LMFDB - Embedded newform 8036.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.287349.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} + 7x - 1 $ x^5 - 6x^3 + 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			1.3
Root			$-1.36870$ 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)

$f(q)$	$=$	$q+0.126667 q^{3} +4.03743 q^{5} -2.98396 q^{9} +O(q^{10})$	$q+0.126667 q^{3} +4.03743 q^{5} -2.98396 q^{9} -3.52602 q^{11} -2.41540 q^{13} +0.511409 q^{15} +0.0534754 q^{17} -4.51925 q^{19} +2.50378 q^{23} +11.3008 q^{25} -0.757969 q^{27} +8.96338 q^{29} -0.562641 q^{31} -0.446630 q^{33} -3.34135 q^{37} -0.305951 q^{39} +1.00000 q^{41} +2.96934 q^{43} -12.0475 q^{45} +6.57187 q^{47} +0.00677356 q^{51} -0.414541 q^{53} -14.2361 q^{55} -0.572439 q^{57} +10.6898 q^{59} -6.49414 q^{61} -9.75201 q^{65} +12.1258 q^{67} +0.317146 q^{69} +10.4897 q^{71} +3.94877 q^{73} +1.43144 q^{75} -9.62951 q^{79} +8.85586 q^{81} -6.61015 q^{83} +0.215903 q^{85} +1.13536 q^{87} +13.8854 q^{89} -0.0712680 q^{93} -18.2462 q^{95} +11.8154 q^{97} +10.5215 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 3 q^{5} + q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 3 * q^5 + q^9	$ 5 q - 2 q^{3} + 3 q^{5} + q^{9} + 6 q^{11} - 7 q^{13} + 9 q^{15} - q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 8 q^{27} + 11 q^{29} - 13 q^{31} - 11 q^{33} + 5 q^{37} + 23 q^{39} + 5 q^{41} + 29 q^{43} - 11 q^{45} - 7 q^{47} - 3 q^{51} + 21 q^{53} - 19 q^{55} + 9 q^{57} - 3 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} - 10 q^{69} + 22 q^{71} + 16 q^{73} + 18 q^{75} + 4 q^{79} - 15 q^{81} + 6 q^{83} + 13 q^{85} - 6 q^{87} + 20 q^{89} - 5 q^{93} - 7 q^{95} + 24 q^{97} + 27 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 3 * q^5 + q^9 + 6 * q^11 - 7 * q^13 + 9 * q^15 - q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 - 8 * q^27 + 11 * q^29 - 13 * q^31 - 11 * q^33 + 5 * q^37 + 23 * q^39 + 5 * q^41 + 29 * q^43 - 11 * q^45 - 7 * q^47 - 3 * q^51 + 21 * q^53 - 19 * q^55 + 9 * q^57 - 3 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 - 10 * q^69 + 22 * q^71 + 16 * q^73 + 18 * q^75 + 4 * q^79 - 15 * q^81 + 6 * q^83 + 13 * q^85 - 6 * q^87 + 20 * q^89 - 5 * q^93 - 7 * q^95 + 24 * q^97 + 27 * 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
$3$	0.126667	0.0731312	0.0365656	−	0.999331i	$-0.488358\pi$
$3$	0.126667	0.0731312	0.0365656	+	0.999331i	$0.488358\pi$
$4$	0	0
$5$	4.03743	1.80559	0.902797	−	0.430067i	$-0.141510\pi$
$5$	4.03743	1.80559	0.902797	+	0.430067i	$0.141510\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.98396	−0.994652
$10$	0	0
$11$	−3.52602	−1.06314	−0.531568	−	0.847016i	$-0.678397\pi$
$11$	−3.52602	−1.06314	−0.531568	+	0.847016i	$0.678397\pi$
$12$	0	0
$13$	−2.41540	−0.669911	−0.334956	−	0.942234i	$-0.608721\pi$
$13$	−2.41540	−0.669911	−0.334956	+	0.942234i	$0.608721\pi$
$14$	0	0
$15$	0.511409	0.132045
$16$	0	0
$17$	0.0534754	0.0129697	0.00648484	−	0.999979i	$-0.497936\pi$
$17$	0.0534754	0.0129697	0.00648484	+	0.999979i	$0.497936\pi$
$18$	0	0
$19$	−4.51925	−1.03679	−0.518393	−	0.855142i	$-0.673470\pi$
$19$	−4.51925	−1.03679	−0.518393	+	0.855142i	$0.673470\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.50378	0.522074	0.261037	−	0.965329i	$-0.415936\pi$
$23$	2.50378	0.522074	0.261037	+	0.965329i	$0.415936\pi$
$24$	0	0
$25$	11.3008	2.26017
$26$	0	0
$27$	−0.757969	−0.145871
$28$	0	0
$29$	8.96338	1.66446	0.832229	−	0.554432i	$-0.187064\pi$
$29$	8.96338	1.66446	0.832229	+	0.554432i	$0.187064\pi$
$30$	0	0
$31$	−0.562641	−0.101053	−0.0505266	−	0.998723i	$-0.516090\pi$
$31$	−0.562641	−0.101053	−0.0505266	+	0.998723i	$0.516090\pi$
$32$	0	0
$33$	−0.446630	−0.0777484
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.34135	−0.549315	−0.274657	−	0.961542i	$-0.588564\pi$
$37$	−3.34135	−0.549315	−0.274657	+	0.961542i	$0.588564\pi$
$38$	0	0
$39$	−0.305951	−0.0489914
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	2.96934	0.452821	0.226410	−	0.974032i	$-0.427301\pi$
$43$	2.96934	0.452821	0.226410	+	0.974032i	$0.427301\pi$
$44$	0	0
$45$	−12.0475	−1.79594
$46$	0	0
$47$	6.57187	0.958605	0.479303	−	0.877650i	$-0.340890\pi$
$47$	6.57187	0.958605	0.479303	+	0.877650i	$0.340890\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.00677356	0.000948488 0
$52$	0	0
$53$	−0.414541	−0.0569416	−0.0284708	−	0.999595i	$-0.509064\pi$
$53$	−0.414541	−0.0569416	−0.0284708	+	0.999595i	$0.509064\pi$
$54$	0	0
$55$	−14.2361	−1.91959
$56$	0	0
$57$	−0.572439	−0.0758214
$58$	0	0
$59$	10.6898	1.39169	0.695844	−	0.718193i	$-0.255031\pi$
$59$	10.6898	1.39169	0.695844	+	0.718193i	$0.255031\pi$
$60$	0	0
$61$	−6.49414	−0.831490	−0.415745	−	0.909481i	$-0.636479\pi$
$61$	−6.49414	−0.831490	−0.415745	+	0.909481i	$0.636479\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−9.75201	−1.20959
$66$	0	0
$67$	12.1258	1.48141	0.740704	−	0.671832i	$-0.234492\pi$
$67$	12.1258	1.48141	0.740704	+	0.671832i	$0.234492\pi$
$68$	0	0
$69$	0.317146	0.0381799
$70$	0	0
$71$	10.4897	1.24489	0.622447	−	0.782662i	$-0.286139\pi$
$71$	10.4897	1.24489	0.622447	+	0.782662i	$0.286139\pi$
$72$	0	0
$73$	3.94877	0.462168	0.231084	−	0.972934i	$-0.425773\pi$
$73$	3.94877	0.462168	0.231084	+	0.972934i	$0.425773\pi$
$74$	0	0
$75$	1.43144	0.165289
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.62951	−1.08340	−0.541702	−	0.840570i	$-0.682220\pi$
$79$	−9.62951	−1.08340	−0.541702	+	0.840570i	$0.682220\pi$
$80$	0	0
$81$	8.85586	0.983984
$82$	0	0
$83$	−6.61015	−0.725559	−0.362779	−	0.931875i	$-0.618172\pi$
$83$	−6.61015	−0.725559	−0.362779	+	0.931875i	$0.618172\pi$
$84$	0	0
$85$	0.215903	0.0234180
$86$	0	0
$87$	1.13536	0.121724
$88$	0	0
$89$	13.8854	1.47185	0.735927	−	0.677061i	$-0.236747\pi$
$89$	13.8854	1.47185	0.735927	+	0.677061i	$0.236747\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.0712680	−0.00739014
$94$	0	0
$95$	−18.2462	−1.87202
$96$	0	0
$97$	11.8154	1.19967	0.599834	−	0.800125i	$-0.295233\pi$
$97$	11.8154	1.19967	0.599834	+	0.800125i	$0.295233\pi$
$98$	0	0
$99$	10.5215	1.05745
$100$	0	0
$101$	4.44606	0.442399	0.221200	−	0.975229i	$-0.429003\pi$
$101$	4.44606	0.442399	0.221200	+	0.975229i	$0.429003\pi$
$102$	0	0
$103$	−12.9239	−1.27343	−0.636717	−	0.771097i	$-0.719708\pi$
$103$	−12.9239	−1.27343	−0.636717	+	0.771097i	$0.719708\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.57971	−0.442737	−0.221369	−	0.975190i	$-0.571052\pi$
$107$	−4.57971	−0.442737	−0.221369	+	0.975190i	$0.571052\pi$
$108$	0	0
$109$	15.1663	1.45267	0.726336	−	0.687340i	$-0.241222\pi$
$109$	15.1663	1.45267	0.726336	+	0.687340i	$0.241222\pi$
$110$	0	0
$111$	−0.423239	−0.0401720
$112$	0	0
$113$	14.4185	1.35637	0.678187	−	0.734889i	$-0.262766\pi$
$113$	14.4185	1.35637	0.678187	+	0.734889i	$0.262766\pi$
$114$	0	0
$115$	10.1088	0.942653
$116$	0	0
$117$	7.20744	0.666328
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.43283	0.130257
$122$	0	0
$123$	0.126667	0.0114212
$124$	0	0
$125$	25.4392	2.27535
$126$	0	0
$127$	17.4061	1.54454	0.772272	−	0.635292i	$-0.219120\pi$
$127$	17.4061	1.54454	0.772272	+	0.635292i	$0.219120\pi$
$128$	0	0
$129$	0.376118	0.0331153
$130$	0	0
$131$	−9.82687	−0.858578	−0.429289	−	0.903167i	$-0.641236\pi$
$131$	−9.82687	−0.858578	−0.429289	+	0.903167i	$0.641236\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.06025	−0.263384
$136$	0	0
$137$	−1.39344	−0.119050	−0.0595248	−	0.998227i	$-0.518959\pi$
$137$	−1.39344	−0.119050	−0.0595248	+	0.998227i	$0.518959\pi$
$138$	0	0
$139$	−11.8863	−1.00818	−0.504092	−	0.863650i	$-0.668173\pi$
$139$	−11.8863	−1.00818	−0.504092	+	0.863650i	$0.668173\pi$
$140$	0	0
$141$	0.832438	0.0701039
$142$	0	0
$143$	8.51675	0.712206
$144$	0	0
$145$	36.1890	3.00534
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.1792	1.57122	0.785609	−	0.618723i	$-0.212350\pi$
$149$	19.1792	1.57122	0.785609	+	0.618723i	$0.212350\pi$
$150$	0	0
$151$	−4.11703	−0.335039	−0.167520	−	0.985869i	$-0.553576\pi$
$151$	−4.11703	−0.335039	−0.167520	+	0.985869i	$0.553576\pi$
$152$	0	0
$153$	−0.159568	−0.0129003
$154$	0	0
$155$	−2.27162	−0.182461
$156$	0	0
$157$	1.55689	0.124253	0.0621267	−	0.998068i	$-0.480212\pi$
$157$	1.55689	0.124253	0.0621267	+	0.998068i	$0.480212\pi$
$158$	0	0
$159$	−0.0525087	−0.00416421
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.62513	0.518920	0.259460	−	0.965754i	$-0.416455\pi$
$163$	6.62513	0.518920	0.259460	+	0.965754i	$0.416455\pi$
$164$	0	0
$165$	−1.80324	−0.140382
$166$	0	0
$167$	5.87052	0.454274	0.227137	−	0.973863i	$-0.427063\pi$
$167$	5.87052	0.454274	0.227137	+	0.973863i	$0.427063\pi$
$168$	0	0
$169$	−7.16585	−0.551219
$170$	0	0
$171$	13.4852	1.03124
$172$	0	0
$173$	22.3062	1.69591	0.847954	−	0.530070i	$-0.177834\pi$
$173$	22.3062	1.69591	0.847954	+	0.530070i	$0.177834\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.35404	0.101776
$178$	0	0
$179$	17.1906	1.28489	0.642445	−	0.766332i	$-0.277920\pi$
$179$	17.1906	1.28489	0.642445	+	0.766332i	$0.277920\pi$
$180$	0	0
$181$	15.6422	1.16267	0.581337	−	0.813663i	$-0.302530\pi$
$181$	15.6422	1.16267	0.581337	+	0.813663i	$0.302530\pi$
$182$	0	0
$183$	−0.822593	−0.0608079
$184$	0	0
$185$	−13.4905	−0.991839
$186$	0	0
$187$	−0.188555	−0.0137885
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.1606	1.38641	0.693207	−	0.720739i	$-0.256197\pi$
$191$	19.1606	1.38641	0.693207	+	0.720739i	$0.256197\pi$
$192$	0	0
$193$	−12.9782	−0.934192	−0.467096	−	0.884207i	$-0.654700\pi$
$193$	−12.9782	−0.934192	−0.467096	+	0.884207i	$0.654700\pi$
$194$	0	0
$195$	−1.23526	−0.0884586
$196$	0	0
$197$	15.9940	1.13953	0.569764	−	0.821808i	$-0.307035\pi$
$197$	15.9940	1.13953	0.569764	+	0.821808i	$0.307035\pi$
$198$	0	0
$199$	−21.2277	−1.50479	−0.752394	−	0.658713i	$-0.771101\pi$
$199$	−21.2277	−1.50479	−0.752394	+	0.658713i	$0.771101\pi$
$200$	0	0
$201$	1.53594	0.108337
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.03743	0.281986
$206$	0	0
$207$	−7.47116	−0.519282
$208$	0	0
$209$	15.9350	1.10224
$210$	0	0
$211$	0.273338	0.0188174	0.00940869	−	0.999956i	$-0.497005\pi$
$211$	0.273338	0.0188174	0.00940869	+	0.999956i	$0.497005\pi$
$212$	0	0
$213$	1.32869	0.0910405
$214$	0	0
$215$	11.9885	0.817610
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.500178	0.0337989
$220$	0	0
$221$	−0.129164	−0.00868853
$222$	0	0
$223$	−5.90787	−0.395620	−0.197810	−	0.980240i	$-0.563383\pi$
$223$	−5.90787	−0.395620	−0.197810	+	0.980240i	$0.563383\pi$
$224$	0	0
$225$	−33.7212	−2.24808
$226$	0	0
$227$	−24.2532	−1.60974	−0.804871	−	0.593450i	$-0.797766\pi$
$227$	−24.2532	−1.60974	−0.804871	+	0.593450i	$0.797766\pi$
$228$	0	0
$229$	−26.2406	−1.73403	−0.867013	−	0.498285i	$-0.833963\pi$
$229$	−26.2406	−1.73403	−0.867013	+	0.498285i	$0.833963\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.47026	−0.554905	−0.277453	−	0.960739i	$-0.589490\pi$
$233$	−8.47026	−0.554905	−0.277453	+	0.960739i	$0.589490\pi$
$234$	0	0
$235$	26.5335	1.73085
$236$	0	0
$237$	−1.21974	−0.0792307
$238$	0	0
$239$	12.2531	0.792586	0.396293	−	0.918124i	$-0.370296\pi$
$239$	12.2531	0.792586	0.396293	+	0.918124i	$0.370296\pi$
$240$	0	0
$241$	12.3179	0.793467	0.396734	−	0.917934i	$-0.370144\pi$
$241$	12.3179	0.793467	0.396734	+	0.917934i	$0.370144\pi$
$242$	0	0
$243$	3.39565	0.217831
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.9158	0.694555
$248$	0	0
$249$	−0.837288	−0.0530610
$250$	0	0
$251$	19.1168	1.20664	0.603322	−	0.797498i	$-0.293844\pi$
$251$	19.1168	1.20664	0.603322	+	0.797498i	$0.293844\pi$
$252$	0	0
$253$	−8.82837	−0.555035
$254$	0	0
$255$	0.0273478	0.00171258
$256$	0	0
$257$	−8.15564	−0.508735	−0.254367	−	0.967108i	$-0.581867\pi$
$257$	−8.15564	−0.508735	−0.254367	+	0.967108i	$0.581867\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−26.7463	−1.65556
$262$	0	0
$263$	−12.6372	−0.779246	−0.389623	−	0.920974i	$-0.627395\pi$
$263$	−12.6372	−0.779246	−0.389623	+	0.920974i	$0.627395\pi$
$264$	0	0
$265$	−1.67368	−0.102813
$266$	0	0
$267$	1.75883	0.107638
$268$	0	0
$269$	−5.37216	−0.327546	−0.163773	−	0.986498i	$-0.552366\pi$
$269$	−5.37216	−0.327546	−0.163773	+	0.986498i	$0.552366\pi$
$270$	0	0
$271$	−22.2397	−1.35096	−0.675482	−	0.737377i	$-0.736064\pi$
$271$	−22.2397	−1.35096	−0.675482	+	0.737377i	$0.736064\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−39.8470	−2.40287
$276$	0	0
$277$	12.0233	0.722411	0.361205	−	0.932486i	$-0.382365\pi$
$277$	12.0233	0.722411	0.361205	+	0.932486i	$0.382365\pi$
$278$	0	0
$279$	1.67889	0.100513
$280$	0	0
$281$	−27.6959	−1.65220	−0.826100	−	0.563524i	$-0.809445\pi$
$281$	−27.6959	−1.65220	−0.826100	+	0.563524i	$0.809445\pi$
$282$	0	0
$283$	−21.0824	−1.25322	−0.626610	−	0.779333i	$-0.715558\pi$
$283$	−21.0824	−1.25322	−0.626610	+	0.779333i	$0.715558\pi$
$284$	0	0
$285$	−2.31118	−0.136903
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9971	−0.999832
$290$	0	0
$291$	1.49662	0.0877331
$292$	0	0
$293$	−0.387194	−0.0226201	−0.0113100	−	0.999936i	$-0.503600\pi$
$293$	−0.387194	−0.0226201	−0.0113100	+	0.999936i	$0.503600\pi$
$294$	0	0
$295$	43.1592	2.51282
$296$	0	0
$297$	2.67262	0.155081
$298$	0	0
$299$	−6.04762	−0.349743
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.563168	0.0323532
$304$	0	0
$305$	−26.2197	−1.50133
$306$	0	0
$307$	21.6093	1.23331	0.616653	−	0.787235i	$-0.288488\pi$
$307$	21.6093	1.23331	0.616653	+	0.787235i	$0.288488\pi$
$308$	0	0
$309$	−1.63704	−0.0931278
$310$	0	0
$311$	−5.82981	−0.330578	−0.165289	−	0.986245i	$-0.552856\pi$
$311$	−5.82981	−0.330578	−0.165289	+	0.986245i	$0.552856\pi$
$312$	0	0
$313$	33.6459	1.90178	0.950889	−	0.309531i	$-0.100172\pi$
$313$	33.6459	1.90178	0.950889	+	0.309531i	$0.100172\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.05722	0.0593796	0.0296898	−	0.999559i	$-0.490548\pi$
$317$	1.05722	0.0593796	0.0296898	+	0.999559i	$0.490548\pi$
$318$	0	0
$319$	−31.6051	−1.76954
$320$	0	0
$321$	−0.580098	−0.0323779
$322$	0	0
$323$	−0.241668	−0.0134468
$324$	0	0
$325$	−27.2961	−1.51411
$326$	0	0
$327$	1.92107	0.106236
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.54210	−0.249656	−0.124828	−	0.992178i	$-0.539838\pi$
$331$	−4.54210	−0.249656	−0.124828	+	0.992178i	$0.539838\pi$
$332$	0	0
$333$	9.97044	0.546377
$334$	0	0
$335$	48.9572	2.67482
$336$	0	0
$337$	−1.87133	−0.101938	−0.0509690	−	0.998700i	$-0.516231\pi$
$337$	−1.87133	−0.101938	−0.0509690	+	0.998700i	$0.516231\pi$
$338$	0	0
$339$	1.82634	0.0991933
$340$	0	0
$341$	1.98388	0.107433
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.28045	0.0689373
$346$	0	0
$347$	−27.5851	−1.48084	−0.740422	−	0.672142i	$-0.765374\pi$
$347$	−27.5851	−1.48084	−0.740422	+	0.672142i	$0.765374\pi$
$348$	0	0
$349$	0.987265	0.0528471	0.0264235	−	0.999651i	$-0.491588\pi$
$349$	0.987265	0.0528471	0.0264235	+	0.999651i	$0.491588\pi$
$350$	0	0
$351$	1.83080	0.0977208
$352$	0	0
$353$	−29.4049	−1.56506	−0.782532	−	0.622610i	$-0.786072\pi$
$353$	−29.4049	−1.56506	−0.782532	+	0.622610i	$0.786072\pi$
$354$	0	0
$355$	42.3513	2.24777
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.11369	−0.217112	−0.108556	−	0.994090i	$-0.534623\pi$
$359$	−4.11369	−0.217112	−0.108556	+	0.994090i	$0.534623\pi$
$360$	0	0
$361$	1.42360	0.0749266
$362$	0	0
$363$	0.181492	0.00952587
$364$	0	0
$365$	15.9429	0.834488
$366$	0	0
$367$	−16.2034	−0.845810	−0.422905	−	0.906174i	$-0.638990\pi$
$367$	−16.2034	−0.845810	−0.422905	+	0.906174i	$0.638990\pi$
$368$	0	0
$369$	−2.98396	−0.155339
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.8956	1.54794	0.773969	−	0.633224i	$-0.218269\pi$
$373$	29.8956	1.54794	0.773969	+	0.633224i	$0.218269\pi$
$374$	0	0
$375$	3.22231	0.166399
$376$	0	0
$377$	−21.6501	−1.11504
$378$	0	0
$379$	−34.7403	−1.78449	−0.892245	−	0.451551i	$-0.850871\pi$
$379$	−34.7403	−1.78449	−0.892245	+	0.451551i	$0.850871\pi$
$380$	0	0
$381$	2.20478	0.112954
$382$	0	0
$383$	20.0199	1.02297	0.511484	−	0.859293i	$-0.329096\pi$
$383$	20.0199	1.02297	0.511484	+	0.859293i	$0.329096\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.86039	−0.450399
$388$	0	0
$389$	19.0731	0.967043	0.483521	−	0.875333i	$-0.339358\pi$
$389$	19.0731	0.967043	0.483521	+	0.875333i	$0.339358\pi$
$390$	0	0
$391$	0.133890	0.00677113
$392$	0	0
$393$	−1.24474	−0.0627888
$394$	0	0
$395$	−38.8785	−1.95619
$396$	0	0
$397$	−5.27701	−0.264845	−0.132423	−	0.991193i	$-0.542276\pi$
$397$	−5.27701	−0.264845	−0.132423	+	0.991193i	$0.542276\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−33.5626	−1.67604	−0.838018	−	0.545643i	$-0.816286\pi$
$401$	−33.5626	−1.67604	−0.838018	+	0.545643i	$0.816286\pi$
$402$	0	0
$403$	1.35900	0.0676967
$404$	0	0
$405$	35.7549	1.77668
$406$	0	0
$407$	11.7817	0.583996
$408$	0	0
$409$	7.67568	0.379538	0.189769	−	0.981829i	$-0.439226\pi$
$409$	7.67568	0.379538	0.189769	+	0.981829i	$0.439226\pi$
$410$	0	0
$411$	−0.176503	−0.00870624
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−26.6880	−1.31006
$416$	0	0
$417$	−1.50560	−0.0737297
$418$	0	0
$419$	−15.9653	−0.779956	−0.389978	−	0.920824i	$-0.627517\pi$
$419$	−15.9653	−0.779956	−0.389978	+	0.920824i	$0.627517\pi$
$420$	0	0
$421$	3.83173	0.186747	0.0933737	−	0.995631i	$-0.470235\pi$
$421$	3.83173	0.186747	0.0933737	+	0.995631i	$0.470235\pi$
$422$	0	0
$423$	−19.6102	−0.953478
$424$	0	0
$425$	0.604317	0.0293137
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.07879	0.0520845
$430$	0	0
$431$	−15.6049	−0.751664	−0.375832	−	0.926688i	$-0.622643\pi$
$431$	−15.6049	−0.751664	−0.375832	+	0.926688i	$0.622643\pi$
$432$	0	0
$433$	−12.5203	−0.601689	−0.300845	−	0.953673i	$-0.597269\pi$
$433$	−12.5203	−0.601689	−0.300845	+	0.953673i	$0.597269\pi$
$434$	0	0
$435$	4.58395	0.219784
$436$	0	0
$437$	−11.3152	−0.541279
$438$	0	0
$439$	−3.29040	−0.157042	−0.0785212	−	0.996912i	$-0.525020\pi$
$439$	−3.29040	−0.157042	−0.0785212	+	0.996912i	$0.525020\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.7668	1.55680	0.778399	−	0.627770i	$-0.216032\pi$
$443$	32.7668	1.55680	0.778399	+	0.627770i	$0.216032\pi$
$444$	0	0
$445$	56.0615	2.65757
$446$	0	0
$447$	2.42937	0.114905
$448$	0	0
$449$	33.6287	1.58704	0.793518	−	0.608547i	$-0.208247\pi$
$449$	33.6287	1.58704	0.793518	+	0.608547i	$0.208247\pi$
$450$	0	0
$451$	−3.52602	−0.166034
$452$	0	0
$453$	−0.521492	−0.0245018
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.80973	−0.178212	−0.0891058	−	0.996022i	$-0.528401\pi$
$457$	−3.80973	−0.178212	−0.0891058	+	0.996022i	$0.528401\pi$
$458$	0	0
$459$	−0.0405327	−0.00189190
$460$	0	0
$461$	8.17907	0.380937	0.190469	−	0.981693i	$-0.438999\pi$
$461$	8.17907	0.380937	0.190469	+	0.981693i	$0.438999\pi$
$462$	0	0
$463$	21.7880	1.01257	0.506287	−	0.862365i	$-0.331018\pi$
$463$	21.7880	1.01257	0.506287	+	0.862365i	$0.331018\pi$
$464$	0	0
$465$	−0.287739	−0.0133436
$466$	0	0
$467$	−36.9535	−1.71001	−0.855003	−	0.518624i	$-0.826445\pi$
$467$	−36.9535	−1.71001	−0.855003	+	0.518624i	$0.826445\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.197207	0.00908680
$472$	0	0
$473$	−10.4700	−0.481410
$474$	0	0
$475$	−51.0713	−2.34331
$476$	0	0
$477$	1.23697	0.0566371
$478$	0	0
$479$	18.3900	0.840259	0.420129	−	0.907464i	$-0.361985\pi$
$479$	18.3900	0.840259	0.420129	+	0.907464i	$0.361985\pi$
$480$	0	0
$481$	8.07069	0.367992
$482$	0	0
$483$	0	0
$484$	0	0
$485$	47.7037	2.16611
$486$	0	0
$487$	16.1947	0.733851	0.366925	−	0.930250i	$-0.380410\pi$
$487$	16.1947	0.733851	0.366925	+	0.930250i	$0.380410\pi$
$488$	0	0
$489$	0.839185	0.0379493
$490$	0	0
$491$	18.5702	0.838061	0.419030	−	0.907972i	$-0.362370\pi$
$491$	18.5702	0.838061	0.419030	+	0.907972i	$0.362370\pi$
$492$	0	0
$493$	0.479320	0.0215875
$494$	0	0
$495$	42.4798	1.90932
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.23630	−0.144877	−0.0724383	−	0.997373i	$-0.523078\pi$
$499$	−3.23630	−0.144877	−0.0724383	+	0.997373i	$0.523078\pi$
$500$	0	0
$501$	0.743600	0.0332216
$502$	0	0
$503$	15.9849	0.712734	0.356367	−	0.934346i	$-0.384015\pi$
$503$	15.9849	0.712734	0.356367	+	0.934346i	$0.384015\pi$
$504$	0	0
$505$	17.9506	0.798793
$506$	0	0
$507$	−0.907676	−0.0403113
$508$	0	0
$509$	−16.1681	−0.716640	−0.358320	−	0.933599i	$-0.616650\pi$
$509$	−16.1681	−0.716640	−0.358320	+	0.933599i	$0.616650\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.42545	0.151237
$514$	0	0
$515$	−52.1795	−2.29931
$516$	0	0
$517$	−23.1725	−1.01913
$518$	0	0
$519$	2.82546	0.124024
$520$	0	0
$521$	19.6766	0.862048	0.431024	−	0.902341i	$-0.358152\pi$
$521$	19.6766	0.862048	0.431024	+	0.902341i	$0.358152\pi$
$522$	0	0
$523$	−17.2052	−0.752332	−0.376166	−	0.926552i	$-0.622758\pi$
$523$	−17.2052	−0.752332	−0.376166	+	0.926552i	$0.622758\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.0300874	−0.00131063
$528$	0	0
$529$	−16.7311	−0.727439
$530$	0	0
$531$	−31.8978	−1.38424
$532$	0	0
$533$	−2.41540	−0.104623
$534$	0	0
$535$	−18.4903	−0.799404
$536$	0	0
$537$	2.17749	0.0939655
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0.0154440	0.000663988 0	0.000331994	−	1.00000i	$-0.499894\pi$
$541$	0.0154440	0.000663988 0	0.000331994		1.00000i	$0.499894\pi$
$542$	0	0
$543$	1.98135	0.0850277
$544$	0	0
$545$	61.2330	2.62294
$546$	0	0
$547$	33.7473	1.44293	0.721464	−	0.692451i	$-0.243469\pi$
$547$	33.7473	1.44293	0.721464	+	0.692451i	$0.243469\pi$
$548$	0	0
$549$	19.3782	0.827043
$550$	0	0
$551$	−40.5077	−1.72569
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.70880	−0.0725344
$556$	0	0
$557$	33.2488	1.40880	0.704399	−	0.709805i	$-0.251217\pi$
$557$	33.2488	1.40880	0.704399	+	0.709805i	$0.251217\pi$
$558$	0	0
$559$	−7.17215	−0.303350
$560$	0	0
$561$	−0.0238837	−0.00100837
$562$	0	0
$563$	20.0275	0.844059	0.422029	−	0.906582i	$-0.361318\pi$
$563$	20.0275	0.844059	0.422029	+	0.906582i	$0.361318\pi$
$564$	0	0
$565$	58.2135	2.44906
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.3964	1.27428	0.637141	−	0.770747i	$-0.280117\pi$
$569$	30.3964	1.27428	0.637141	+	0.770747i	$0.280117\pi$
$570$	0	0
$571$	−26.0843	−1.09160	−0.545798	−	0.837917i	$-0.683773\pi$
$571$	−26.0843	−1.09160	−0.545798	+	0.837917i	$0.683773\pi$
$572$	0	0
$573$	2.42702	0.101390
$574$	0	0
$575$	28.2948	1.17998
$576$	0	0
$577$	−40.3390	−1.67934	−0.839668	−	0.543101i	$-0.817250\pi$
$577$	−40.3390	−1.67934	−0.839668	+	0.543101i	$0.817250\pi$
$578$	0	0
$579$	−1.64391	−0.0683186
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.46168	0.0605367
$584$	0	0
$585$	29.0996	1.20312
$586$	0	0
$587$	−1.30040	−0.0536733	−0.0268367	−	0.999640i	$-0.508543\pi$
$587$	−1.30040	−0.0536733	−0.0268367	+	0.999640i	$0.508543\pi$
$588$	0	0
$589$	2.54271	0.104771
$590$	0	0
$591$	2.02592	0.0833351
$592$	0	0
$593$	−31.1353	−1.27857	−0.639286	−	0.768969i	$-0.720770\pi$
$593$	−31.1353	−1.27857	−0.639286	+	0.768969i	$0.720770\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.68884	−0.110047
$598$	0	0
$599$	11.3986	0.465734	0.232867	−	0.972509i	$-0.425189\pi$
$599$	11.3986	0.465734	0.232867	+	0.972509i	$0.425189\pi$
$600$	0	0
$601$	11.7031	0.477380	0.238690	−	0.971096i	$-0.423282\pi$
$601$	11.7031	0.477380	0.238690	+	0.971096i	$0.423282\pi$
$602$	0	0
$603$	−36.1830	−1.47348
$604$	0	0
$605$	5.78495	0.235192
$606$	0	0
$607$	−33.8918	−1.37562	−0.687812	−	0.725888i	$-0.741429\pi$
$607$	−33.8918	−1.37562	−0.687812	+	0.725888i	$0.741429\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.8737	−0.642180
$612$	0	0
$613$	−37.4202	−1.51139	−0.755693	−	0.654925i	$-0.772700\pi$
$613$	−37.4202	−1.51139	−0.755693	+	0.654925i	$0.772700\pi$
$614$	0	0
$615$	0.511409	0.0206220
$616$	0	0
$617$	24.1568	0.972518	0.486259	−	0.873815i	$-0.338361\pi$
$617$	24.1568	0.972518	0.486259	+	0.873815i	$0.338361\pi$
$618$	0	0
$619$	−14.0504	−0.564732	−0.282366	−	0.959307i	$-0.591119\pi$
$619$	−14.0504	−0.564732	−0.282366	+	0.959307i	$0.591119\pi$
$620$	0	0
$621$	−1.89779	−0.0761556
$622$	0	0
$623$	0	0
$624$	0	0
$625$	46.2049	1.84820
$626$	0	0
$627$	2.01843	0.0806085
$628$	0	0
$629$	−0.178680	−0.00712443
$630$	0	0
$631$	9.31402	0.370785	0.185393	−	0.982665i	$-0.440644\pi$
$631$	9.31402	0.370785	0.185393	+	0.982665i	$0.440644\pi$
$632$	0	0
$633$	0.0346229	0.00137614
$634$	0	0
$635$	70.2760	2.78882
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−31.3007	−1.23824
$640$	0	0
$641$	7.28680	0.287811	0.143906	−	0.989591i	$-0.454034\pi$
$641$	7.28680	0.287811	0.143906	+	0.989591i	$0.454034\pi$
$642$	0	0
$643$	0.967833	0.0381676	0.0190838	−	0.999818i	$-0.493925\pi$
$643$	0.967833	0.0381676	0.0190838	+	0.999818i	$0.493925\pi$
$644$	0	0
$645$	1.51855	0.0597928
$646$	0	0
$647$	−10.7171	−0.421333	−0.210666	−	0.977558i	$-0.567563\pi$
$647$	−10.7171	−0.421333	−0.210666	+	0.977558i	$0.567563\pi$
$648$	0	0
$649$	−37.6923	−1.47955
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.0378	0.471077	0.235539	−	0.971865i	$-0.424315\pi$
$653$	12.0378	0.471077	0.235539	+	0.971865i	$0.424315\pi$
$654$	0	0
$655$	−39.6753	−1.55024
$656$	0	0
$657$	−11.7829	−0.459697
$658$	0	0
$659$	−42.3539	−1.64987	−0.824936	−	0.565226i	$-0.808789\pi$
$659$	−42.3539	−1.64987	−0.824936	+	0.565226i	$0.808789\pi$
$660$	0	0
$661$	2.16165	0.0840783	0.0420391	−	0.999116i	$-0.486615\pi$
$661$	2.16165	0.0840783	0.0420391	+	0.999116i	$0.486615\pi$
$662$	0	0
$663$	−0.0163609	−0.000635403 0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	22.4423	0.868970
$668$	0	0
$669$	−0.748332	−0.0289322
$670$	0	0
$671$	22.8985	0.883987
$672$	0	0
$673$	−0.796393	−0.0306987	−0.0153494	−	0.999882i	$-0.504886\pi$
$673$	−0.796393	−0.0306987	−0.0153494	+	0.999882i	$0.504886\pi$
$674$	0	0
$675$	−8.56570	−0.329694
$676$	0	0
$677$	−29.3629	−1.12851	−0.564254	−	0.825601i	$-0.690836\pi$
$677$	−29.3629	−1.12851	−0.564254	+	0.825601i	$0.690836\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.07208	−0.117722
$682$	0	0
$683$	39.2777	1.50292	0.751459	−	0.659780i	$-0.229351\pi$
$683$	39.2777	1.50292	0.751459	+	0.659780i	$0.229351\pi$
$684$	0	0
$685$	−5.62591	−0.214955
$686$	0	0
$687$	−3.32381	−0.126811
$688$	0	0
$689$	1.00128	0.0381458
$690$	0	0
$691$	35.3284	1.34396	0.671979	−	0.740570i	$-0.265445\pi$
$691$	35.3284	1.34396	0.671979	+	0.740570i	$0.265445\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−47.9901	−1.82037
$696$	0	0
$697$	0.0534754	0.00202552
$698$	0	0
$699$	−1.07290	−0.0405809
$700$	0	0
$701$	−13.6523	−0.515640	−0.257820	−	0.966193i	$-0.583004\pi$
$701$	−13.6523	−0.515640	−0.257820	+	0.966193i	$0.583004\pi$
$702$	0	0
$703$	15.1004	0.569522
$704$	0	0
$705$	3.36091	0.126579
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−11.4663	−0.430626	−0.215313	−	0.976545i	$-0.569077\pi$
$709$	−11.4663	−0.430626	−0.215313	+	0.976545i	$0.569077\pi$
$710$	0	0
$711$	28.7340	1.07761
$712$	0	0
$713$	−1.40873	−0.0527572
$714$	0	0
$715$	34.3858	1.28596
$716$	0	0
$717$	1.55206	0.0579628
$718$	0	0
$719$	−0.0868167	−0.00323772	−0.00161886	−	0.999999i	$-0.500515\pi$
$719$	−0.0868167	−0.00323772	−0.00161886	+	0.999999i	$0.500515\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.56027	0.0580272
$724$	0	0
$725$	101.294	3.76196
$726$	0	0
$727$	4.28840	0.159048	0.0795239	−	0.996833i	$-0.474660\pi$
$727$	4.28840	0.159048	0.0795239	+	0.996833i	$0.474660\pi$
$728$	0	0
$729$	−26.1375	−0.968054
$730$	0	0
$731$	0.158787	0.00587294
$732$	0	0
$733$	20.7270	0.765569	0.382785	−	0.923838i	$-0.374965\pi$
$733$	20.7270	0.765569	0.382785	+	0.923838i	$0.374965\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−42.7560	−1.57494
$738$	0	0
$739$	−5.79602	−0.213210	−0.106605	−	0.994301i	$-0.533998\pi$
$739$	−5.79602	−0.213210	−0.106605	+	0.994301i	$0.533998\pi$
$740$	0	0
$741$	1.38267	0.0507936
$742$	0	0
$743$	1.74866	0.0641520	0.0320760	−	0.999485i	$-0.489788\pi$
$743$	1.74866	0.0641520	0.0320760	+	0.999485i	$0.489788\pi$
$744$	0	0
$745$	77.4345	2.83698
$746$	0	0
$747$	19.7244	0.721678
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0.113270	0.00413330	0.00206665	−	0.999998i	$-0.499342\pi$
$751$	0.113270	0.00413330	0.00206665	+	0.999998i	$0.499342\pi$
$752$	0	0
$753$	2.42147	0.0882433
$754$	0	0
$755$	−16.6222	−0.604945
$756$	0	0
$757$	29.7134	1.07995	0.539976	−	0.841680i	$-0.318433\pi$
$757$	29.7134	1.07995	0.539976	+	0.841680i	$0.318433\pi$
$758$	0	0
$759$	−1.11826	−0.0405904
$760$	0	0
$761$	45.1150	1.63542	0.817708	−	0.575633i	$-0.195244\pi$
$761$	45.1150	1.63542	0.817708	+	0.575633i	$0.195244\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.644245	−0.0232927
$766$	0	0
$767$	−25.8200	−0.932307
$768$	0	0
$769$	−32.2415	−1.16266	−0.581329	−	0.813669i	$-0.697467\pi$
$769$	−32.2415	−1.16266	−0.581329	+	0.813669i	$0.697467\pi$
$770$	0	0
$771$	−1.03305	−0.0372044
$772$	0	0
$773$	23.6254	0.849747	0.424874	−	0.905253i	$-0.360318\pi$
$773$	23.6254	0.849747	0.424874	+	0.905253i	$0.360318\pi$
$774$	0	0
$775$	−6.35832	−0.228397
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.51925	−0.161919
$780$	0	0
$781$	−36.9868	−1.32349
$782$	0	0
$783$	−6.79397	−0.242797
$784$	0	0
$785$	6.28584	0.224351
$786$	0	0
$787$	4.48114	0.159735	0.0798676	−	0.996805i	$-0.474550\pi$
$787$	4.48114	0.159735	0.0798676	+	0.996805i	$0.474550\pi$
$788$	0	0
$789$	−1.60072	−0.0569872
$790$	0	0
$791$	0	0
$792$	0	0
$793$	15.6860	0.557024
$794$	0	0
$795$	−0.212000	−0.00751887
$796$	0	0
$797$	7.42314	0.262941	0.131470	−	0.991320i	$-0.458030\pi$
$797$	7.42314	0.262941	0.131470	+	0.991320i	$0.458030\pi$
$798$	0	0
$799$	0.351433	0.0124328
$800$	0	0
$801$	−41.4336	−1.46398
$802$	0	0
$803$	−13.9234	−0.491348
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.680475	−0.0239538
$808$	0	0
$809$	14.9314	0.524961	0.262480	−	0.964937i	$-0.415460\pi$
$809$	14.9314	0.524961	0.262480	+	0.964937i	$0.415460\pi$
$810$	0	0
$811$	−44.8200	−1.57384	−0.786922	−	0.617052i	$-0.788327\pi$
$811$	−44.8200	−1.57384	−0.786922	+	0.617052i	$0.788327\pi$
$812$	0	0
$813$	−2.81703	−0.0987976
$814$	0	0
$815$	26.7485	0.936959
$816$	0	0
$817$	−13.4192	−0.469478
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.5228	−0.925654	−0.462827	−	0.886449i	$-0.653165\pi$
$821$	−26.5228	−0.925654	−0.462827	+	0.886449i	$0.653165\pi$
$822$	0	0
$823$	40.9182	1.42632	0.713160	−	0.701002i	$-0.247263\pi$
$823$	40.9182	1.42632	0.713160	+	0.701002i	$0.247263\pi$
$824$	0	0
$825$	−5.04730	−0.175725
$826$	0	0
$827$	4.31952	0.150204	0.0751022	−	0.997176i	$-0.476072\pi$
$827$	4.31952	0.150204	0.0751022	+	0.997176i	$0.476072\pi$
$828$	0	0
$829$	51.7234	1.79643	0.898214	−	0.439558i	$-0.144865\pi$
$829$	51.7234	1.79643	0.898214	+	0.439558i	$0.144865\pi$
$830$	0	0
$831$	1.52296	0.0528308
$832$	0	0
$833$	0	0
$834$	0	0
$835$	23.7018	0.820235
$836$	0	0
$837$	0.426464	0.0147408
$838$	0	0
$839$	34.5939	1.19431	0.597157	−	0.802125i	$-0.296297\pi$
$839$	34.5939	1.19431	0.597157	+	0.802125i	$0.296297\pi$
$840$	0	0
$841$	51.3422	1.77042
$842$	0	0
$843$	−3.50816	−0.120827
$844$	0	0
$845$	−28.9316	−0.995278
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.67045	−0.0916494
$850$	0	0
$851$	−8.36600	−0.286783
$852$	0	0
$853$	−9.33241	−0.319536	−0.159768	−	0.987155i	$-0.551075\pi$
$853$	−9.33241	−0.319536	−0.159768	+	0.987155i	$0.551075\pi$
$854$	0	0
$855$	54.4457	1.86200
$856$	0	0
$857$	29.1930	0.997215	0.498607	−	0.866828i	$-0.333845\pi$
$857$	29.1930	0.997215	0.498607	+	0.866828i	$0.333845\pi$
$858$	0	0
$859$	9.72856	0.331934	0.165967	−	0.986131i	$-0.446925\pi$
$859$	9.72856	0.331934	0.165967	+	0.986131i	$0.446925\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.606028	−0.0206294	−0.0103147	−	0.999947i	$-0.503283\pi$
$863$	−0.606028	−0.0206294	−0.0103147	+	0.999947i	$0.503283\pi$
$864$	0	0
$865$	90.0597	3.06212
$866$	0	0
$867$	−2.15298	−0.0731189
$868$	0	0
$869$	33.9539	1.15181
$870$	0	0
$871$	−29.2887	−0.992411
$872$	0	0
$873$	−35.2565	−1.19325
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−39.1441	−1.32180	−0.660900	−	0.750474i	$-0.729825\pi$
$877$	−39.1441	−1.32180	−0.660900	+	0.750474i	$0.729825\pi$
$878$	0	0
$879$	−0.0490446	−0.00165423
$880$	0	0
$881$	45.0373	1.51735	0.758673	−	0.651471i	$-0.225848\pi$
$881$	45.0373	1.51735	0.758673	+	0.651471i	$0.225848\pi$
$882$	0	0
$883$	42.3193	1.42416	0.712078	−	0.702100i	$-0.247754\pi$
$883$	42.3193	1.42416	0.712078	+	0.702100i	$0.247754\pi$
$884$	0	0
$885$	5.46684	0.183766
$886$	0	0
$887$	25.2033	0.846244	0.423122	−	0.906073i	$-0.360934\pi$
$887$	25.2033	0.846244	0.423122	+	0.906073i	$0.360934\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−31.2259	−1.04611
$892$	0	0
$893$	−29.6999	−0.993869
$894$	0	0
$895$	69.4060	2.31999
$896$	0	0
$897$	−0.766034	−0.0255771
$898$	0	0
$899$	−5.04316	−0.168199
$900$	0	0
$901$	−0.0221677	−0.000738515 0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	63.1542	2.09932
$906$	0	0
$907$	−16.9134	−0.561602	−0.280801	−	0.959766i	$-0.590600\pi$
$907$	−16.9134	−0.561602	−0.280801	+	0.959766i	$0.590600\pi$
$908$	0	0
$909$	−13.2668	−0.440033
$910$	0	0
$911$	−14.1664	−0.469355	−0.234678	−	0.972073i	$-0.575403\pi$
$911$	−14.1664	−0.469355	−0.234678	+	0.972073i	$0.575403\pi$
$912$	0	0
$913$	23.3075	0.771367
$914$	0	0
$915$	−3.32116	−0.109794
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−14.9329	−0.492591	−0.246296	−	0.969195i	$-0.579213\pi$
$919$	−14.9329	−0.492591	−0.246296	+	0.969195i	$0.579213\pi$
$920$	0	0
$921$	2.73718	0.0901931
$922$	0	0
$923$	−25.3367	−0.833968
$924$	0	0
$925$	−37.7601	−1.24154
$926$	0	0
$927$	38.5645	1.26662
$928$	0	0
$929$	−17.4443	−0.572328	−0.286164	−	0.958181i	$-0.592380\pi$
$929$	−17.4443	−0.572328	−0.286164	+	0.958181i	$0.592380\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−0.738444	−0.0241756
$934$	0	0
$935$	−0.761279	−0.0248965
$936$	0	0
$937$	−32.1148	−1.04914	−0.524572	−	0.851366i	$-0.675775\pi$
$937$	−32.1148	−1.04914	−0.524572	+	0.851366i	$0.675775\pi$
$938$	0	0
$939$	4.26182	0.139079
$940$	0	0
$941$	29.4780	0.960955	0.480478	−	0.877007i	$-0.340463\pi$
$941$	29.4780	0.960955	0.480478	+	0.877007i	$0.340463\pi$
$942$	0	0
$943$	2.50378	0.0815342
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.8238	1.22911	0.614554	−	0.788875i	$-0.289336\pi$
$947$	37.8238	1.22911	0.614554	+	0.788875i	$0.289336\pi$
$948$	0	0
$949$	−9.53785	−0.309612
$950$	0	0
$951$	0.133915	0.00434250
$952$	0	0
$953$	23.3765	0.757240	0.378620	−	0.925552i	$-0.376399\pi$
$953$	23.3765	0.757240	0.378620	+	0.925552i	$0.376399\pi$
$954$	0	0
$955$	77.3597	2.50330
$956$	0	0
$957$	−4.00332	−0.129409
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.6834	−0.989788
$962$	0	0
$963$	13.6656	0.440369
$964$	0	0
$965$	−52.3986	−1.68677
$966$	0	0
$967$	10.2162	0.328530	0.164265	−	0.986416i	$-0.447475\pi$
$967$	10.2162	0.328530	0.164265	+	0.986416i	$0.447475\pi$
$968$	0	0
$969$	−0.0306114	−0.000983380 0
$970$	0	0
$971$	−17.4432	−0.559779	−0.279889	−	0.960032i	$-0.590298\pi$
$971$	−17.4432	−0.559779	−0.279889	+	0.960032i	$0.590298\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.45751	−0.110729
$976$	0	0
$977$	43.8180	1.40186	0.700931	−	0.713229i	$-0.252768\pi$
$977$	43.8180	1.40186	0.700931	+	0.713229i	$0.252768\pi$
$978$	0	0
$979$	−48.9604	−1.56478
$980$	0	0
$981$	−45.2557	−1.44490
$982$	0	0
$983$	39.2429	1.25166	0.625828	−	0.779961i	$-0.284761\pi$
$983$	39.2429	1.25166	0.625828	+	0.779961i	$0.284761\pi$
$984$	0	0
$985$	64.5748	2.05752
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.43457	0.236406
$990$	0	0
$991$	−8.97731	−0.285174	−0.142587	−	0.989782i	$-0.545542\pi$
$991$	−8.97731	−0.285174	−0.142587	+	0.989782i	$0.545542\pi$
$992$	0	0
$993$	−0.575334	−0.0182577
$994$	0	0
$995$	−85.7052	−2.71704
$996$	0	0
$997$	−13.6449	−0.432139	−0.216070	−	0.976378i	$-0.569324\pi$
$997$	−13.6449	−0.432139	−0.216070	+	0.976378i	$0.569324\pi$
$998$	0	0
$999$	2.53264	0.0801292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.k.1.3		5
7.6	odd	2		1148.2.a.d.1.3	✓	5
28.27	even	2		4592.2.a.bc.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.d.1.3	✓	5	7.6	odd	2
4592.2.a.bc.1.3		5	28.27	even	2
8036.2.a.k.1.3		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)

\(f(q)\)	\(=\)	\(q+0.126667 q^{3} +4.03743 q^{5} -2.98396 q^{9} +O(q^{10})\)	\(q+0.126667 q^{3} +4.03743 q^{5} -2.98396 q^{9} -3.52602 q^{11} -2.41540 q^{13} +0.511409 q^{15} +0.0534754 q^{17} -4.51925 q^{19} +2.50378 q^{23} +11.3008 q^{25} -0.757969 q^{27} +8.96338 q^{29} -0.562641 q^{31} -0.446630 q^{33} -3.34135 q^{37} -0.305951 q^{39} +1.00000 q^{41} +2.96934 q^{43} -12.0475 q^{45} +6.57187 q^{47} +0.00677356 q^{51} -0.414541 q^{53} -14.2361 q^{55} -0.572439 q^{57} +10.6898 q^{59} -6.49414 q^{61} -9.75201 q^{65} +12.1258 q^{67} +0.317146 q^{69} +10.4897 q^{71} +3.94877 q^{73} +1.43144 q^{75} -9.62951 q^{79} +8.85586 q^{81} -6.61015 q^{83} +0.215903 q^{85} +1.13536 q^{87} +13.8854 q^{89} -0.0712680 q^{93} -18.2462 q^{95} +11.8154 q^{97} +10.5215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 3 q^{5} + q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 3 * q^5 + q^9	\( 5 q - 2 q^{3} + 3 q^{5} + q^{9} + 6 q^{11} - 7 q^{13} + 9 q^{15} - q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 8 q^{27} + 11 q^{29} - 13 q^{31} - 11 q^{33} + 5 q^{37} + 23 q^{39} + 5 q^{41} + 29 q^{43} - 11 q^{45} - 7 q^{47} - 3 q^{51} + 21 q^{53} - 19 q^{55} + 9 q^{57} - 3 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} - 10 q^{69} + 22 q^{71} + 16 q^{73} + 18 q^{75} + 4 q^{79} - 15 q^{81} + 6 q^{83} + 13 q^{85} - 6 q^{87} + 20 q^{89} - 5 q^{93} - 7 q^{95} + 24 q^{97} + 27 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 3 * q^5 + q^9 + 6 * q^11 - 7 * q^13 + 9 * q^15 - q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 - 8 * q^27 + 11 * q^29 - 13 * q^31 - 11 * q^33 + 5 * q^37 + 23 * q^39 + 5 * q^41 + 29 * q^43 - 11 * q^45 - 7 * q^47 - 3 * q^51 + 21 * q^53 - 19 * q^55 + 9 * q^57 - 3 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 - 10 * q^69 + 22 * q^71 + 16 * q^73 + 18 * q^75 + 4 * q^79 - 15 * q^81 + 6 * q^83 + 13 * q^85 - 6 * q^87 + 20 * q^89 - 5 * q^93 - 7 * q^95 + 24 * q^97 + 27 * q^99

Introduction

L-functions

Modular forms

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Database

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