LMFDB - Embedded newform 8036.2.a.l.1.2

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:	$5$
Coefficient field:	5.5.470117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + 7x + 1 $ x^5 - 2x^4 - 6x^3 + 8x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.2
Root			$-0.475832$ 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)

$f(q)$	$=$	$q-0.475832 q^{3} -1.19617 q^{5} -2.77358 q^{9} +O(q^{10})$	$q-0.475832 q^{3} -1.19617 q^{5} -2.77358 q^{9} -4.51385 q^{11} -1.46760 q^{13} +0.569176 q^{15} +4.96975 q^{17} +3.69193 q^{19} +4.36269 q^{23} -3.56918 q^{25} +2.74726 q^{27} -3.12850 q^{29} +7.79167 q^{31} +2.14783 q^{33} +9.52861 q^{37} +0.698329 q^{39} -1.00000 q^{41} -11.7158 q^{43} +3.31768 q^{45} -6.93311 q^{47} -2.36477 q^{51} +8.63019 q^{53} +5.39933 q^{55} -1.75674 q^{57} +2.99301 q^{59} -0.624504 q^{61} +1.75549 q^{65} +10.3304 q^{67} -2.07591 q^{69} +1.26235 q^{71} +6.30820 q^{73} +1.69833 q^{75} -10.4848 q^{79} +7.01352 q^{81} +7.11184 q^{83} -5.94467 q^{85} +1.48864 q^{87} +10.5356 q^{89} -3.70753 q^{93} -4.41618 q^{95} -14.6608 q^{97} +12.5195 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} + q^{5} + q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 + q^5 + q^9	$ 5 q + 2 q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - 9 q^{15} + 3 q^{17} + 4 q^{19} - 8 q^{23} - 6 q^{25} + 8 q^{27} - 9 q^{29} + 11 q^{31} - 5 q^{33} - 11 q^{37} - 17 q^{39} - 5 q^{41} - 27 q^{43} + 3 q^{45} + 3 q^{47} - 3 q^{51} - 19 q^{53} + 13 q^{55} - 11 q^{57} + 15 q^{59} + 7 q^{65} - 21 q^{67} - 14 q^{69} - 16 q^{71} + 10 q^{73} - 12 q^{75} - 14 q^{79} - 7 q^{81} + 2 q^{83} - 21 q^{85} + 36 q^{87} - 6 q^{89} + 17 q^{93} + 9 q^{95} - 20 q^{97} - 17 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 + q^5 + q^9 - 2 * q^11 + q^13 - 9 * q^15 + 3 * q^17 + 4 * q^19 - 8 * q^23 - 6 * q^25 + 8 * q^27 - 9 * q^29 + 11 * q^31 - 5 * q^33 - 11 * q^37 - 17 * q^39 - 5 * q^41 - 27 * q^43 + 3 * q^45 + 3 * q^47 - 3 * q^51 - 19 * q^53 + 13 * q^55 - 11 * q^57 + 15 * q^59 + 7 * q^65 - 21 * q^67 - 14 * q^69 - 16 * q^71 + 10 * q^73 - 12 * q^75 - 14 * q^79 - 7 * q^81 + 2 * q^83 - 21 * q^85 + 36 * q^87 - 6 * q^89 + 17 * q^93 + 9 * q^95 - 20 * q^97 - 17 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.475832	−0.274722	−0.137361	−	0.990521i	$-0.543862\pi$
$3$	−0.475832	−0.274722	−0.137361	+	0.990521i	$0.543862\pi$
$4$	0	0
$5$	−1.19617	−0.534944	−0.267472	−	0.963566i	$-0.586188\pi$
$5$	−1.19617	−0.534944	−0.267472	+	0.963566i	$0.586188\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.77358	−0.924528
$10$	0	0
$11$	−4.51385	−1.36098	−0.680488	−	0.732759i	$-0.738232\pi$
$11$	−4.51385	−1.36098	−0.680488	+	0.732759i	$0.738232\pi$
$12$	0	0
$13$	−1.46760	−0.407038	−0.203519	−	0.979071i	$-0.565238\pi$
$13$	−1.46760	−0.407038	−0.203519	+	0.979071i	$0.565238\pi$
$14$	0	0
$15$	0.569176	0.146961
$16$	0	0
$17$	4.96975	1.20534	0.602671	−	0.797990i	$-0.294103\pi$
$17$	4.96975	1.20534	0.602671	+	0.797990i	$0.294103\pi$
$18$	0	0
$19$	3.69193	0.846987	0.423493	−	0.905899i	$-0.360804\pi$
$19$	3.69193	0.846987	0.423493	+	0.905899i	$0.360804\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.36269	0.909683	0.454842	−	0.890572i	$-0.349696\pi$
$23$	4.36269	0.909683	0.454842	+	0.890572i	$0.349696\pi$
$24$	0	0
$25$	−3.56918	−0.713835
$26$	0	0
$27$	2.74726	0.528710
$28$	0	0
$29$	−3.12850	−0.580948	−0.290474	−	0.956883i	$-0.593813\pi$
$29$	−3.12850	−0.580948	−0.290474	+	0.956883i	$0.593813\pi$
$30$	0	0
$31$	7.79167	1.39943	0.699713	−	0.714424i	$-0.253311\pi$
$31$	7.79167	1.39943	0.699713	+	0.714424i	$0.253311\pi$
$32$	0	0
$33$	2.14783	0.373890
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.52861	1.56649	0.783247	−	0.621711i	$-0.213562\pi$
$37$	9.52861	1.56649	0.783247	+	0.621711i	$0.213562\pi$
$38$	0	0
$39$	0.698329	0.111822
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−11.7158	−1.78664	−0.893319	−	0.449424i	$-0.851629\pi$
$43$	−11.7158	−1.78664	−0.893319	+	0.449424i	$0.851629\pi$
$44$	0	0
$45$	3.31768	0.494570
$46$	0	0
$47$	−6.93311	−1.01130	−0.505649	−	0.862739i	$-0.668747\pi$
$47$	−6.93311	−1.01130	−0.505649	+	0.862739i	$0.668747\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.36477	−0.331134
$52$	0	0
$53$	8.63019	1.18545	0.592724	−	0.805406i	$-0.298052\pi$
$53$	8.63019	1.18545	0.592724	+	0.805406i	$0.298052\pi$
$54$	0	0
$55$	5.39933	0.728046
$56$	0	0
$57$	−1.75674	−0.232686
$58$	0	0
$59$	2.99301	0.389656	0.194828	−	0.980837i	$-0.437585\pi$
$59$	2.99301	0.389656	0.194828	+	0.980837i	$0.437585\pi$
$60$	0	0
$61$	−0.624504	−0.0799595	−0.0399798	−	0.999200i	$-0.512729\pi$
$61$	−0.624504	−0.0799595	−0.0399798	+	0.999200i	$0.512729\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.75549	0.217742
$66$	0	0
$67$	10.3304	1.26205	0.631027	−	0.775761i	$-0.282634\pi$
$67$	10.3304	1.26205	0.631027	+	0.775761i	$0.282634\pi$
$68$	0	0
$69$	−2.07591	−0.249910
$70$	0	0
$71$	1.26235	0.149814	0.0749068	−	0.997191i	$-0.476134\pi$
$71$	1.26235	0.149814	0.0749068	+	0.997191i	$0.476134\pi$
$72$	0	0
$73$	6.30820	0.738319	0.369159	−	0.929366i	$-0.379646\pi$
$73$	6.30820	0.738319	0.369159	+	0.929366i	$0.379646\pi$
$74$	0	0
$75$	1.69833	0.196106
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.4848	−1.17964	−0.589819	−	0.807536i	$-0.700801\pi$
$79$	−10.4848	−1.17964	−0.589819	+	0.807536i	$0.700801\pi$
$80$	0	0
$81$	7.01352	0.779280
$82$	0	0
$83$	7.11184	0.780626	0.390313	−	0.920682i	$-0.372367\pi$
$83$	7.11184	0.780626	0.390313	+	0.920682i	$0.372367\pi$
$84$	0	0
$85$	−5.94467	−0.644790
$86$	0	0
$87$	1.48864	0.159599
$88$	0	0
$89$	10.5356	1.11677	0.558386	−	0.829581i	$-0.311421\pi$
$89$	10.5356	1.11677	0.558386	+	0.829581i	$0.311421\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.70753	−0.384453
$94$	0	0
$95$	−4.41618	−0.453090
$96$	0	0
$97$	−14.6608	−1.48858	−0.744292	−	0.667855i	$-0.767213\pi$
$97$	−14.6608	−1.48858	−0.744292	+	0.667855i	$0.767213\pi$
$98$	0	0
$99$	12.5195	1.25826
$100$	0	0
$101$	−1.56697	−0.155919	−0.0779596	−	0.996957i	$-0.524841\pi$
$101$	−1.56697	−0.155919	−0.0779596	+	0.996957i	$0.524841\pi$
$102$	0	0
$103$	0.719686	0.0709127	0.0354564	−	0.999371i	$-0.488712\pi$
$103$	0.719686	0.0709127	0.0354564	+	0.999371i	$0.488712\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.62861	−0.254117	−0.127059	−	0.991895i	$-0.540554\pi$
$107$	−2.62861	−0.254117	−0.127059	+	0.991895i	$0.540554\pi$
$108$	0	0
$109$	−12.6880	−1.21529	−0.607645	−	0.794208i	$-0.707886\pi$
$109$	−12.6880	−1.21529	−0.607645	+	0.794208i	$0.707886\pi$
$110$	0	0
$111$	−4.53402	−0.430350
$112$	0	0
$113$	−10.7838	−1.01446	−0.507229	−	0.861811i	$-0.669330\pi$
$113$	−10.7838	−1.01446	−0.507229	+	0.861811i	$0.669330\pi$
$114$	0	0
$115$	−5.21852	−0.486629
$116$	0	0
$117$	4.07050	0.376318
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.37483	0.852258
$122$	0	0
$123$	0.475832	0.0429043
$124$	0	0
$125$	10.2502	0.916805
$126$	0	0
$127$	−11.1678	−0.990979	−0.495489	−	0.868614i	$-0.665011\pi$
$127$	−11.1678	−0.990979	−0.495489	+	0.868614i	$0.665011\pi$
$128$	0	0
$129$	5.57474	0.490828
$130$	0	0
$131$	−4.60067	−0.401962	−0.200981	−	0.979595i	$-0.564413\pi$
$131$	−4.60067	−0.401962	−0.200981	+	0.979595i	$0.564413\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.28619	−0.282830
$136$	0	0
$137$	3.44168	0.294042	0.147021	−	0.989133i	$-0.453031\pi$
$137$	3.44168	0.294042	0.147021	+	0.989133i	$0.453031\pi$
$138$	0	0
$139$	2.13549	0.181130	0.0905650	−	0.995891i	$-0.471133\pi$
$139$	2.13549	0.181130	0.0905650	+	0.995891i	$0.471133\pi$
$140$	0	0
$141$	3.29900	0.277826
$142$	0	0
$143$	6.62450	0.553969
$144$	0	0
$145$	3.74222	0.310774
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.58702	0.621553	0.310777	−	0.950483i	$-0.399411\pi$
$149$	7.58702	0.621553	0.310777	+	0.950483i	$0.399411\pi$
$150$	0	0
$151$	−11.7612	−0.957112	−0.478556	−	0.878057i	$-0.658840\pi$
$151$	−11.7612	−0.957112	−0.478556	+	0.878057i	$0.658840\pi$
$152$	0	0
$153$	−13.7840	−1.11437
$154$	0	0
$155$	−9.32017	−0.748614
$156$	0	0
$157$	−12.3221	−0.983412	−0.491706	−	0.870761i	$-0.663627\pi$
$157$	−12.3221	−0.983412	−0.491706	+	0.870761i	$0.663627\pi$
$158$	0	0
$159$	−4.10652	−0.325669
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.3974	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$163$	−20.3974	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$164$	0	0
$165$	−2.56918	−0.200010
$166$	0	0
$167$	−6.37176	−0.493062	−0.246531	−	0.969135i	$-0.579291\pi$
$167$	−6.37176	−0.493062	−0.246531	+	0.969135i	$0.579291\pi$
$168$	0	0
$169$	−10.8462	−0.834320
$170$	0	0
$171$	−10.2399	−0.783063
$172$	0	0
$173$	−21.9841	−1.67142	−0.835708	−	0.549174i	$-0.814943\pi$
$173$	−21.9841	−1.67142	−0.835708	+	0.549174i	$0.814943\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.42417	−0.107047
$178$	0	0
$179$	−4.33452	−0.323977	−0.161989	−	0.986793i	$-0.551791\pi$
$179$	−4.33452	−0.323977	−0.161989	+	0.986793i	$0.551791\pi$
$180$	0	0
$181$	−24.8399	−1.84633	−0.923167	−	0.384399i	$-0.874409\pi$
$181$	−24.8399	−1.84633	−0.923167	+	0.384399i	$0.874409\pi$
$182$	0	0
$183$	0.297159	0.0219666
$184$	0	0
$185$	−11.3978	−0.837986
$186$	0	0
$187$	−22.4327	−1.64044
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.5998	1.41819	0.709097	−	0.705111i	$-0.249103\pi$
$191$	19.5998	1.41819	0.709097	+	0.705111i	$0.249103\pi$
$192$	0	0
$193$	19.0722	1.37285	0.686425	−	0.727201i	$-0.259179\pi$
$193$	19.0722	1.37285	0.686425	+	0.727201i	$0.259179\pi$
$194$	0	0
$195$	−0.835321	−0.0598186
$196$	0	0
$197$	−13.9686	−0.995224	−0.497612	−	0.867400i	$-0.665790\pi$
$197$	−13.9686	−0.995224	−0.497612	+	0.867400i	$0.665790\pi$
$198$	0	0
$199$	−10.6845	−0.757402	−0.378701	−	0.925519i	$-0.623629\pi$
$199$	−10.6845	−0.757402	−0.378701	+	0.925519i	$0.623629\pi$
$200$	0	0
$201$	−4.91552	−0.346714
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.19617	0.0835442
$206$	0	0
$207$	−12.1003	−0.841027
$208$	0	0
$209$	−16.6648	−1.15273
$210$	0	0
$211$	8.39554	0.577973	0.288986	−	0.957333i	$-0.406682\pi$
$211$	8.39554	0.577973	0.288986	+	0.957333i	$0.406682\pi$
$212$	0	0
$213$	−0.600667	−0.0411571
$214$	0	0
$215$	14.0141	0.955750
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3.00164	−0.202832
$220$	0	0
$221$	−7.29359	−0.490620
$222$	0	0
$223$	13.0262	0.872299	0.436150	−	0.899874i	$-0.356342\pi$
$223$	13.0262	0.872299	0.436150	+	0.899874i	$0.356342\pi$
$224$	0	0
$225$	9.89941	0.659961
$226$	0	0
$227$	11.5789	0.768519	0.384260	−	0.923225i	$-0.374457\pi$
$227$	11.5789	0.768519	0.384260	+	0.923225i	$0.374457\pi$
$228$	0	0
$229$	14.8025	0.978178	0.489089	−	0.872234i	$-0.337329\pi$
$229$	14.8025	0.978178	0.489089	+	0.872234i	$0.337329\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.2130	1.06215	0.531075	−	0.847325i	$-0.321788\pi$
$233$	16.2130	1.06215	0.531075	+	0.847325i	$0.321788\pi$
$234$	0	0
$235$	8.29318	0.540987
$236$	0	0
$237$	4.98903	0.324072
$238$	0	0
$239$	−16.1097	−1.04205	−0.521025	−	0.853542i	$-0.674450\pi$
$239$	−16.1097	−1.04205	−0.521025	+	0.853542i	$0.674450\pi$
$240$	0	0
$241$	25.3504	1.63296	0.816482	−	0.577372i	$-0.195922\pi$
$241$	25.3504	1.63296	0.816482	+	0.577372i	$0.195922\pi$
$242$	0	0
$243$	−11.5790	−0.742795
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.41826	−0.344756
$248$	0	0
$249$	−3.38404	−0.214455
$250$	0	0
$251$	14.4484	0.911973	0.455987	−	0.889987i	$-0.349286\pi$
$251$	14.4484	0.911973	0.455987	+	0.889987i	$0.349286\pi$
$252$	0	0
$253$	−19.6925	−1.23806
$254$	0	0
$255$	2.82867	0.177138
$256$	0	0
$257$	−30.2527	−1.88711	−0.943557	−	0.331210i	$-0.892543\pi$
$257$	−30.2527	−1.88711	−0.943557	+	0.331210i	$0.892543\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.67716	0.537103
$262$	0	0
$263$	−5.36884	−0.331057	−0.165529	−	0.986205i	$-0.552933\pi$
$263$	−5.36884	−0.331057	−0.165529	+	0.986205i	$0.552933\pi$
$264$	0	0
$265$	−10.3232	−0.634148
$266$	0	0
$267$	−5.01318	−0.306802
$268$	0	0
$269$	−14.9910	−0.914020	−0.457010	−	0.889462i	$-0.651080\pi$
$269$	−14.9910	−0.914020	−0.457010	+	0.889462i	$0.651080\pi$
$270$	0	0
$271$	−14.8762	−0.903662	−0.451831	−	0.892104i	$-0.649229\pi$
$271$	−14.8762	−0.903662	−0.451831	+	0.892104i	$0.649229\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.1107	0.971513
$276$	0	0
$277$	−16.7649	−1.00730	−0.503652	−	0.863907i	$-0.668011\pi$
$277$	−16.7649	−1.00730	−0.503652	+	0.863907i	$0.668011\pi$
$278$	0	0
$279$	−21.6109	−1.29381
$280$	0	0
$281$	9.48615	0.565896	0.282948	−	0.959135i	$-0.408688\pi$
$281$	9.48615	0.565896	0.282948	+	0.959135i	$0.408688\pi$
$282$	0	0
$283$	−23.3364	−1.38721	−0.693603	−	0.720357i	$-0.743978\pi$
$283$	−23.3364	−1.38721	−0.693603	+	0.720357i	$0.743978\pi$
$284$	0	0
$285$	2.10136	0.124474
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.69846	0.452850
$290$	0	0
$291$	6.97610	0.408946
$292$	0	0
$293$	−5.24146	−0.306209	−0.153105	−	0.988210i	$-0.548927\pi$
$293$	−5.24146	−0.306209	−0.153105	+	0.988210i	$0.548927\pi$
$294$	0	0
$295$	−3.58015	−0.208444
$296$	0	0
$297$	−12.4007	−0.719562
$298$	0	0
$299$	−6.40266	−0.370275
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.745614	0.0428344
$304$	0	0
$305$	0.747013	0.0427738
$306$	0	0
$307$	12.8756	0.734851	0.367426	−	0.930053i	$-0.380239\pi$
$307$	12.8756	0.734851	0.367426	+	0.930053i	$0.380239\pi$
$308$	0	0
$309$	−0.342450	−0.0194813
$310$	0	0
$311$	33.5225	1.90089	0.950443	−	0.310899i	$-0.100630\pi$
$311$	33.5225	1.90089	0.950443	+	0.310899i	$0.100630\pi$
$312$	0	0
$313$	8.48895	0.479824	0.239912	−	0.970795i	$-0.422881\pi$
$313$	8.48895	0.479824	0.239912	+	0.970795i	$0.422881\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.70236	−0.544939	−0.272470	−	0.962164i	$-0.587840\pi$
$317$	−9.70236	−0.544939	−0.272470	+	0.962164i	$0.587840\pi$
$318$	0	0
$319$	14.1216	0.790657
$320$	0	0
$321$	1.25078	0.0698116
$322$	0	0
$323$	18.3480	1.02091
$324$	0	0
$325$	5.23811	0.290558
$326$	0	0
$327$	6.03736	0.333867
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.58702	0.0872306	0.0436153	−	0.999048i	$-0.486112\pi$
$331$	1.58702	0.0872306	0.0436153	+	0.999048i	$0.486112\pi$
$332$	0	0
$333$	−26.4284	−1.44827
$334$	0	0
$335$	−12.3569	−0.675128
$336$	0	0
$337$	−4.97001	−0.270734	−0.135367	−	0.990796i	$-0.543221\pi$
$337$	−4.97001	−0.270734	−0.135367	+	0.990796i	$0.543221\pi$
$338$	0	0
$339$	5.13130	0.278694
$340$	0	0
$341$	−35.1704	−1.90459
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.48314	0.133688
$346$	0	0
$347$	13.3942	0.719037	0.359518	−	0.933138i	$-0.382941\pi$
$347$	13.3942	0.719037	0.359518	+	0.933138i	$0.382941\pi$
$348$	0	0
$349$	−15.6258	−0.836432	−0.418216	−	0.908348i	$-0.637345\pi$
$349$	−15.6258	−0.836432	−0.418216	+	0.908348i	$0.637345\pi$
$350$	0	0
$351$	−4.03186	−0.215205
$352$	0	0
$353$	−29.1802	−1.55310	−0.776552	−	0.630053i	$-0.783033\pi$
$353$	−29.1802	−1.55310	−0.776552	+	0.630053i	$0.783033\pi$
$354$	0	0
$355$	−1.50999	−0.0801418
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.2780	1.86190	0.930951	−	0.365144i	$-0.118980\pi$
$359$	35.2780	1.86190	0.930951	+	0.365144i	$0.118980\pi$
$360$	0	0
$361$	−5.36965	−0.282613
$362$	0	0
$363$	−4.46085	−0.234134
$364$	0	0
$365$	−7.54568	−0.394959
$366$	0	0
$367$	36.0756	1.88313	0.941564	−	0.336833i	$-0.109356\pi$
$367$	36.0756	1.88313	0.941564	+	0.336833i	$0.109356\pi$
$368$	0	0
$369$	2.77358	0.144387
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.34975	−0.173444	−0.0867218	−	0.996233i	$-0.527639\pi$
$373$	−3.34975	−0.173444	−0.0867218	+	0.996233i	$0.527639\pi$
$374$	0	0
$375$	−4.87737	−0.251866
$376$	0	0
$377$	4.59137	0.236468
$378$	0	0
$379$	−32.2304	−1.65556	−0.827782	−	0.561050i	$-0.810398\pi$
$379$	−32.2304	−1.65556	−0.827782	+	0.561050i	$0.810398\pi$
$380$	0	0
$381$	5.31398	0.272243
$382$	0	0
$383$	−8.81598	−0.450475	−0.225238	−	0.974304i	$-0.572316\pi$
$383$	−8.81598	−0.450475	−0.225238	+	0.974304i	$0.572316\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	32.4947	1.65180
$388$	0	0
$389$	−20.4606	−1.03739	−0.518697	−	0.854958i	$-0.673583\pi$
$389$	−20.4606	−1.03739	−0.518697	+	0.854958i	$0.673583\pi$
$390$	0	0
$391$	21.6815	1.09648
$392$	0	0
$393$	2.18915	0.110428
$394$	0	0
$395$	12.5417	0.631040
$396$	0	0
$397$	13.4021	0.672631	0.336315	−	0.941749i	$-0.390819\pi$
$397$	13.4021	0.672631	0.336315	+	0.941749i	$0.390819\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.6935	1.18320	0.591598	−	0.806233i	$-0.298497\pi$
$401$	23.6935	1.18320	0.591598	+	0.806233i	$0.298497\pi$
$402$	0	0
$403$	−11.4350	−0.569619
$404$	0	0
$405$	−8.38936	−0.416871
$406$	0	0
$407$	−43.0107	−2.13196
$408$	0	0
$409$	1.63250	0.0807218	0.0403609	−	0.999185i	$-0.487149\pi$
$409$	1.63250	0.0807218	0.0403609	+	0.999185i	$0.487149\pi$
$410$	0	0
$411$	−1.63766	−0.0807799
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−8.50698	−0.417591
$416$	0	0
$417$	−1.01614	−0.0497604
$418$	0	0
$419$	−17.3900	−0.849559	−0.424779	−	0.905297i	$-0.639648\pi$
$419$	−17.3900	−0.849559	−0.424779	+	0.905297i	$0.639648\pi$
$420$	0	0
$421$	−6.25998	−0.305093	−0.152546	−	0.988296i	$-0.548747\pi$
$421$	−6.25998	−0.305093	−0.152546	+	0.988296i	$0.548747\pi$
$422$	0	0
$423$	19.2296	0.934973
$424$	0	0
$425$	−17.7379	−0.860416
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.15215	−0.152187
$430$	0	0
$431$	28.3389	1.36504	0.682519	−	0.730868i	$-0.260885\pi$
$431$	28.3389	1.36504	0.682519	+	0.730868i	$0.260885\pi$
$432$	0	0
$433$	9.51273	0.457153	0.228576	−	0.973526i	$-0.426593\pi$
$433$	9.51273	0.457153	0.228576	+	0.973526i	$0.426593\pi$
$434$	0	0
$435$	−1.78067	−0.0853765
$436$	0	0
$437$	16.1067	0.770490
$438$	0	0
$439$	17.5414	0.837203	0.418601	−	0.908170i	$-0.362520\pi$
$439$	17.5414	0.837203	0.418601	+	0.908170i	$0.362520\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.6698	−1.02956	−0.514781	−	0.857321i	$-0.672127\pi$
$443$	−21.6698	−1.02956	−0.514781	+	0.857321i	$0.672127\pi$
$444$	0	0
$445$	−12.6024	−0.597410
$446$	0	0
$447$	−3.61015	−0.170754
$448$	0	0
$449$	18.9887	0.896133	0.448066	−	0.894000i	$-0.352113\pi$
$449$	18.9887	0.896133	0.448066	+	0.894000i	$0.352113\pi$
$450$	0	0
$451$	4.51385	0.212549
$452$	0	0
$453$	5.59635	0.262939
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.9902	−1.02866	−0.514329	−	0.857593i	$-0.671959\pi$
$457$	−21.9902	−1.02866	−0.514329	+	0.857593i	$0.671959\pi$
$458$	0	0
$459$	13.6532	0.637276
$460$	0	0
$461$	−18.8535	−0.878097	−0.439048	−	0.898463i	$-0.644684\pi$
$461$	−18.8535	−0.878097	−0.439048	+	0.898463i	$0.644684\pi$
$462$	0	0
$463$	−3.20547	−0.148971	−0.0744853	−	0.997222i	$-0.523731\pi$
$463$	−3.20547	−0.148971	−0.0744853	+	0.997222i	$0.523731\pi$
$464$	0	0
$465$	4.43484	0.205661
$466$	0	0
$467$	−6.96252	−0.322187	−0.161093	−	0.986939i	$-0.551502\pi$
$467$	−6.96252	−0.322187	−0.161093	+	0.986939i	$0.551502\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.86326	0.270165
$472$	0	0
$473$	52.8832	2.43157
$474$	0	0
$475$	−13.1771	−0.604609
$476$	0	0
$477$	−23.9366	−1.09598
$478$	0	0
$479$	−19.8629	−0.907560	−0.453780	−	0.891114i	$-0.649925\pi$
$479$	−19.8629	−0.907560	−0.453780	+	0.891114i	$0.649925\pi$
$480$	0	0
$481$	−13.9841	−0.637622
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.5369	0.796308
$486$	0	0
$487$	−8.40158	−0.380712	−0.190356	−	0.981715i	$-0.560964\pi$
$487$	−8.40158	−0.380712	−0.190356	+	0.981715i	$0.560964\pi$
$488$	0	0
$489$	9.70573	0.438908
$490$	0	0
$491$	−12.9454	−0.584218	−0.292109	−	0.956385i	$-0.594357\pi$
$491$	−12.9454	−0.584218	−0.292109	+	0.956385i	$0.594357\pi$
$492$	0	0
$493$	−15.5479	−0.700241
$494$	0	0
$495$	−14.9755	−0.673099
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17.7557	−0.794853	−0.397427	−	0.917634i	$-0.630097\pi$
$499$	−17.7557	−0.794853	−0.397427	+	0.917634i	$0.630097\pi$
$500$	0	0
$501$	3.03189	0.135455
$502$	0	0
$503$	−8.42005	−0.375431	−0.187716	−	0.982223i	$-0.560108\pi$
$503$	−8.42005	−0.375431	−0.187716	+	0.982223i	$0.560108\pi$
$504$	0	0
$505$	1.87436	0.0834080
$506$	0	0
$507$	5.16095	0.229206
$508$	0	0
$509$	27.1793	1.20470	0.602351	−	0.798231i	$-0.294231\pi$
$509$	27.1793	1.20470	0.602351	+	0.798231i	$0.294231\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	10.1427	0.447810
$514$	0	0
$515$	−0.860867	−0.0379343
$516$	0	0
$517$	31.2950	1.37635
$518$	0	0
$519$	10.4607	0.459175
$520$	0	0
$521$	38.4643	1.68515	0.842576	−	0.538578i	$-0.181038\pi$
$521$	38.4643	1.68515	0.842576	+	0.538578i	$0.181038\pi$
$522$	0	0
$523$	38.7671	1.69517	0.847583	−	0.530664i	$-0.178057\pi$
$523$	38.7671	1.69517	0.847583	+	0.530664i	$0.178057\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	38.7227	1.68679
$528$	0	0
$529$	−3.96696	−0.172476
$530$	0	0
$531$	−8.30136	−0.360248
$532$	0	0
$533$	1.46760	0.0635686
$534$	0	0
$535$	3.14427	0.135938
$536$	0	0
$537$	2.06251	0.0890037
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.22511	−0.0526717	−0.0263359	−	0.999653i	$-0.508384\pi$
$541$	−1.22511	−0.0526717	−0.0263359	+	0.999653i	$0.508384\pi$
$542$	0	0
$543$	11.8196	0.507228
$544$	0	0
$545$	15.1770	0.650112
$546$	0	0
$547$	−26.3496	−1.12663	−0.563314	−	0.826243i	$-0.690474\pi$
$547$	−26.3496	−1.12663	−0.563314	+	0.826243i	$0.690474\pi$
$548$	0	0
$549$	1.73211	0.0739248
$550$	0	0
$551$	−11.5502	−0.492055
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.42346	0.230213
$556$	0	0
$557$	−19.8215	−0.839864	−0.419932	−	0.907556i	$-0.637946\pi$
$557$	−19.8215	−0.839864	−0.419932	+	0.907556i	$0.637946\pi$
$558$	0	0
$559$	17.1940	0.727229
$560$	0	0
$561$	10.6742	0.450666
$562$	0	0
$563$	15.7443	0.663543	0.331772	−	0.943360i	$-0.392354\pi$
$563$	15.7443	0.663543	0.331772	+	0.943360i	$0.392354\pi$
$564$	0	0
$565$	12.8993	0.542678
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−43.3943	−1.81918	−0.909592	−	0.415502i	$-0.863606\pi$
$569$	−43.3943	−1.81918	−0.909592	+	0.415502i	$0.863606\pi$
$570$	0	0
$571$	−33.9825	−1.42212	−0.711062	−	0.703129i	$-0.751786\pi$
$571$	−33.9825	−1.42212	−0.711062	+	0.703129i	$0.751786\pi$
$572$	0	0
$573$	−9.32623	−0.389609
$574$	0	0
$575$	−15.5712	−0.649364
$576$	0	0
$577$	−21.1670	−0.881195	−0.440598	−	0.897705i	$-0.645233\pi$
$577$	−21.1670	−0.881195	−0.440598	+	0.897705i	$0.645233\pi$
$578$	0	0
$579$	−9.07519	−0.377152
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−38.9554	−1.61337
$584$	0	0
$585$	−4.86901	−0.201309
$586$	0	0
$587$	−2.78343	−0.114885	−0.0574423	−	0.998349i	$-0.518295\pi$
$587$	−2.78343	−0.114885	−0.0574423	+	0.998349i	$0.518295\pi$
$588$	0	0
$589$	28.7663	1.18530
$590$	0	0
$591$	6.64673	0.273410
$592$	0	0
$593$	−23.5233	−0.965985	−0.482992	−	0.875625i	$-0.660450\pi$
$593$	−23.5233	−0.965985	−0.482992	+	0.875625i	$0.660450\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.08402	0.208075
$598$	0	0
$599$	−20.8380	−0.851417	−0.425709	−	0.904860i	$-0.639975\pi$
$599$	−20.8380	−0.851417	−0.425709	+	0.904860i	$0.639975\pi$
$600$	0	0
$601$	−25.5628	−1.04273	−0.521365	−	0.853334i	$-0.674577\pi$
$601$	−25.5628	−1.04273	−0.521365	+	0.853334i	$0.674577\pi$
$602$	0	0
$603$	−28.6521	−1.16680
$604$	0	0
$605$	−11.2139	−0.455910
$606$	0	0
$607$	−37.2169	−1.51059	−0.755294	−	0.655386i	$-0.772506\pi$
$607$	−37.2169	−1.51059	−0.755294	+	0.655386i	$0.772506\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.1750	0.411636
$612$	0	0
$613$	−31.6927	−1.28006	−0.640028	−	0.768351i	$-0.721077\pi$
$613$	−31.6927	−1.28006	−0.640028	+	0.768351i	$0.721077\pi$
$614$	0	0
$615$	−0.569176	−0.0229514
$616$	0	0
$617$	−24.4239	−0.983270	−0.491635	−	0.870801i	$-0.663601\pi$
$617$	−24.4239	−0.983270	−0.491635	+	0.870801i	$0.663601\pi$
$618$	0	0
$619$	6.01600	0.241803	0.120902	−	0.992664i	$-0.461421\pi$
$619$	6.01600	0.241803	0.120902	+	0.992664i	$0.461421\pi$
$620$	0	0
$621$	11.9854	0.480958
$622$	0	0
$623$	0	0
$624$	0	0
$625$	5.58490	0.223396
$626$	0	0
$627$	7.92966	0.316680
$628$	0	0
$629$	47.3549	1.88816
$630$	0	0
$631$	12.3588	0.491995	0.245997	−	0.969270i	$-0.420885\pi$
$631$	12.3588	0.491995	0.245997	+	0.969270i	$0.420885\pi$
$632$	0	0
$633$	−3.99487	−0.158782
$634$	0	0
$635$	13.3585	0.530118
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.50124	−0.138507
$640$	0	0
$641$	−9.58897	−0.378741	−0.189371	−	0.981906i	$-0.560645\pi$
$641$	−9.58897	−0.378741	−0.189371	+	0.981906i	$0.560645\pi$
$642$	0	0
$643$	2.27552	0.0897379	0.0448689	−	0.998993i	$-0.485713\pi$
$643$	2.27552	0.0897379	0.0448689	+	0.998993i	$0.485713\pi$
$644$	0	0
$645$	−6.66834	−0.262566
$646$	0	0
$647$	11.3295	0.445408	0.222704	−	0.974886i	$-0.428512\pi$
$647$	11.3295	0.445408	0.222704	+	0.974886i	$0.428512\pi$
$648$	0	0
$649$	−13.5100	−0.530313
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.7539	−1.20349	−0.601747	−	0.798687i	$-0.705528\pi$
$653$	−30.7539	−1.20349	−0.601747	+	0.798687i	$0.705528\pi$
$654$	0	0
$655$	5.50318	0.215027
$656$	0	0
$657$	−17.4963	−0.682596
$658$	0	0
$659$	−26.8574	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$659$	−26.8574	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$660$	0	0
$661$	−18.9154	−0.735723	−0.367861	−	0.929881i	$-0.619910\pi$
$661$	−18.9154	−0.735723	−0.367861	+	0.929881i	$0.619910\pi$
$662$	0	0
$663$	3.47052	0.134784
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−13.6487	−0.528479
$668$	0	0
$669$	−6.19829	−0.239640
$670$	0	0
$671$	2.81892	0.108823
$672$	0	0
$673$	−15.5337	−0.598780	−0.299390	−	0.954131i	$-0.596783\pi$
$673$	−15.5337	−0.598780	−0.299390	+	0.954131i	$0.596783\pi$
$674$	0	0
$675$	−9.80545	−0.377412
$676$	0	0
$677$	26.8789	1.03304	0.516521	−	0.856275i	$-0.327227\pi$
$677$	26.8789	1.03304	0.516521	+	0.856275i	$0.327227\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5.50962	−0.211129
$682$	0	0
$683$	−40.7859	−1.56063	−0.780314	−	0.625388i	$-0.784941\pi$
$683$	−40.7859	−1.56063	−0.780314	+	0.625388i	$0.784941\pi$
$684$	0	0
$685$	−4.11683	−0.157296
$686$	0	0
$687$	−7.04351	−0.268727
$688$	0	0
$689$	−12.6656	−0.482522
$690$	0	0
$691$	−45.5508	−1.73283	−0.866417	−	0.499322i	$-0.833583\pi$
$691$	−45.5508	−1.73283	−0.866417	+	0.499322i	$0.833583\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.55441	−0.0968944
$696$	0	0
$697$	−4.96975	−0.188243
$698$	0	0
$699$	−7.71467	−0.291796
$700$	0	0
$701$	12.1849	0.460218	0.230109	−	0.973165i	$-0.426092\pi$
$701$	12.1849	0.460218	0.230109	+	0.973165i	$0.426092\pi$
$702$	0	0
$703$	35.1790	1.32680
$704$	0	0
$705$	−3.94616	−0.148621
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−33.4995	−1.25810	−0.629050	−	0.777365i	$-0.716556\pi$
$709$	−33.4995	−1.25810	−0.629050	+	0.777365i	$0.716556\pi$
$710$	0	0
$711$	29.0806	1.09061
$712$	0	0
$713$	33.9926	1.27303
$714$	0	0
$715$	−7.92404	−0.296342
$716$	0	0
$717$	7.66551	0.286274
$718$	0	0
$719$	−29.3346	−1.09399	−0.546997	−	0.837134i	$-0.684229\pi$
$719$	−29.3346	−1.09399	−0.546997	+	0.837134i	$0.684229\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−12.0625	−0.448611
$724$	0	0
$725$	11.1662	0.414701
$726$	0	0
$727$	−38.9297	−1.44382	−0.721912	−	0.691985i	$-0.756736\pi$
$727$	−38.9297	−1.44382	−0.721912	+	0.691985i	$0.756736\pi$
$728$	0	0
$729$	−15.5309	−0.575218
$730$	0	0
$731$	−58.2245	−2.15351
$732$	0	0
$733$	27.3084	1.00866	0.504328	−	0.863512i	$-0.331740\pi$
$733$	27.3084	1.00866	0.504328	+	0.863512i	$0.331740\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−46.6297	−1.71763
$738$	0	0
$739$	20.6677	0.760273	0.380137	−	0.924930i	$-0.375877\pi$
$739$	20.6677	0.760273	0.380137	+	0.924930i	$0.375877\pi$
$740$	0	0
$741$	2.57818	0.0947119
$742$	0	0
$743$	39.1654	1.43684	0.718420	−	0.695610i	$-0.244866\pi$
$743$	39.1654	1.43684	0.718420	+	0.695610i	$0.244866\pi$
$744$	0	0
$745$	−9.07537	−0.332496
$746$	0	0
$747$	−19.7253	−0.721711
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.61890	0.168546	0.0842731	−	0.996443i	$-0.473143\pi$
$751$	4.61890	0.168546	0.0842731	+	0.996443i	$0.473143\pi$
$752$	0	0
$753$	−6.87500	−0.250539
$754$	0	0
$755$	14.0684	0.512001
$756$	0	0
$757$	−3.91662	−0.142352	−0.0711759	−	0.997464i	$-0.522675\pi$
$757$	−3.91662	−0.142352	−0.0711759	+	0.997464i	$0.522675\pi$
$758$	0	0
$759$	9.37033	0.340121
$760$	0	0
$761$	4.08928	0.148236	0.0741181	−	0.997249i	$-0.476386\pi$
$761$	4.08928	0.148236	0.0741181	+	0.997249i	$0.476386\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	16.4880	0.596127
$766$	0	0
$767$	−4.39252	−0.158605
$768$	0	0
$769$	2.41895	0.0872297	0.0436148	−	0.999048i	$-0.486113\pi$
$769$	2.41895	0.0872297	0.0436148	+	0.999048i	$0.486113\pi$
$770$	0	0
$771$	14.3952	0.518432
$772$	0	0
$773$	22.8874	0.823204	0.411602	−	0.911364i	$-0.364969\pi$
$773$	22.8874	0.823204	0.411602	+	0.911364i	$0.364969\pi$
$774$	0	0
$775$	−27.8099	−0.998960
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.69193	−0.132277
$780$	0	0
$781$	−5.69806	−0.203893
$782$	0	0
$783$	−8.59479	−0.307153
$784$	0	0
$785$	14.7394	0.526070
$786$	0	0
$787$	47.2149	1.68303	0.841516	−	0.540233i	$-0.181664\pi$
$787$	47.2149	1.68303	0.841516	+	0.540233i	$0.181664\pi$
$788$	0	0
$789$	2.55467	0.0909486
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.916519	0.0325465
$794$	0	0
$795$	4.91210	0.174214
$796$	0	0
$797$	−50.4571	−1.78728	−0.893642	−	0.448781i	$-0.851858\pi$
$797$	−50.4571	−1.78728	−0.893642	+	0.448781i	$0.851858\pi$
$798$	0	0
$799$	−34.4558	−1.21896
$800$	0	0
$801$	−29.2214	−1.03249
$802$	0	0
$803$	−28.4742	−1.00483
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.13322	0.251101
$808$	0	0
$809$	−28.8477	−1.01423	−0.507116	−	0.861878i	$-0.669288\pi$
$809$	−28.8477	−1.01423	−0.507116	+	0.861878i	$0.669288\pi$
$810$	0	0
$811$	25.2208	0.885623	0.442811	−	0.896615i	$-0.353981\pi$
$811$	25.2208	0.885623	0.442811	+	0.896615i	$0.353981\pi$
$812$	0	0
$813$	7.07855	0.248256
$814$	0	0
$815$	24.3987	0.854651
$816$	0	0
$817$	−43.2538	−1.51326
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.10919	0.317913	0.158957	−	0.987286i	$-0.449187\pi$
$821$	9.10919	0.317913	0.158957	+	0.987286i	$0.449187\pi$
$822$	0	0
$823$	19.5664	0.682042	0.341021	−	0.940056i	$-0.389227\pi$
$823$	19.5664	0.682042	0.341021	+	0.940056i	$0.389227\pi$
$824$	0	0
$825$	−7.66600	−0.266896
$826$	0	0
$827$	23.8177	0.828224	0.414112	−	0.910226i	$-0.364092\pi$
$827$	23.8177	0.828224	0.414112	+	0.910226i	$0.364092\pi$
$828$	0	0
$829$	47.2073	1.63958	0.819789	−	0.572666i	$-0.194091\pi$
$829$	47.2073	1.63958	0.819789	+	0.572666i	$0.194091\pi$
$830$	0	0
$831$	7.97727	0.276728
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.62171	0.263760
$836$	0	0
$837$	21.4057	0.739890
$838$	0	0
$839$	9.02383	0.311537	0.155769	−	0.987794i	$-0.450215\pi$
$839$	9.02383	0.311537	0.155769	+	0.987794i	$0.450215\pi$
$840$	0	0
$841$	−19.2125	−0.662500
$842$	0	0
$843$	−4.51382	−0.155464
$844$	0	0
$845$	12.9739	0.446314
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.1042	0.381096
$850$	0	0
$851$	41.5704	1.42501
$852$	0	0
$853$	3.32467	0.113835	0.0569173	−	0.998379i	$-0.481873\pi$
$853$	3.32467	0.113835	0.0569173	+	0.998379i	$0.481873\pi$
$854$	0	0
$855$	12.2486	0.418895
$856$	0	0
$857$	30.2358	1.03283	0.516417	−	0.856337i	$-0.327265\pi$
$857$	30.2358	1.03283	0.516417	+	0.856337i	$0.327265\pi$
$858$	0	0
$859$	−48.7343	−1.66279	−0.831397	−	0.555679i	$-0.812458\pi$
$859$	−48.7343	−1.66279	−0.831397	+	0.555679i	$0.812458\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.35784	0.318544	0.159272	−	0.987235i	$-0.449085\pi$
$863$	9.35784	0.318544	0.159272	+	0.987235i	$0.449085\pi$
$864$	0	0
$865$	26.2967	0.894114
$866$	0	0
$867$	−3.66317	−0.124408
$868$	0	0
$869$	47.3270	1.60546
$870$	0	0
$871$	−15.1608	−0.513704
$872$	0	0
$873$	40.6631	1.37624
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−48.4956	−1.63758	−0.818790	−	0.574094i	$-0.805354\pi$
$877$	−48.4956	−1.63758	−0.818790	+	0.574094i	$0.805354\pi$
$878$	0	0
$879$	2.49406	0.0841224
$880$	0	0
$881$	−2.12371	−0.0715495	−0.0357747	−	0.999360i	$-0.511390\pi$
$881$	−2.12371	−0.0715495	−0.0357747	+	0.999360i	$0.511390\pi$
$882$	0	0
$883$	52.6435	1.77159	0.885797	−	0.464072i	$-0.153612\pi$
$883$	52.6435	1.77159	0.885797	+	0.464072i	$0.153612\pi$
$884$	0	0
$885$	1.70355	0.0572642
$886$	0	0
$887$	41.6617	1.39886	0.699432	−	0.714699i	$-0.253436\pi$
$887$	41.6617	1.39886	0.699432	+	0.714699i	$0.253436\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−31.6580	−1.06058
$892$	0	0
$893$	−25.5966	−0.856556
$894$	0	0
$895$	5.18483	0.173310
$896$	0	0
$897$	3.04659	0.101723
$898$	0	0
$899$	−24.3763	−0.812993
$900$	0	0
$901$	42.8899	1.42887
$902$	0	0
$903$	0	0
$904$	0	0
$905$	29.7127	0.987685
$906$	0	0
$907$	−25.5547	−0.848530	−0.424265	−	0.905538i	$-0.639467\pi$
$907$	−25.5547	−0.848530	−0.424265	+	0.905538i	$0.639467\pi$
$908$	0	0
$909$	4.34612	0.144152
$910$	0	0
$911$	53.2918	1.76564	0.882818	−	0.469715i	$-0.155643\pi$
$911$	53.2918	1.76564	0.882818	+	0.469715i	$0.155643\pi$
$912$	0	0
$913$	−32.1018	−1.06241
$914$	0	0
$915$	−0.355453	−0.0117509
$916$	0	0
$917$	0	0
$918$	0	0
$919$	55.0164	1.81482	0.907412	−	0.420242i	$-0.138055\pi$
$919$	55.0164	1.81482	0.907412	+	0.420242i	$0.138055\pi$
$920$	0	0
$921$	−6.12664	−0.201880
$922$	0	0
$923$	−1.85262	−0.0609798
$924$	0	0
$925$	−34.0093	−1.11822
$926$	0	0
$927$	−1.99611	−0.0655608
$928$	0	0
$929$	50.6341	1.66125	0.830625	−	0.556832i	$-0.187983\pi$
$929$	50.6341	1.66125	0.830625	+	0.556832i	$0.187983\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−15.9511	−0.522215
$934$	0	0
$935$	26.8334	0.877545
$936$	0	0
$937$	−8.64462	−0.282407	−0.141204	−	0.989981i	$-0.545097\pi$
$937$	−8.64462	−0.282407	−0.141204	+	0.989981i	$0.545097\pi$
$938$	0	0
$939$	−4.03932	−0.131818
$940$	0	0
$941$	9.76003	0.318168	0.159084	−	0.987265i	$-0.449146\pi$
$941$	9.76003	0.318168	0.159084	+	0.987265i	$0.449146\pi$
$942$	0	0
$943$	−4.36269	−0.142069
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.9190	1.23220	0.616100	−	0.787668i	$-0.288712\pi$
$947$	37.9190	1.23220	0.616100	+	0.787668i	$0.288712\pi$
$948$	0	0
$949$	−9.25788	−0.300523
$950$	0	0
$951$	4.61670	0.149707
$952$	0	0
$953$	16.2393	0.526042	0.263021	−	0.964790i	$-0.415281\pi$
$953$	16.2393	0.526042	0.263021	+	0.964790i	$0.415281\pi$
$954$	0	0
$955$	−23.4447	−0.758653
$956$	0	0
$957$	−6.71950	−0.217211
$958$	0	0
$959$	0	0
$960$	0	0
$961$	29.7102	0.958393
$962$	0	0
$963$	7.29067	0.234939
$964$	0	0
$965$	−22.8136	−0.734397
$966$	0	0
$967$	1.32778	0.0426987	0.0213493	−	0.999772i	$-0.493204\pi$
$967$	1.32778	0.0426987	0.0213493	+	0.999772i	$0.493204\pi$
$968$	0	0
$969$	−8.73056	−0.280466
$970$	0	0
$971$	39.4967	1.26751	0.633755	−	0.773534i	$-0.281513\pi$
$971$	39.4967	1.26751	0.633755	+	0.773534i	$0.281513\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.49246	−0.0798226
$976$	0	0
$977$	−30.3135	−0.969814	−0.484907	−	0.874566i	$-0.661147\pi$
$977$	−30.3135	−0.969814	−0.484907	+	0.874566i	$0.661147\pi$
$978$	0	0
$979$	−47.5561	−1.51990
$980$	0	0
$981$	35.1913	1.12357
$982$	0	0
$983$	−10.7778	−0.343757	−0.171879	−	0.985118i	$-0.554984\pi$
$983$	−10.7778	−0.343757	−0.171879	+	0.985118i	$0.554984\pi$
$984$	0	0
$985$	16.7089	0.532389
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−51.1122	−1.62527
$990$	0	0
$991$	−26.6428	−0.846336	−0.423168	−	0.906051i	$-0.639082\pi$
$991$	−26.6428	−0.846336	−0.423168	+	0.906051i	$0.639082\pi$
$992$	0	0
$993$	−0.755156	−0.0239642
$994$	0	0
$995$	12.7804	0.405167
$996$	0	0
$997$	−10.8206	−0.342693	−0.171346	−	0.985211i	$-0.554812\pi$
$997$	−10.8206	−0.342693	−0.171346	+	0.985211i	$0.554812\pi$
$998$	0	0
$999$	26.1775	0.828221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.l.1.2		5
7.6	odd	2		1148.2.a.c.1.4	✓	5
28.27	even	2		4592.2.a.be.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.c.1.4	✓	5	7.6	odd	2
4592.2.a.be.1.2		5	28.27	even	2
8036.2.a.l.1.2		5	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)

\(f(q)\)	\(=\)	\(q-0.475832 q^{3} -1.19617 q^{5} -2.77358 q^{9} +O(q^{10})\)	\(q-0.475832 q^{3} -1.19617 q^{5} -2.77358 q^{9} -4.51385 q^{11} -1.46760 q^{13} +0.569176 q^{15} +4.96975 q^{17} +3.69193 q^{19} +4.36269 q^{23} -3.56918 q^{25} +2.74726 q^{27} -3.12850 q^{29} +7.79167 q^{31} +2.14783 q^{33} +9.52861 q^{37} +0.698329 q^{39} -1.00000 q^{41} -11.7158 q^{43} +3.31768 q^{45} -6.93311 q^{47} -2.36477 q^{51} +8.63019 q^{53} +5.39933 q^{55} -1.75674 q^{57} +2.99301 q^{59} -0.624504 q^{61} +1.75549 q^{65} +10.3304 q^{67} -2.07591 q^{69} +1.26235 q^{71} +6.30820 q^{73} +1.69833 q^{75} -10.4848 q^{79} +7.01352 q^{81} +7.11184 q^{83} -5.94467 q^{85} +1.48864 q^{87} +10.5356 q^{89} -3.70753 q^{93} -4.41618 q^{95} -14.6608 q^{97} +12.5195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} + q^{5} + q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 + q^5 + q^9	\( 5 q + 2 q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - 9 q^{15} + 3 q^{17} + 4 q^{19} - 8 q^{23} - 6 q^{25} + 8 q^{27} - 9 q^{29} + 11 q^{31} - 5 q^{33} - 11 q^{37} - 17 q^{39} - 5 q^{41} - 27 q^{43} + 3 q^{45} + 3 q^{47} - 3 q^{51} - 19 q^{53} + 13 q^{55} - 11 q^{57} + 15 q^{59} + 7 q^{65} - 21 q^{67} - 14 q^{69} - 16 q^{71} + 10 q^{73} - 12 q^{75} - 14 q^{79} - 7 q^{81} + 2 q^{83} - 21 q^{85} + 36 q^{87} - 6 q^{89} + 17 q^{93} + 9 q^{95} - 20 q^{97} - 17 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 + q^5 + q^9 - 2 * q^11 + q^13 - 9 * q^15 + 3 * q^17 + 4 * q^19 - 8 * q^23 - 6 * q^25 + 8 * q^27 - 9 * q^29 + 11 * q^31 - 5 * q^33 - 11 * q^37 - 17 * q^39 - 5 * q^41 - 27 * q^43 + 3 * q^45 + 3 * q^47 - 3 * q^51 - 19 * q^53 + 13 * q^55 - 11 * q^57 + 15 * q^59 + 7 * q^65 - 21 * q^67 - 14 * q^69 - 16 * q^71 + 10 * q^73 - 12 * q^75 - 14 * q^79 - 7 * q^81 + 2 * q^83 - 21 * q^85 + 36 * q^87 - 6 * q^89 + 17 * q^93 + 9 * q^95 - 20 * q^97 - 17 * q^99

Introduction

L-functions

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Database

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