135320
$804$	0	0
$805$	0	0
$806$	0	0
$807$	72.7701	2.56163
$808$	0	0
$809$	40.3726	1.41942	0.709712	−	0.704492i	$-0.248825\pi$
$809$	40.3726	1.41942	0.709712	+	0.704492i	$0.248825\pi$
$810$	0	0
$811$	46.0779	1.61801	0.809007	−	0.587799i	$-0.200006\pi$
$811$	46.0779	1.61801	0.809007	+	0.587799i	$0.200006\pi$
$812$	0	0
$813$	−38.1910	−1.33942
$814$	0	0
$815$	−5.93879	−0.208027
$816$	0	0
$817$	−5.12735	−0.179383
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.1739	0.564471	0.282236	−	0.959345i	$-0.408924\pi$
$821$	16.1739	0.564471	0.282236	+	0.959345i	$0.408924\pi$
$822$	0	0
$823$	5.62680	0.196138	0.0980689	−	0.995180i	$-0.468733\pi$
$823$	5.62680	0.196138	0.0980689	+	0.995180i	$0.468733\pi$
$824$	0	0
$825$	3.15275	0.109765
$826$	0	0
$827$	−8.75878	−0.304573	−0.152286	−	0.988336i	$-0.548664\pi$
$827$	−8.75878	−0.304573	−0.152286	+	0.988336i	$0.548664\pi$
$828$	0	0
$829$	19.6059	0.680942	0.340471	−	0.940255i	$-0.389413\pi$
$829$	19.6059	0.680942	0.340471	+	0.940255i	$0.389413\pi$
$830$	0	0
$831$	−0.950462	−0.0329712
$832$	0	0
$833$	0	0
$834$	0	0
$835$	23.2531	0.804708
$836$	0	0
$837$	17.5065	0.605112
$838$	0	0
$839$	9.03213	0.311824	0.155912	−	0.987771i	$-0.450168\pi$
$839$	9.03213	0.311824	0.155912	+	0.987771i	$0.450168\pi$
$840$	0	0
$841$	−27.4162	−0.945385
$842$	0	0
$843$	−23.5636	−0.811573
$844$	0	0
$845$	−25.7397	−0.885472
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−61.9257	−2.12528
$850$	0	0
$851$	18.9203	0.648579
$852$	0	0
$853$	−23.1999	−0.794348	−0.397174	−	0.917743i	$-0.630009\pi$
$853$	−23.1999	−0.794348	−0.397174	+	0.917743i	$0.630009\pi$
$854$	0	0
$855$	2.90925	0.0994941
$856$	0	0
$857$	5.56663	0.190153	0.0950763	−	0.995470i	$-0.469691\pi$
$857$	5.56663	0.190153	0.0950763	+	0.995470i	$0.469691\pi$
$858$	0	0
$859$	−26.6041	−0.907719	−0.453860	−	0.891073i	$-0.649953\pi$
$859$	−26.6041	−0.907719	−0.453860	+	0.891073i	$0.649953\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.26472	−0.179213	−0.0896066	−	0.995977i	$-0.528561\pi$
$863$	−5.26472	−0.179213	−0.0896066	+	0.995977i	$0.528561\pi$
$864$	0	0
$865$	−20.5158	−0.697559
$866$	0	0
$867$	2.50899	0.0852097
$868$	0	0
$869$	4.93533	0.167420
$870$	0	0
$871$	21.6217	0.732622
$872$	0	0
$873$	−14.0677	−0.476121
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.8920	1.00938	0.504691	−	0.863300i	$-0.331606\pi$
$877$	29.8920	1.00938	0.504691	+	0.863300i	$0.331606\pi$
$878$	0	0
$879$	50.9153	1.71733
$880$	0	0
$881$	−6.34231	−0.213678	−0.106839	−	0.994276i	$-0.534073\pi$
$881$	−6.34231	−0.213678	−0.106839	+	0.994276i	$0.534073\pi$
$882$	0	0
$883$	−15.9746	−0.537587	−0.268794	−	0.963198i	$-0.586625\pi$
$883$	−15.9746	−0.537587	−0.268794	+	0.963198i	$0.586625\pi$
$884$	0	0
$885$	−4.87126	−0.163745
$886$	0	0
$887$	−13.5568	−0.455192	−0.227596	−	0.973756i	$-0.573087\pi$
$887$	−13.5568	−0.455192	−0.227596	+	0.973756i	$0.573087\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.40636	0.181120
$892$	0	0
$893$	−3.52437	−0.117939
$894$	0	0
$895$	−24.2002	−0.808923
$896$	0	0
$897$	−28.5670	−0.953826
$898$	0	0
$899$	−10.4013	−0.346902
$900$	0	0
$901$	−36.3753	−1.21184
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.61913	−0.0870627
$906$	0	0
$907$	28.4506	0.944688	0.472344	−	0.881414i	$-0.343408\pi$
$907$	28.4506	0.944688	0.472344	+	0.881414i	$0.343408\pi$
$908$	0	0
$909$	5.13604	0.170352
$910$	0	0
$911$	27.8892	0.924011	0.462006	−	0.886877i	$-0.347130\pi$
$911$	27.8892	0.924011	0.462006	+	0.886877i	$0.347130\pi$
$912$	0	0
$913$	3.79768	0.125685
$914$	0	0
$915$	20.6228	0.681769
$916$	0	0
$917$	0	0
$918$	0	0
$919$	29.3422	0.967910	0.483955	−	0.875093i	$-0.339200\pi$
$919$	29.3422	0.967910	0.483955	+	0.875093i	$0.339200\pi$
$920$	0	0
$921$	−20.2572	−0.667499
$922$	0	0
$923$	−56.1185	−1.84716
$924$	0	0
$925$	23.3314	0.767132
$926$	0	0
$927$	−4.69080	−0.154066
$928$	0	0
$929$	45.7393	1.50066	0.750329	−	0.661064i	$-0.229895\pi$
$929$	45.7393	1.50066	0.750329	+	0.661064i	$0.229895\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−58.0316	−1.89987
$934$	0	0
$935$	−3.09443	−0.101199
$936$	0	0
$937$	8.24433	0.269331	0.134665	−	0.990891i	$-0.457004\pi$
$937$	8.24433	0.269331	0.134665	+	0.990891i	$0.457004\pi$
$938$	0	0
$939$	−11.9343	−0.389462
$940$	0	0
$941$	−10.2362	−0.333691	−0.166846	−	0.985983i	$-0.553358\pi$
$941$	−10.2362	−0.333691	−0.166846	+	0.985983i	$0.553358\pi$
$942$	0	0
$943$	−2.30001	−0.0748986
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.0375	0.358669	0.179335	−	0.983788i	$-0.442606\pi$
$947$	11.0375	0.358669	0.179335	+	0.983788i	$0.442606\pi$
$948$	0	0
$949$	4.28459	0.139084
$950$	0	0
$951$	6.15595	0.199620
$952$	0	0
$953$	0.620938	0.0201141	0.0100571	−	0.999949i	$-0.496799\pi$
$953$	0.620938	0.0201141	0.0100571	+	0.999949i	$0.496799\pi$
$954$	0	0
$955$	−32.7344	−1.05926
$956$	0	0
$957$	−1.39895	−0.0452216
$958$	0	0
$959$	0	0
$960$	0	0
$961$	37.3064	1.20343
$962$	0	0
$963$	−12.4723	−0.401913
$964$	0	0
$965$	6.39616	0.205900
$966$	0	0
$967$	15.9164	0.511836	0.255918	−	0.966698i	$-0.417622\pi$
$967$	15.9164	0.511836	0.255918	+	0.966698i	$0.417622\pi$
$968$	0	0
$969$	9.19850	0.295498
$970$	0	0
$971$	−34.7069	−1.11380	−0.556899	−	0.830580i	$-0.688009\pi$
$971$	−34.7069	−1.11380	−0.556899	+	0.830580i	$0.688009\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−35.2272	−1.12817
$976$	0	0
$977$	−24.1455	−0.772484	−0.386242	−	0.922397i	$-0.626227\pi$
$977$	−24.1455	−0.772484	−0.386242	+	0.922397i	$0.626227\pi$
$978$	0	0
$979$	3.37614	0.107902
$980$	0	0
$981$	−11.4294	−0.364913
$982$	0	0
$983$	7.08919	0.226110	0.113055	−	0.993589i	$-0.463936\pi$
$983$	7.08919	0.226110	0.113055	+	0.993589i	$0.463936\pi$
$984$	0	0
$985$	27.2881	0.869471
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.2724	−0.390239
$990$	0	0
$991$	49.1063	1.55991	0.779956	−	0.625834i	$-0.215241\pi$
$991$	49.1063	1.55991	0.779956	+	0.625834i	$0.215241\pi$
$992$	0	0
$993$	−25.9740	−0.824260
$994$	0	0
$995$	8.86979	0.281191
$996$	0	0
$997$	−23.8456	−0.755198	−0.377599	−	0.925969i	$-0.623250\pi$
$997$	−23.8456	−0.755198	−0.377599	+	0.925969i	$0.623250\pi$
$998$	0	0
$999$	17.4247	0.551293

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.m.1.7		8
7.2	even	3		1148.2.i.d.165.2	✓	16
7.4	even	3		1148.2.i.d.821.2	yes	16
7.6	odd	2		8036.2.a.n.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.d.165.2	✓	16	7.2	even	3
1148.2.i.d.821.2	yes	16	7.4	even	3
8036.2.a.m.1.7		8	1.1	even	1	trivial
8036.2.a.n.1.2		8	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)

\(f(q)\)	\(=\)	\(q+2.24904 q^{3} -1.47097 q^{5} +2.05817 q^{9} +O(q^{10})\)	\(q+2.24904 q^{3} -1.47097 q^{5} +2.05817 q^{9} -0.494253 q^{11} +5.52254 q^{13} -3.30827 q^{15} -4.25624 q^{17} -0.960935 q^{19} -2.30001 q^{23} -2.83624 q^{25} -2.11820 q^{27} +1.25851 q^{29} -8.26477 q^{31} -1.11159 q^{33} -8.22617 q^{37} +12.4204 q^{39} +1.00000 q^{41} +5.33579 q^{43} -3.02752 q^{45} +3.66765 q^{47} -9.57245 q^{51} +8.54635 q^{53} +0.727033 q^{55} -2.16118 q^{57} +1.47245 q^{59} -6.23371 q^{61} -8.12350 q^{65} +3.91517 q^{67} -5.17281 q^{69} -10.1617 q^{71} +0.775836 q^{73} -6.37881 q^{75} -9.98543 q^{79} -10.9384 q^{81} -7.68368 q^{83} +6.26081 q^{85} +2.83043 q^{87} -6.83079 q^{89} -18.5878 q^{93} +1.41351 q^{95} -6.83506 q^{97} -1.01726 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * q^99

Introduction

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