LMFDB - Embedded newform 680.2.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [680,2,Mod(1,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 680 = 2^{3} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.42982733745$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{3} +1.00000 q^{5} -4.73205 q^{7} -2.46410 q^{9} -2.73205 q^{11} +0.732051 q^{15} -1.00000 q^{17} -5.46410 q^{19} -3.46410 q^{21} -3.26795 q^{23} +1.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} +9.66025 q^{31} -2.00000 q^{33} -4.73205 q^{35} -0.535898 q^{37} +10.3923 q^{41} +8.92820 q^{43} -2.46410 q^{45} -11.4641 q^{47} +15.3923 q^{49} -0.732051 q^{51} -2.00000 q^{53} -2.73205 q^{55} -4.00000 q^{57} +9.46410 q^{59} -12.9282 q^{61} +11.6603 q^{63} -7.46410 q^{67} -2.39230 q^{69} -10.7321 q^{71} -7.46410 q^{73} +0.732051 q^{75} +12.9282 q^{77} -16.5885 q^{79} +4.46410 q^{81} -8.92820 q^{83} -1.00000 q^{85} -2.53590 q^{87} +13.4641 q^{89} +7.07180 q^{93} -5.46410 q^{95} +0.928203 q^{97} +6.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{15} - 2 q^{17} - 4 q^{19} - 10 q^{23} + 2 q^{25} - 8 q^{27} + 2 q^{31} - 4 q^{33} - 6 q^{35} - 8 q^{37} + 4 q^{43} + 2 q^{45} - 16 q^{47} + 10 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^15 - 2 * q^17 - 4 * q^19 - 10 * q^23 + 2 * q^25 - 8 * q^27 + 2 * q^31 - 4 * q^33 - 6 * q^35 - 8 * q^37 + 4 * q^43 + 2 * q^45 - 16 * q^47 + 10 * q^49 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 8 * q^57 + 12 * q^59 - 12 * q^61 + 6 * q^63 - 8 * q^67 + 16 * q^69 - 18 * q^71 - 8 * q^73 - 2 * q^75 + 12 * q^77 - 2 * q^79 + 2 * q^81 - 4 * q^83 - 2 * q^85 - 12 * q^87 + 20 * q^89 + 28 * q^93 - 4 * q^95 - 12 * q^97 + 10 * q^99

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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.73205	−1.78855	−0.894274	−	0.447521i	$-0.852307\pi$
$7$	−4.73205	−1.78855	−0.894274	+	0.447521i	$0.852307\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−2.73205	−0.823744	−0.411872	−	0.911242i	$-0.635125\pi$
$11$	−2.73205	−0.823744	−0.411872	+	0.911242i	$0.635125\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0.732051	0.189015
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	−3.26795	−0.681415	−0.340707	−	0.940169i	$-0.610666\pi$
$23$	−3.26795	−0.681415	−0.340707	+	0.940169i	$0.610666\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	9.66025	1.73503	0.867516	−	0.497409i	$-0.165715\pi$
$31$	9.66025	1.73503	0.867516	+	0.497409i	$0.165715\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−4.73205	−0.799863
$36$	0	0
$37$	−0.535898	−0.0881012	−0.0440506	−	0.999029i	$-0.514026\pi$
$37$	−0.535898	−0.0881012	−0.0440506	+	0.999029i	$0.514026\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.3923	1.62301	0.811503	−	0.584349i	$-0.198650\pi$
$41$	10.3923	1.62301	0.811503	+	0.584349i	$0.198650\pi$
$42$	0	0
$43$	8.92820	1.36154	0.680769	−	0.732498i	$-0.261646\pi$
$43$	8.92820	1.36154	0.680769	+	0.732498i	$0.261646\pi$
$44$	0	0
$45$	−2.46410	−0.367327
$46$	0	0
$47$	−11.4641	−1.67221	−0.836106	−	0.548569i	$-0.815173\pi$
$47$	−11.4641	−1.67221	−0.836106	+	0.548569i	$0.815173\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	−0.732051	−0.102508
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−2.73205	−0.368390
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	9.46410	1.23212	0.616061	−	0.787699i	$-0.288728\pi$
$59$	9.46410	1.23212	0.616061	+	0.787699i	$0.288728\pi$
$60$	0	0
$61$	−12.9282	−1.65529	−0.827643	−	0.561254i	$-0.810319\pi$
$61$	−12.9282	−1.65529	−0.827643	+	0.561254i	$0.810319\pi$
$62$	0	0
$63$	11.6603	1.46905
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.46410	−0.911885	−0.455943	−	0.890009i	$-0.650698\pi$
$67$	−7.46410	−0.911885	−0.455943	+	0.890009i	$0.650698\pi$
$68$	0	0
$69$	−2.39230	−0.288000
$70$	0	0
$71$	−10.7321	−1.27366	−0.636830	−	0.771004i	$-0.719755\pi$
$71$	−10.7321	−1.27366	−0.636830	+	0.771004i	$0.719755\pi$
$72$	0	0
$73$	−7.46410	−0.873607	−0.436804	−	0.899557i	$-0.643889\pi$
$73$	−7.46410	−0.873607	−0.436804	+	0.899557i	$0.643889\pi$
$74$	0	0
$75$	0.732051	0.0845299
$76$	0	0
$77$	12.9282	1.47331
$78$	0	0
$79$	−16.5885	−1.86635	−0.933174	−	0.359426i	$-0.882973\pi$
$79$	−16.5885	−1.86635	−0.933174	+	0.359426i	$0.882973\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−8.92820	−0.979998	−0.489999	−	0.871723i	$-0.663003\pi$
$83$	−8.92820	−0.979998	−0.489999	+	0.871723i	$0.663003\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	−2.53590	−0.271877
$88$	0	0
$89$	13.4641	1.42719	0.713596	−	0.700557i	$-0.247065\pi$
$89$	13.4641	1.42719	0.713596	+	0.700557i	$0.247065\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.07180	0.733311
$94$	0	0
$95$	−5.46410	−0.560605
$96$	0	0
$97$	0.928203	0.0942448	0.0471224	−	0.998889i	$-0.484995\pi$
$97$	0.928203	0.0942448	0.0471224	+	0.998889i	$0.484995\pi$
$98$	0	0
$99$	6.73205	0.676597
$100$	0	0
$101$	8.39230	0.835066	0.417533	−	0.908662i	$-0.362895\pi$
$101$	8.39230	0.835066	0.417533	+	0.908662i	$0.362895\pi$
$102$	0	0
$103$	3.46410	0.341328	0.170664	−	0.985329i	$-0.445409\pi$
$103$	3.46410	0.341328	0.170664	+	0.985329i	$0.445409\pi$
$104$	0	0
$105$	−3.46410	−0.338062
$106$	0	0
$107$	4.73205	0.457465	0.228732	−	0.973489i	$-0.426542\pi$
$107$	4.73205	0.457465	0.228732	+	0.973489i	$0.426542\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−0.392305	−0.0372359
$112$	0	0
$113$	−3.46410	−0.325875	−0.162938	−	0.986636i	$-0.552097\pi$
$113$	−3.46410	−0.325875	−0.162938	+	0.986636i	$0.552097\pi$
$114$	0	0
$115$	−3.26795	−0.304738
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.73205	0.433786
$120$	0	0
$121$	−3.53590	−0.321445
$122$	0	0
$123$	7.60770	0.685963
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.92820	−0.437307	−0.218654	−	0.975803i	$-0.570166\pi$
$127$	−4.92820	−0.437307	−0.218654	+	0.975803i	$0.570166\pi$
$128$	0	0
$129$	6.53590	0.575454
$130$	0	0
$131$	−1.66025	−0.145057	−0.0725285	−	0.997366i	$-0.523107\pi$
$131$	−1.66025	−0.145057	−0.0725285	+	0.997366i	$0.523107\pi$
$132$	0	0
$133$	25.8564	2.24203
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	1.07180	0.0915698	0.0457849	−	0.998951i	$-0.485421\pi$
$137$	1.07180	0.0915698	0.0457849	+	0.998951i	$0.485421\pi$
$138$	0	0
$139$	12.5885	1.06774	0.533870	−	0.845567i	$-0.320737\pi$
$139$	12.5885	1.06774	0.533870	+	0.845567i	$0.320737\pi$
$140$	0	0
$141$	−8.39230	−0.706760
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.46410	−0.287678
$146$	0	0
$147$	11.2679	0.929365
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−8.39230	−0.682956	−0.341478	−	0.939890i	$-0.610927\pi$
$151$	−8.39230	−0.682956	−0.341478	+	0.939890i	$0.610927\pi$
$152$	0	0
$153$	2.46410	0.199211
$154$	0	0
$155$	9.66025	0.775930
$156$	0	0
$157$	−20.9282	−1.67025	−0.835126	−	0.550058i	$-0.814605\pi$
$157$	−20.9282	−1.67025	−0.835126	+	0.550058i	$0.814605\pi$
$158$	0	0
$159$	−1.46410	−0.116111
$160$	0	0
$161$	15.4641	1.21874
$162$	0	0
$163$	8.73205	0.683947	0.341974	−	0.939710i	$-0.388905\pi$
$163$	8.73205	0.683947	0.341974	+	0.939710i	$0.388905\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	3.66025	0.283239	0.141619	−	0.989921i	$-0.454769\pi$
$167$	3.66025	0.283239	0.141619	+	0.989921i	$0.454769\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	13.4641	1.02963
$172$	0	0
$173$	−10.3923	−0.790112	−0.395056	−	0.918657i	$-0.629275\pi$
$173$	−10.3923	−0.790112	−0.395056	+	0.918657i	$0.629275\pi$
$174$	0	0
$175$	−4.73205	−0.357709
$176$	0	0
$177$	6.92820	0.520756
$178$	0	0
$179$	9.46410	0.707380	0.353690	−	0.935363i	$-0.384927\pi$
$179$	9.46410	0.707380	0.353690	+	0.935363i	$0.384927\pi$
$180$	0	0
$181$	9.32051	0.692788	0.346394	−	0.938089i	$-0.387406\pi$
$181$	9.32051	0.692788	0.346394	+	0.938089i	$0.387406\pi$
$182$	0	0
$183$	−9.46410	−0.699607
$184$	0	0
$185$	−0.535898	−0.0394000
$186$	0	0
$187$	2.73205	0.199787
$188$	0	0
$189$	18.9282	1.37682
$190$	0	0
$191$	9.07180	0.656412	0.328206	−	0.944606i	$-0.393556\pi$
$191$	9.07180	0.656412	0.328206	+	0.944606i	$0.393556\pi$
$192$	0	0
$193$	−7.46410	−0.537278	−0.268639	−	0.963241i	$-0.586574\pi$
$193$	−7.46410	−0.537278	−0.268639	+	0.963241i	$0.586574\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.3923	−1.88037	−0.940187	−	0.340659i	$-0.889350\pi$
$197$	−26.3923	−1.88037	−0.940187	+	0.340659i	$0.889350\pi$
$198$	0	0
$199$	−10.3397	−0.732965	−0.366483	−	0.930425i	$-0.619438\pi$
$199$	−10.3397	−0.732965	−0.366483	+	0.930425i	$0.619438\pi$
$200$	0	0
$201$	−5.46410	−0.385408
$202$	0	0
$203$	16.3923	1.15051
$204$	0	0
$205$	10.3923	0.725830
$206$	0	0
$207$	8.05256	0.559692
$208$	0	0
$209$	14.9282	1.03261
$210$	0	0
$211$	−12.1962	−0.839618	−0.419809	−	0.907613i	$-0.637903\pi$
$211$	−12.1962	−0.839618	−0.419809	+	0.907613i	$0.637903\pi$
$212$	0	0
$213$	−7.85641	−0.538312
$214$	0	0
$215$	8.92820	0.608898
$216$	0	0
$217$	−45.7128	−3.10319
$218$	0	0
$219$	−5.46410	−0.369230
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.85641	0.526104	0.263052	−	0.964782i	$-0.415271\pi$
$223$	7.85641	0.526104	0.263052	+	0.964782i	$0.415271\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	12.7321	0.845056	0.422528	−	0.906350i	$-0.361143\pi$
$227$	12.7321	0.845056	0.422528	+	0.906350i	$0.361143\pi$
$228$	0	0
$229$	−4.39230	−0.290252	−0.145126	−	0.989413i	$-0.546359\pi$
$229$	−4.39230	−0.290252	−0.145126	+	0.989413i	$0.546359\pi$
$230$	0	0
$231$	9.46410	0.622692
$232$	0	0
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	−11.4641	−0.747836
$236$	0	0
$237$	−12.1436	−0.788811
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−11.4641	−0.738468	−0.369234	−	0.929337i	$-0.620380\pi$
$241$	−11.4641	−0.738468	−0.369234	+	0.929337i	$0.620380\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	15.3923	0.983378
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.53590	−0.414196
$250$	0	0
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	0	0
$253$	8.92820	0.561311
$254$	0	0
$255$	−0.732051	−0.0458428
$256$	0	0
$257$	2.14359	0.133714	0.0668568	−	0.997763i	$-0.478703\pi$
$257$	2.14359	0.133714	0.0668568	+	0.997763i	$0.478703\pi$
$258$	0	0
$259$	2.53590	0.157573
$260$	0	0
$261$	8.53590	0.528359
$262$	0	0
$263$	20.9282	1.29049	0.645244	−	0.763976i	$-0.276756\pi$
$263$	20.9282	1.29049	0.645244	+	0.763976i	$0.276756\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	9.85641	0.603202
$268$	0	0
$269$	30.7846	1.87697	0.938485	−	0.345319i	$-0.112229\pi$
$269$	30.7846	1.87697	0.938485	+	0.345319i	$0.112229\pi$
$270$	0	0
$271$	16.7846	1.01959	0.509796	−	0.860295i	$-0.329721\pi$
$271$	16.7846	1.01959	0.509796	+	0.860295i	$0.329721\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.73205	−0.164749
$276$	0	0
$277$	−3.07180	−0.184566	−0.0922832	−	0.995733i	$-0.529417\pi$
$277$	−3.07180	−0.184566	−0.0922832	+	0.995733i	$0.529417\pi$
$278$	0	0
$279$	−23.8038	−1.42510
$280$	0	0
$281$	11.0718	0.660488	0.330244	−	0.943896i	$-0.392869\pi$
$281$	11.0718	0.660488	0.330244	+	0.943896i	$0.392869\pi$
$282$	0	0
$283$	−8.05256	−0.478675	−0.239337	−	0.970936i	$-0.576930\pi$
$283$	−8.05256	−0.478675	−0.239337	+	0.970936i	$0.576930\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−49.1769	−2.90282
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0.679492	0.0398325
$292$	0	0
$293$	−8.92820	−0.521591	−0.260796	−	0.965394i	$-0.583985\pi$
$293$	−8.92820	−0.521591	−0.260796	+	0.965394i	$0.583985\pi$
$294$	0	0
$295$	9.46410	0.551021
$296$	0	0
$297$	10.9282	0.634119
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−42.2487	−2.43518
$302$	0	0
$303$	6.14359	0.352940
$304$	0	0
$305$	−12.9282	−0.740267
$306$	0	0
$307$	−5.32051	−0.303657	−0.151829	−	0.988407i	$-0.548516\pi$
$307$	−5.32051	−0.303657	−0.151829	+	0.988407i	$0.548516\pi$
$308$	0	0
$309$	2.53590	0.144262
$310$	0	0
$311$	3.12436	0.177166	0.0885830	−	0.996069i	$-0.471766\pi$
$311$	3.12436	0.177166	0.0885830	+	0.996069i	$0.471766\pi$
$312$	0	0
$313$	−32.2487	−1.82280	−0.911402	−	0.411516i	$-0.864999\pi$
$313$	−32.2487	−1.82280	−0.911402	+	0.411516i	$0.864999\pi$
$314$	0	0
$315$	11.6603	0.656981
$316$	0	0
$317$	−3.07180	−0.172529	−0.0862646	−	0.996272i	$-0.527493\pi$
$317$	−3.07180	−0.172529	−0.0862646	+	0.996272i	$0.527493\pi$
$318$	0	0
$319$	9.46410	0.529888
$320$	0	0
$321$	3.46410	0.193347
$322$	0	0
$323$	5.46410	0.304031
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.39230	0.242895
$328$	0	0
$329$	54.2487	2.99083
$330$	0	0
$331$	19.3205	1.06195	0.530976	−	0.847387i	$-0.321826\pi$
$331$	19.3205	1.06195	0.530976	+	0.847387i	$0.321826\pi$
$332$	0	0
$333$	1.32051	0.0723634
$334$	0	0
$335$	−7.46410	−0.407807
$336$	0	0
$337$	−16.9282	−0.922138	−0.461069	−	0.887364i	$-0.652534\pi$
$337$	−16.9282	−0.922138	−0.461069	+	0.887364i	$0.652534\pi$
$338$	0	0
$339$	−2.53590	−0.137731
$340$	0	0
$341$	−26.3923	−1.42922
$342$	0	0
$343$	−39.7128	−2.14429
$344$	0	0
$345$	−2.39230	−0.128797
$346$	0	0
$347$	−4.05256	−0.217553	−0.108776	−	0.994066i	$-0.534693\pi$
$347$	−4.05256	−0.217553	−0.108776	+	0.994066i	$0.534693\pi$
$348$	0	0
$349$	−32.9282	−1.76261	−0.881303	−	0.472551i	$-0.843333\pi$
$349$	−32.9282	−1.76261	−0.881303	+	0.472551i	$0.843333\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.7846	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$353$	26.7846	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$354$	0	0
$355$	−10.7321	−0.569598
$356$	0	0
$357$	3.46410	0.183340
$358$	0	0
$359$	−9.46410	−0.499496	−0.249748	−	0.968311i	$-0.580348\pi$
$359$	−9.46410	−0.499496	−0.249748	+	0.968311i	$0.580348\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	0	0
$363$	−2.58846	−0.135859
$364$	0	0
$365$	−7.46410	−0.390689
$366$	0	0
$367$	−15.6603	−0.817459	−0.408729	−	0.912656i	$-0.634028\pi$
$367$	−15.6603	−0.817459	−0.408729	+	0.912656i	$0.634028\pi$
$368$	0	0
$369$	−25.6077	−1.33308
$370$	0	0
$371$	9.46410	0.491352
$372$	0	0
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	0	0
$375$	0.732051	0.0378029
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−32.5885	−1.67396	−0.836978	−	0.547236i	$-0.815680\pi$
$379$	−32.5885	−1.67396	−0.836978	+	0.547236i	$0.815680\pi$
$380$	0	0
$381$	−3.60770	−0.184828
$382$	0	0
$383$	−26.7846	−1.36863	−0.684315	−	0.729187i	$-0.739899\pi$
$383$	−26.7846	−1.36863	−0.684315	+	0.729187i	$0.739899\pi$
$384$	0	0
$385$	12.9282	0.658882
$386$	0	0
$387$	−22.0000	−1.11832
$388$	0	0
$389$	−30.2487	−1.53367	−0.766835	−	0.641844i	$-0.778170\pi$
$389$	−30.2487	−1.53367	−0.766835	+	0.641844i	$0.778170\pi$
$390$	0	0
$391$	3.26795	0.165267
$392$	0	0
$393$	−1.21539	−0.0613083
$394$	0	0
$395$	−16.5885	−0.834656
$396$	0	0
$397$	33.7128	1.69200	0.845999	−	0.533185i	$-0.179005\pi$
$397$	33.7128	1.69200	0.845999	+	0.533185i	$0.179005\pi$
$398$	0	0
$399$	18.9282	0.947595
$400$	0	0
$401$	−6.78461	−0.338807	−0.169404	−	0.985547i	$-0.554184\pi$
$401$	−6.78461	−0.338807	−0.169404	+	0.985547i	$0.554184\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	4.46410	0.221823
$406$	0	0
$407$	1.46410	0.0725728
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	0.784610	0.0387019
$412$	0	0
$413$	−44.7846	−2.20371
$414$	0	0
$415$	−8.92820	−0.438268
$416$	0	0
$417$	9.21539	0.451280
$418$	0	0
$419$	18.7321	0.915121	0.457560	−	0.889179i	$-0.348723\pi$
$419$	18.7321	0.915121	0.457560	+	0.889179i	$0.348723\pi$
$420$	0	0
$421$	−22.5359	−1.09833	−0.549166	−	0.835713i	$-0.685055\pi$
$421$	−22.5359	−1.09833	−0.549166	+	0.835713i	$0.685055\pi$
$422$	0	0
$423$	28.2487	1.37350
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	61.1769	2.96056
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.73205	−0.324271	−0.162136	−	0.986768i	$-0.551838\pi$
$431$	−6.73205	−0.324271	−0.162136	+	0.986768i	$0.551838\pi$
$432$	0	0
$433$	−2.92820	−0.140720	−0.0703602	−	0.997522i	$-0.522415\pi$
$433$	−2.92820	−0.140720	−0.0703602	+	0.997522i	$0.522415\pi$
$434$	0	0
$435$	−2.53590	−0.121587
$436$	0	0
$437$	17.8564	0.854188
$438$	0	0
$439$	−9.66025	−0.461059	−0.230529	−	0.973065i	$-0.574046\pi$
$439$	−9.66025	−0.461059	−0.230529	+	0.973065i	$0.574046\pi$
$440$	0	0
$441$	−37.9282	−1.80610
$442$	0	0
$443$	11.4641	0.544676	0.272338	−	0.962202i	$-0.412203\pi$
$443$	11.4641	0.544676	0.272338	+	0.962202i	$0.412203\pi$
$444$	0	0
$445$	13.4641	0.638260
$446$	0	0
$447$	13.1769	0.623247
$448$	0	0
$449$	−16.2487	−0.766824	−0.383412	−	0.923577i	$-0.625251\pi$
$449$	−16.2487	−0.766824	−0.383412	+	0.923577i	$0.625251\pi$
$450$	0	0
$451$	−28.3923	−1.33694
$452$	0	0
$453$	−6.14359	−0.288651
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	7.07180	0.329366	0.164683	−	0.986347i	$-0.447340\pi$
$461$	7.07180	0.329366	0.164683	+	0.986347i	$0.447340\pi$
$462$	0	0
$463$	38.3923	1.78424	0.892121	−	0.451797i	$-0.149217\pi$
$463$	38.3923	1.78424	0.892121	+	0.451797i	$0.149217\pi$
$464$	0	0
$465$	7.07180	0.327947
$466$	0	0
$467$	11.0718	0.512342	0.256171	−	0.966632i	$-0.417539\pi$
$467$	11.0718	0.512342	0.256171	+	0.966632i	$0.417539\pi$
$468$	0	0
$469$	35.3205	1.63095
$470$	0	0
$471$	−15.3205	−0.705932
$472$	0	0
$473$	−24.3923	−1.12156
$474$	0	0
$475$	−5.46410	−0.250710
$476$	0	0
$477$	4.92820	0.225647
$478$	0	0
$479$	−16.1962	−0.740021	−0.370011	−	0.929028i	$-0.620646\pi$
$479$	−16.1962	−0.740021	−0.370011	+	0.929028i	$0.620646\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	11.3205	0.515101
$484$	0	0
$485$	0.928203	0.0421475
$486$	0	0
$487$	19.2679	0.873114	0.436557	−	0.899677i	$-0.356198\pi$
$487$	19.2679	0.873114	0.436557	+	0.899677i	$0.356198\pi$
$488$	0	0
$489$	6.39230	0.289070
$490$	0	0
$491$	6.24871	0.282000	0.141000	−	0.990010i	$-0.454968\pi$
$491$	6.24871	0.282000	0.141000	+	0.990010i	$0.454968\pi$
$492$	0	0
$493$	3.46410	0.156015
$494$	0	0
$495$	6.73205	0.302583
$496$	0	0
$497$	50.7846	2.27800
$498$	0	0
$499$	26.7321	1.19669	0.598345	−	0.801238i	$-0.295825\pi$
$499$	26.7321	1.19669	0.598345	+	0.801238i	$0.295825\pi$
$500$	0	0
$501$	2.67949	0.119711
$502$	0	0
$503$	−21.8038	−0.972186	−0.486093	−	0.873907i	$-0.661578\pi$
$503$	−21.8038	−0.972186	−0.486093	+	0.873907i	$0.661578\pi$
$504$	0	0
$505$	8.39230	0.373453
$506$	0	0
$507$	−9.51666	−0.422650
$508$	0	0
$509$	−23.8564	−1.05742	−0.528708	−	0.848804i	$-0.677323\pi$
$509$	−23.8564	−1.05742	−0.528708	+	0.848804i	$0.677323\pi$
$510$	0	0
$511$	35.3205	1.56249
$512$	0	0
$513$	21.8564	0.964984
$514$	0	0
$515$	3.46410	0.152647
$516$	0	0
$517$	31.3205	1.37747
$518$	0	0
$519$	−7.60770	−0.333941
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	−24.5359	−1.07288	−0.536440	−	0.843938i	$-0.680231\pi$
$523$	−24.5359	−1.07288	−0.536440	+	0.843938i	$0.680231\pi$
$524$	0	0
$525$	−3.46410	−0.151186
$526$	0	0
$527$	−9.66025	−0.420807
$528$	0	0
$529$	−12.3205	−0.535674
$530$	0	0
$531$	−23.3205	−1.01202
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4.73205	0.204584
$536$	0	0
$537$	6.92820	0.298974
$538$	0	0
$539$	−42.0526	−1.81133
$540$	0	0
$541$	33.3205	1.43256	0.716280	−	0.697813i	$-0.245843\pi$
$541$	33.3205	1.43256	0.716280	+	0.697813i	$0.245843\pi$
$542$	0	0
$543$	6.82309	0.292807
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	−3.26795	−0.139727	−0.0698637	−	0.997557i	$-0.522256\pi$
$547$	−3.26795	−0.139727	−0.0698637	+	0.997557i	$0.522256\pi$
$548$	0	0
$549$	31.8564	1.35960
$550$	0	0
$551$	18.9282	0.806369
$552$	0	0
$553$	78.4974	3.33805
$554$	0	0
$555$	−0.392305	−0.0166524
$556$	0	0
$557$	−2.92820	−0.124072	−0.0620360	−	0.998074i	$-0.519759\pi$
$557$	−2.92820	−0.124072	−0.0620360	+	0.998074i	$0.519759\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	13.7128	0.577926	0.288963	−	0.957340i	$-0.406690\pi$
$563$	13.7128	0.577926	0.288963	+	0.957340i	$0.406690\pi$
$564$	0	0
$565$	−3.46410	−0.145736
$566$	0	0
$567$	−21.1244	−0.887140
$568$	0	0
$569$	−6.78461	−0.284426	−0.142213	−	0.989836i	$-0.545422\pi$
$569$	−6.78461	−0.284426	−0.142213	+	0.989836i	$0.545422\pi$
$570$	0	0
$571$	−31.1244	−1.30251	−0.651257	−	0.758857i	$-0.725758\pi$
$571$	−31.1244	−1.30251	−0.651257	+	0.758857i	$0.725758\pi$
$572$	0	0
$573$	6.64102	0.277432
$574$	0	0
$575$	−3.26795	−0.136283
$576$	0	0
$577$	−31.7128	−1.32022	−0.660111	−	0.751168i	$-0.729491\pi$
$577$	−31.7128	−1.32022	−0.660111	+	0.751168i	$0.729491\pi$
$578$	0	0
$579$	−5.46410	−0.227080
$580$	0	0
$581$	42.2487	1.75277
$582$	0	0
$583$	5.46410	0.226300
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.6410	1.67743	0.838717	−	0.544567i	$-0.183306\pi$
$587$	40.6410	1.67743	0.838717	+	0.544567i	$0.183306\pi$
$588$	0	0
$589$	−52.7846	−2.17495
$590$	0	0
$591$	−19.3205	−0.794740
$592$	0	0
$593$	25.7128	1.05590	0.527949	−	0.849276i	$-0.322961\pi$
$593$	25.7128	1.05590	0.527949	+	0.849276i	$0.322961\pi$
$594$	0	0
$595$	4.73205	0.193995
$596$	0	0
$597$	−7.56922	−0.309788
$598$	0	0
$599$	−1.07180	−0.0437924	−0.0218962	−	0.999760i	$-0.506970\pi$
$599$	−1.07180	−0.0437924	−0.0218962	+	0.999760i	$0.506970\pi$
$600$	0	0
$601$	−25.3205	−1.03285	−0.516423	−	0.856334i	$-0.672737\pi$
$601$	−25.3205	−1.03285	−0.516423	+	0.856334i	$0.672737\pi$
$602$	0	0
$603$	18.3923	0.748993
$604$	0	0
$605$	−3.53590	−0.143755
$606$	0	0
$607$	−46.3013	−1.87931	−0.939655	−	0.342123i	$-0.888854\pi$
$607$	−46.3013	−1.87931	−0.939655	+	0.342123i	$0.888854\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	7.60770	0.306772
$616$	0	0
$617$	−1.32051	−0.0531617	−0.0265808	−	0.999647i	$-0.508462\pi$
$617$	−1.32051	−0.0531617	−0.0265808	+	0.999647i	$0.508462\pi$
$618$	0	0
$619$	−19.8038	−0.795984	−0.397992	−	0.917389i	$-0.630293\pi$
$619$	−19.8038	−0.795984	−0.397992	+	0.917389i	$0.630293\pi$
$620$	0	0
$621$	13.0718	0.524553
$622$	0	0
$623$	−63.7128	−2.55260
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.9282	0.436430
$628$	0	0
$629$	0.535898	0.0213677
$630$	0	0
$631$	−4.39230	−0.174855	−0.0874274	−	0.996171i	$-0.527865\pi$
$631$	−4.39230	−0.174855	−0.0874274	+	0.996171i	$0.527865\pi$
$632$	0	0
$633$	−8.92820	−0.354864
$634$	0	0
$635$	−4.92820	−0.195570
$636$	0	0
$637$	0	0
$638$	0	0
$639$	26.4449	1.04614
$640$	0	0
$641$	21.6077	0.853453	0.426726	−	0.904381i	$-0.359667\pi$
$641$	21.6077	0.853453	0.426726	+	0.904381i	$0.359667\pi$
$642$	0	0
$643$	29.8038	1.17535	0.587675	−	0.809097i	$-0.300044\pi$
$643$	29.8038	1.17535	0.587675	+	0.809097i	$0.300044\pi$
$644$	0	0
$645$	6.53590	0.257351
$646$	0	0
$647$	34.3923	1.35210	0.676051	−	0.736855i	$-0.263690\pi$
$647$	34.3923	1.35210	0.676051	+	0.736855i	$0.263690\pi$
$648$	0	0
$649$	−25.8564	−1.01495
$650$	0	0
$651$	−33.4641	−1.31156
$652$	0	0
$653$	16.5359	0.647100	0.323550	−	0.946211i	$-0.395124\pi$
$653$	16.5359	0.647100	0.323550	+	0.946211i	$0.395124\pi$
$654$	0	0
$655$	−1.66025	−0.0648715
$656$	0	0
$657$	18.3923	0.717552
$658$	0	0
$659$	29.8564	1.16304	0.581520	−	0.813532i	$-0.302458\pi$
$659$	29.8564	1.16304	0.581520	+	0.813532i	$0.302458\pi$
$660$	0	0
$661$	20.1436	0.783495	0.391747	−	0.920073i	$-0.371871\pi$
$661$	20.1436	0.783495	0.391747	+	0.920073i	$0.371871\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	25.8564	1.00267
$666$	0	0
$667$	11.3205	0.438332
$668$	0	0
$669$	5.75129	0.222358
$670$	0	0
$671$	35.3205	1.36353
$672$	0	0
$673$	−14.3923	−0.554783	−0.277391	−	0.960757i	$-0.589470\pi$
$673$	−14.3923	−0.554783	−0.277391	+	0.960757i	$0.589470\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	23.0718	0.886721	0.443361	−	0.896343i	$-0.353786\pi$
$677$	23.0718	0.886721	0.443361	+	0.896343i	$0.353786\pi$
$678$	0	0
$679$	−4.39230	−0.168561
$680$	0	0
$681$	9.32051	0.357163
$682$	0	0
$683$	−37.1244	−1.42052	−0.710262	−	0.703937i	$-0.751424\pi$
$683$	−37.1244	−1.42052	−0.710262	+	0.703937i	$0.751424\pi$
$684$	0	0
$685$	1.07180	0.0409512
$686$	0	0
$687$	−3.21539	−0.122675
$688$	0	0
$689$	0	0
$690$	0	0
$691$	46.8372	1.78177	0.890885	−	0.454229i	$-0.150085\pi$
$691$	46.8372	1.78177	0.890885	+	0.454229i	$0.150085\pi$
$692$	0	0
$693$	−31.8564	−1.21012
$694$	0	0
$695$	12.5885	0.477507
$696$	0	0
$697$	−10.3923	−0.393637
$698$	0	0
$699$	−19.0333	−0.719906
$700$	0	0
$701$	37.1769	1.40415	0.702076	−	0.712102i	$-0.252257\pi$
$701$	37.1769	1.40415	0.702076	+	0.712102i	$0.252257\pi$
$702$	0	0
$703$	2.92820	0.110439
$704$	0	0
$705$	−8.39230	−0.316072
$706$	0	0
$707$	−39.7128	−1.49355
$708$	0	0
$709$	−15.1769	−0.569981	−0.284990	−	0.958530i	$-0.591990\pi$
$709$	−15.1769	−0.569981	−0.284990	+	0.958530i	$0.591990\pi$
$710$	0	0
$711$	40.8756	1.53296
$712$	0	0
$713$	−31.5692	−1.18228
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.78461	−0.328067
$718$	0	0
$719$	−46.7321	−1.74281	−0.871406	−	0.490563i	$-0.836791\pi$
$719$	−46.7321	−1.74281	−0.871406	+	0.490563i	$0.836791\pi$
$720$	0	0
$721$	−16.3923	−0.610481
$722$	0	0
$723$	−8.39230	−0.312113
$724$	0	0
$725$	−3.46410	−0.128654
$726$	0	0
$727$	20.2487	0.750983	0.375492	−	0.926826i	$-0.377474\pi$
$727$	20.2487	0.750983	0.375492	+	0.926826i	$0.377474\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−8.92820	−0.330222
$732$	0	0
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	0	0
$735$	11.2679	0.415625
$736$	0	0
$737$	20.3923	0.751160
$738$	0	0
$739$	38.2487	1.40700	0.703501	−	0.710694i	$-0.251619\pi$
$739$	38.2487	1.40700	0.703501	+	0.710694i	$0.251619\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.9808	0.989828	0.494914	−	0.868942i	$-0.335200\pi$
$743$	26.9808	0.989828	0.494914	+	0.868942i	$0.335200\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	22.0000	0.804938
$748$	0	0
$749$	−22.3923	−0.818197
$750$	0	0
$751$	−4.87564	−0.177915	−0.0889574	−	0.996035i	$-0.528353\pi$
$751$	−4.87564	−0.177915	−0.0889574	+	0.996035i	$0.528353\pi$
$752$	0	0
$753$	5.07180	0.184827
$754$	0	0
$755$	−8.39230	−0.305427
$756$	0	0
$757$	52.7846	1.91849	0.959245	−	0.282577i	$-0.0911893\pi$
$757$	52.7846	1.91849	0.959245	+	0.282577i	$0.0911893\pi$
$758$	0	0
$759$	6.53590	0.237238
$760$	0	0
$761$	8.39230	0.304221	0.152110	−	0.988364i	$-0.451393\pi$
$761$	8.39230	0.304221	0.152110	+	0.988364i	$0.451393\pi$
$762$	0	0
$763$	−28.3923	−1.02787
$764$	0	0
$765$	2.46410	0.0890898
$766$	0	0
$767$	0	0
$768$	0	0
$769$	41.1769	1.48488	0.742439	−	0.669914i	$-0.233669\pi$
$769$	41.1769	1.48488	0.742439	+	0.669914i	$0.233669\pi$
$770$	0	0
$771$	1.56922	0.0565141
$772$	0	0
$773$	30.9282	1.11241	0.556205	−	0.831045i	$-0.312257\pi$
$773$	30.9282	1.11241	0.556205	+	0.831045i	$0.312257\pi$
$774$	0	0
$775$	9.66025	0.347007
$776$	0	0
$777$	1.85641	0.0665982
$778$	0	0
$779$	−56.7846	−2.03452
$780$	0	0
$781$	29.3205	1.04917
$782$	0	0
$783$	13.8564	0.495188
$784$	0	0
$785$	−20.9282	−0.746960
$786$	0	0
$787$	−0.732051	−0.0260948	−0.0130474	−	0.999915i	$-0.504153\pi$
$787$	−0.732051	−0.0260948	−0.0130474	+	0.999915i	$0.504153\pi$
$788$	0	0
$789$	15.3205	0.545425
$790$	0	0
$791$	16.3923	0.582843
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1.46410	−0.0519263
$796$	0	0
$797$	4.92820	0.174566	0.0872830	−	0.996184i	$-0.472182\pi$
$797$	4.92820	0.174566	0.0872830	+	0.996184i	$0.472182\pi$
$798$	0	0
$799$	11.4641	0.405571
$800$	0	0
$801$	−33.1769	−1.17225
$802$	0	0
$803$	20.3923	0.719629
$804$	0	0
$805$	15.4641	0.545038
$806$	0	0
$807$	22.5359	0.793301
$808$	0	0
$809$	−15.8564	−0.557482	−0.278741	−	0.960366i	$-0.589917\pi$
$809$	−15.8564	−0.557482	−0.278741	+	0.960366i	$0.589917\pi$
$810$	0	0
$811$	−21.2679	−0.746819	−0.373409	−	0.927667i	$-0.621811\pi$
$811$	−21.2679	−0.746819	−0.373409	+	0.927667i	$0.621811\pi$
$812$	0	0
$813$	12.2872	0.430930
$814$	0	0
$815$	8.73205	0.305870
$816$	0	0
$817$	−48.7846	−1.70676
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.6410	0.859977	0.429989	−	0.902834i	$-0.358518\pi$
$821$	24.6410	0.859977	0.429989	+	0.902834i	$0.358518\pi$
$822$	0	0
$823$	−22.1962	−0.773709	−0.386855	−	0.922141i	$-0.626438\pi$
$823$	−22.1962	−0.773709	−0.386855	+	0.922141i	$0.626438\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	24.0526	0.836389	0.418195	−	0.908357i	$-0.362663\pi$
$827$	24.0526	0.836389	0.418195	+	0.908357i	$0.362663\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	−2.24871	−0.0780069
$832$	0	0
$833$	−15.3923	−0.533312
$834$	0	0
$835$	3.66025	0.126668
$836$	0	0
$837$	−38.6410	−1.33563
$838$	0	0
$839$	−39.1244	−1.35072	−0.675361	−	0.737487i	$-0.736012\pi$
$839$	−39.1244	−1.35072	−0.675361	+	0.737487i	$0.736012\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	8.10512	0.279155
$844$	0	0
$845$	−13.0000	−0.447214
$846$	0	0
$847$	16.7321	0.574920
$848$	0	0
$849$	−5.89488	−0.202312
$850$	0	0
$851$	1.75129	0.0600334
$852$	0	0
$853$	−15.1769	−0.519648	−0.259824	−	0.965656i	$-0.583664\pi$
$853$	−15.1769	−0.519648	−0.259824	+	0.965656i	$0.583664\pi$
$854$	0	0
$855$	13.4641	0.460463
$856$	0	0
$857$	29.6077	1.01138	0.505690	−	0.862715i	$-0.331238\pi$
$857$	29.6077	1.01138	0.505690	+	0.862715i	$0.331238\pi$
$858$	0	0
$859$	−21.4641	−0.732346	−0.366173	−	0.930547i	$-0.619332\pi$
$859$	−21.4641	−0.732346	−0.366173	+	0.930547i	$0.619332\pi$
$860$	0	0
$861$	−36.0000	−1.22688
$862$	0	0
$863$	−27.4641	−0.934889	−0.467444	−	0.884022i	$-0.654825\pi$
$863$	−27.4641	−0.934889	−0.467444	+	0.884022i	$0.654825\pi$
$864$	0	0
$865$	−10.3923	−0.353349
$866$	0	0
$867$	0.732051	0.0248617
$868$	0	0
$869$	45.3205	1.53739
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.28719	−0.0774096
$874$	0	0
$875$	−4.73205	−0.159973
$876$	0	0
$877$	20.6410	0.696998	0.348499	−	0.937309i	$-0.386692\pi$
$877$	20.6410	0.696998	0.348499	+	0.937309i	$0.386692\pi$
$878$	0	0
$879$	−6.53590	−0.220450
$880$	0	0
$881$	−17.3205	−0.583543	−0.291771	−	0.956488i	$-0.594245\pi$
$881$	−17.3205	−0.583543	−0.291771	+	0.956488i	$0.594245\pi$
$882$	0	0
$883$	−21.3205	−0.717492	−0.358746	−	0.933435i	$-0.616796\pi$
$883$	−21.3205	−0.717492	−0.358746	+	0.933435i	$0.616796\pi$
$884$	0	0
$885$	6.92820	0.232889
$886$	0	0
$887$	−47.6603	−1.60027	−0.800137	−	0.599817i	$-0.795240\pi$
$887$	−47.6603	−1.60027	−0.800137	+	0.599817i	$0.795240\pi$
$888$	0	0
$889$	23.3205	0.782145
$890$	0	0
$891$	−12.1962	−0.408586
$892$	0	0
$893$	62.6410	2.09620
$894$	0	0
$895$	9.46410	0.316350
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−33.4641	−1.11609
$900$	0	0
$901$	2.00000	0.0666297
$902$	0	0
$903$	−30.9282	−1.02923
$904$	0	0
$905$	9.32051	0.309824
$906$	0	0
$907$	1.01924	0.0338432	0.0169216	−	0.999857i	$-0.494613\pi$
$907$	1.01924	0.0338432	0.0169216	+	0.999857i	$0.494613\pi$
$908$	0	0
$909$	−20.6795	−0.685895
$910$	0	0
$911$	8.58846	0.284548	0.142274	−	0.989827i	$-0.454559\pi$
$911$	8.58846	0.284548	0.142274	+	0.989827i	$0.454559\pi$
$912$	0	0
$913$	24.3923	0.807267
$914$	0	0
$915$	−9.46410	−0.312874
$916$	0	0
$917$	7.85641	0.259441
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−3.89488	−0.128341
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−0.535898	−0.0176202
$926$	0	0
$927$	−8.53590	−0.280356
$928$	0	0
$929$	8.24871	0.270631	0.135316	−	0.990803i	$-0.456795\pi$
$929$	8.24871	0.270631	0.135316	+	0.990803i	$0.456795\pi$
$930$	0	0
$931$	−84.1051	−2.75643
$932$	0	0
$933$	2.28719	0.0748791
$934$	0	0
$935$	2.73205	0.0893476
$936$	0	0
$937$	−52.9282	−1.72909	−0.864545	−	0.502556i	$-0.832393\pi$
$937$	−52.9282	−1.72909	−0.864545	+	0.502556i	$0.832393\pi$
$938$	0	0
$939$	−23.6077	−0.770408
$940$	0	0
$941$	−1.32051	−0.0430473	−0.0215237	−	0.999768i	$-0.506852\pi$
$941$	−1.32051	−0.0430473	−0.0215237	+	0.999768i	$0.506852\pi$
$942$	0	0
$943$	−33.9615	−1.10594
$944$	0	0
$945$	18.9282	0.615734
$946$	0	0
$947$	39.2679	1.27604	0.638018	−	0.770021i	$-0.279754\pi$
$947$	39.2679	1.27604	0.638018	+	0.770021i	$0.279754\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−2.24871	−0.0729195
$952$	0	0
$953$	−6.14359	−0.199011	−0.0995053	−	0.995037i	$-0.531726\pi$
$953$	−6.14359	−0.199011	−0.0995053	+	0.995037i	$0.531726\pi$
$954$	0	0
$955$	9.07180	0.293556
$956$	0	0
$957$	6.92820	0.223957
$958$	0	0
$959$	−5.07180	−0.163777
$960$	0	0
$961$	62.3205	2.01034
$962$	0	0
$963$	−11.6603	−0.375746
$964$	0	0
$965$	−7.46410	−0.240278
$966$	0	0
$967$	47.8564	1.53896	0.769479	−	0.638672i	$-0.220516\pi$
$967$	47.8564	1.53896	0.769479	+	0.638672i	$0.220516\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−40.1051	−1.28703	−0.643517	−	0.765432i	$-0.722526\pi$
$971$	−40.1051	−1.28703	−0.643517	+	0.765432i	$0.722526\pi$
$972$	0	0
$973$	−59.5692	−1.90970
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.4256	1.70924	0.854619	−	0.519256i	$-0.173791\pi$
$977$	53.4256	1.70924	0.854619	+	0.519256i	$0.173791\pi$
$978$	0	0
$979$	−36.7846	−1.17564
$980$	0	0
$981$	−14.7846	−0.472036
$982$	0	0
$983$	32.4449	1.03483	0.517415	−	0.855734i	$-0.326894\pi$
$983$	32.4449	1.03483	0.517415	+	0.855734i	$0.326894\pi$
$984$	0	0
$985$	−26.3923	−0.840929
$986$	0	0
$987$	39.7128	1.26407
$988$	0	0
$989$	−29.1769	−0.927772
$990$	0	0
$991$	27.9090	0.886558	0.443279	−	0.896384i	$-0.353815\pi$
$991$	27.9090	0.886558	0.443279	+	0.896384i	$0.353815\pi$
$992$	0	0
$993$	14.1436	0.448833
$994$	0	0
$995$	−10.3397	−0.327792
$996$	0	0
$997$	−28.5359	−0.903741	−0.451870	−	0.892084i	$-0.649243\pi$
$997$	−28.5359	−0.903741	−0.451870	+	0.892084i	$0.649243\pi$
$998$	0	0
$999$	2.14359	0.0678203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	680.2.a.d.1.2	✓	2
3.2	odd	2		6120.2.a.ba.1.1		2
4.3	odd	2		1360.2.a.n.1.1		2
5.2	odd	4		3400.2.e.g.2449.2		4
5.3	odd	4		3400.2.e.g.2449.3		4
5.4	even	2		3400.2.a.j.1.1		2
8.3	odd	2		5440.2.a.bc.1.2		2
8.5	even	2		5440.2.a.bk.1.1		2
20.19	odd	2		6800.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.a.d.1.2	✓	2	1.1	even	1	trivial
1360.2.a.n.1.1		2	4.3	odd	2
3400.2.a.j.1.1		2	5.4	even	2
3400.2.e.g.2449.2		4	5.2	odd	4
3400.2.e.g.2449.3		4	5.3	odd	4
5440.2.a.bc.1.2		2	8.3	odd	2
5440.2.a.bk.1.1		2	8.5	even	2
6120.2.a.ba.1.1		2	3.2	odd	2
6800.2.a.bb.1.2		2	20.19	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} +1.00000 q^{5} -4.73205 q^{7} -2.46410 q^{9} -2.73205 q^{11} +0.732051 q^{15} -1.00000 q^{17} -5.46410 q^{19} -3.46410 q^{21} -3.26795 q^{23} +1.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} +9.66025 q^{31} -2.00000 q^{33} -4.73205 q^{35} -0.535898 q^{37} +10.3923 q^{41} +8.92820 q^{43} -2.46410 q^{45} -11.4641 q^{47} +15.3923 q^{49} -0.732051 q^{51} -2.00000 q^{53} -2.73205 q^{55} -4.00000 q^{57} +9.46410 q^{59} -12.9282 q^{61} +11.6603 q^{63} -7.46410 q^{67} -2.39230 q^{69} -10.7321 q^{71} -7.46410 q^{73} +0.732051 q^{75} +12.9282 q^{77} -16.5885 q^{79} +4.46410 q^{81} -8.92820 q^{83} -1.00000 q^{85} -2.53590 q^{87} +13.4641 q^{89} +7.07180 q^{93} -5.46410 q^{95} +0.928203 q^{97} +6.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{15} - 2 q^{17} - 4 q^{19} - 10 q^{23} + 2 q^{25} - 8 q^{27} + 2 q^{31} - 4 q^{33} - 6 q^{35} - 8 q^{37} + 4 q^{43} + 2 q^{45} - 16 q^{47} + 10 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^15 - 2 * q^17 - 4 * q^19 - 10 * q^23 + 2 * q^25 - 8 * q^27 + 2 * q^31 - 4 * q^33 - 6 * q^35 - 8 * q^37 + 4 * q^43 + 2 * q^45 - 16 * q^47 + 10 * q^49 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 8 * q^57 + 12 * q^59 - 12 * q^61 + 6 * q^63 - 8 * q^67 + 16 * q^69 - 18 * q^71 - 8 * q^73 - 2 * q^75 + 12 * q^77 - 2 * q^79 + 2 * q^81 - 4 * q^83 - 2 * q^85 - 12 * q^87 + 20 * q^89 + 28 * q^93 - 4 * q^95 - 12 * q^97 + 10 * q^99

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