LMFDB - Embedded newform 4352.2.a.bf.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4352,2,Mod(1,4352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4352.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4352 = 2^{8} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4352.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,12,0,8,0,0,0,0,0,12,0,8,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.7508949597$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.71515595776.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 13x^{6} + 55x^{4} - 79x^{2} + 16 $ x^8 - 13x^6 + 55x^4 - 79*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.37995$ of defining polynomial
Character	$\chi$	$=$	4352.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.10562 q^{3} -1.58069 q^{5} +5.01127 q^{7} +1.43364 q^{9} +2.65428 q^{11} +6.34060 q^{13} -3.32834 q^{15} +1.00000 q^{17} +2.44834 q^{19} +10.5518 q^{21} -2.83821 q^{23} -2.50141 q^{25} -3.29817 q^{27} -1.58069 q^{29} +1.68293 q^{31} +5.58891 q^{33} -7.92129 q^{35} +1.16881 q^{37} +13.3509 q^{39} +3.50141 q^{41} -6.52280 q^{43} -2.26614 q^{45} +8.82975 q^{47} +18.1129 q^{49} +2.10562 q^{51} +7.37263 q^{53} -4.19561 q^{55} +5.15528 q^{57} -10.7815 q^{59} -1.58069 q^{61} +7.18434 q^{63} -10.0225 q^{65} -9.82097 q^{67} -5.97619 q^{69} -2.51268 q^{71} +2.52114 q^{73} -5.26701 q^{75} +13.3013 q^{77} +0.815662 q^{79} -11.2456 q^{81} -8.22246 q^{83} -1.58069 q^{85} -3.32834 q^{87} -1.45337 q^{89} +31.7745 q^{91} +3.54362 q^{93} -3.87008 q^{95} -7.15528 q^{97} +3.80527 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{7} + 8 q^{9} + 12 q^{15} + 8 q^{17} + 16 q^{23} + 8 q^{25} + 24 q^{31} - 8 q^{33} + 12 q^{39} + 4 q^{47} + 8 q^{49} + 12 q^{55} - 8 q^{57} + 40 q^{63} - 24 q^{65} + 36 q^{71} - 8 q^{73} + 24 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) $ 8 * q + 12 * q^7 + 8 * q^9 + 12 * q^15 + 8 * q^17 + 16 * q^23 + 8 * q^25 + 24 * q^31 - 8 * q^33 + 12 * q^39 + 4 * q^47 + 8 * q^49 + 12 * q^55 - 8 * q^57 + 40 * q^63 - 24 * q^65 + 36 * q^71 - 8 * q^73 + 24 * q^79 + 40 * q^81 + 12 * q^87 - 8 * q^89 + 32 * q^95 - 8 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.10562	1.21568	0.607840	−	0.794059i	$-0.292036\pi$
$3$	2.10562	1.21568	0.607840	+	0.794059i	$0.292036\pi$
$4$	0	0
$5$	−1.58069	−0.706908	−0.353454	−	0.935452i	$-0.614993\pi$
$5$	−1.58069	−0.706908	−0.353454	+	0.935452i	$0.614993\pi$
$6$	0	0
$7$	5.01127	1.89408	0.947042	−	0.321110i	$-0.104056\pi$
$7$	5.01127	1.89408	0.947042	+	0.321110i	$0.104056\pi$
$8$	0	0
$9$	1.43364	0.477878
$10$	0	0
$11$	2.65428	0.800297	0.400148	−	0.916450i	$-0.368959\pi$
$11$	2.65428	0.800297	0.400148	+	0.916450i	$0.368959\pi$
$12$	0	0
$13$	6.34060	1.75857	0.879283	−	0.476300i	$-0.158022\pi$
$13$	6.34060	1.75857	0.879283	+	0.476300i	$0.158022\pi$
$14$	0	0
$15$	−3.32834	−0.859374
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.44834	0.561688	0.280844	−	0.959753i	$-0.409386\pi$
$19$	2.44834	0.561688	0.280844	+	0.959753i	$0.409386\pi$
$20$	0	0
$21$	10.5518	2.30260
$22$	0	0
$23$	−2.83821	−0.591808	−0.295904	−	0.955218i	$-0.595621\pi$
$23$	−2.83821	−0.591808	−0.295904	+	0.955218i	$0.595621\pi$
$24$	0	0
$25$	−2.50141	−0.500281
$26$	0	0
$27$	−3.29817	−0.634733
$28$	0	0
$29$	−1.58069	−0.293528	−0.146764	−	0.989172i	$-0.546886\pi$
$29$	−1.58069	−0.293528	−0.146764	+	0.989172i	$0.546886\pi$
$30$	0	0
$31$	1.68293	0.302264	0.151132	−	0.988514i	$-0.451708\pi$
$31$	1.68293	0.302264	0.151132	+	0.988514i	$0.451708\pi$
$32$	0	0
$33$	5.58891	0.972905
$34$	0	0
$35$	−7.92129	−1.33894
$36$	0	0
$37$	1.16881	0.192151	0.0960757	−	0.995374i	$-0.469371\pi$
$37$	1.16881	0.192151	0.0960757	+	0.995374i	$0.469371\pi$
$38$	0	0
$39$	13.3509	2.13785
$40$	0	0
$41$	3.50141	0.546828	0.273414	−	0.961897i	$-0.411847\pi$
$41$	3.50141	0.546828	0.273414	+	0.961897i	$0.411847\pi$
$42$	0	0
$43$	−6.52280	−0.994718	−0.497359	−	0.867545i	$-0.665697\pi$
$43$	−6.52280	−0.994718	−0.497359	+	0.867545i	$0.665697\pi$
$44$	0	0
$45$	−2.26614	−0.337816
$46$	0	0
$47$	8.82975	1.28795	0.643975	−	0.765046i	$-0.277284\pi$
$47$	8.82975	1.28795	0.643975	+	0.765046i	$0.277284\pi$
$48$	0	0
$49$	18.1129	2.58755
$50$	0	0
$51$	2.10562	0.294846
$52$	0	0
$53$	7.37263	1.01271	0.506354	−	0.862326i	$-0.330993\pi$
$53$	7.37263	1.01271	0.506354	+	0.862326i	$0.330993\pi$
$54$	0	0
$55$	−4.19561	−0.565736
$56$	0	0
$57$	5.15528	0.682833
$58$	0	0
$59$	−10.7815	−1.40363	−0.701817	−	0.712357i	$-0.747628\pi$
$59$	−10.7815	−1.40363	−0.701817	+	0.712357i	$0.747628\pi$
$60$	0	0
$61$	−1.58069	−0.202387	−0.101194	−	0.994867i	$-0.532266\pi$
$61$	−1.58069	−0.202387	−0.101194	+	0.994867i	$0.532266\pi$
$62$	0	0
$63$	7.18434	0.905141
$64$	0	0
$65$	−10.0225	−1.24314
$66$	0	0
$67$	−9.82097	−1.19982	−0.599911	−	0.800067i	$-0.704797\pi$
$67$	−9.82097	−1.19982	−0.599911	+	0.800067i	$0.704797\pi$
$68$	0	0
$69$	−5.97619	−0.719449
$70$	0	0
$71$	−2.51268	−0.298200	−0.149100	−	0.988822i	$-0.547638\pi$
$71$	−2.51268	−0.298200	−0.149100	+	0.988822i	$0.547638\pi$
$72$	0	0
$73$	2.52114	0.295077	0.147539	−	0.989056i	$-0.452865\pi$
$73$	2.52114	0.295077	0.147539	+	0.989056i	$0.452865\pi$
$74$	0	0
$75$	−5.26701	−0.608182
$76$	0	0
$77$	13.3013	1.51583
$78$	0	0
$79$	0.815662	0.0917692	0.0458846	−	0.998947i	$-0.485389\pi$
$79$	0.815662	0.0917692	0.0458846	+	0.998947i	$0.485389\pi$
$80$	0	0
$81$	−11.2456	−1.24951
$82$	0	0
$83$	−8.22246	−0.902532	−0.451266	−	0.892390i	$-0.649027\pi$
$83$	−8.22246	−0.902532	−0.451266	+	0.892390i	$0.649027\pi$
$84$	0	0
$85$	−1.58069	−0.171450
$86$	0	0
$87$	−3.32834	−0.356836
$88$	0	0
$89$	−1.45337	−0.154057	−0.0770286	−	0.997029i	$-0.524543\pi$
$89$	−1.45337	−0.154057	−0.0770286	+	0.997029i	$0.524543\pi$
$90$	0	0
$91$	31.7745	3.33087
$92$	0	0
$93$	3.54362	0.367456
$94$	0	0
$95$	−3.87008	−0.397062
$96$	0	0
$97$	−7.15528	−0.726508	−0.363254	−	0.931690i	$-0.618334\pi$
$97$	−7.15528	−0.726508	−0.363254	+	0.931690i	$0.618334\pi$
$98$	0	0
$99$	3.80527	0.382444
$100$	0	0
$101$	−8.67822	−0.863515	−0.431758	−	0.901990i	$-0.642106\pi$
$101$	−8.67822	−0.863515	−0.431758	+	0.901990i	$0.642106\pi$
$102$	0	0
$103$	7.13273	0.702809	0.351404	−	0.936224i	$-0.385704\pi$
$103$	7.13273	0.702809	0.351404	+	0.936224i	$0.385704\pi$
$104$	0	0
$105$	−16.6792	−1.62773
$106$	0	0
$107$	−5.26701	−0.509181	−0.254590	−	0.967049i	$-0.581941\pi$
$107$	−5.26701	−0.509181	−0.254590	+	0.967049i	$0.581941\pi$
$108$	0	0
$109$	−8.35100	−0.799880	−0.399940	−	0.916541i	$-0.630969\pi$
$109$	−8.35100	−0.799880	−0.399940	+	0.916541i	$0.630969\pi$
$110$	0	0
$111$	2.46107	0.233595
$112$	0	0
$113$	−8.02255	−0.754698	−0.377349	−	0.926071i	$-0.623164\pi$
$113$	−8.02255	−0.754698	−0.377349	+	0.926071i	$0.623164\pi$
$114$	0	0
$115$	4.48634	0.418354
$116$	0	0
$117$	9.09010	0.840380
$118$	0	0
$119$	5.01127	0.459383
$120$	0	0
$121$	−3.95478	−0.359525
$122$	0	0
$123$	7.37263	0.664768
$124$	0	0
$125$	11.8574	1.06056
$126$	0	0
$127$	13.5014	1.19806	0.599028	−	0.800728i	$-0.295554\pi$
$127$	13.5014	1.19806	0.599028	+	0.800728i	$0.295554\pi$
$128$	0	0
$129$	−13.7345	−1.20926
$130$	0	0
$131$	−5.90498	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$131$	−5.90498	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$132$	0	0
$133$	12.2693	1.06388
$134$	0	0
$135$	5.21340	0.448698
$136$	0	0
$137$	−1.56636	−0.133824	−0.0669118	−	0.997759i	$-0.521315\pi$
$137$	−1.56636	−0.133824	−0.0669118	+	0.997759i	$0.521315\pi$
$138$	0	0
$139$	−3.29225	−0.279245	−0.139623	−	0.990205i	$-0.544589\pi$
$139$	−3.29225	−0.279245	−0.139623	+	0.990205i	$0.544589\pi$
$140$	0	0
$141$	18.5921	1.56574
$142$	0	0
$143$	16.8297	1.40737
$144$	0	0
$145$	2.49859	0.207497
$146$	0	0
$147$	38.1388	3.14564
$148$	0	0
$149$	−2.70203	−0.221359	−0.110679	−	0.993856i	$-0.535303\pi$
$149$	−2.70203	−0.221359	−0.110679	+	0.993856i	$0.535303\pi$
$150$	0	0
$151$	−2.88982	−0.235170	−0.117585	−	0.993063i	$-0.537515\pi$
$151$	−2.88982	−0.235170	−0.117585	+	0.993063i	$0.537515\pi$
$152$	0	0
$153$	1.43364	0.115903
$154$	0	0
$155$	−2.66020	−0.213673
$156$	0	0
$157$	−1.01421	−0.0809428	−0.0404714	−	0.999181i	$-0.512886\pi$
$157$	−1.01421	−0.0809428	−0.0404714	+	0.999181i	$0.512886\pi$
$158$	0	0
$159$	15.5240	1.23113
$160$	0	0
$161$	−14.2230	−1.12093
$162$	0	0
$163$	15.5259	1.21608	0.608042	−	0.793905i	$-0.291955\pi$
$163$	15.5259	1.21608	0.608042	+	0.793905i	$0.291955\pi$
$164$	0	0
$165$	−8.83436	−0.687754
$166$	0	0
$167$	15.9011	1.23046	0.615232	−	0.788346i	$-0.289062\pi$
$167$	15.9011	1.23046	0.615232	+	0.788346i	$0.289062\pi$
$168$	0	0
$169$	27.2032	2.09255
$170$	0	0
$171$	3.51003	0.268419
$172$	0	0
$173$	−21.1752	−1.60992	−0.804959	−	0.593331i	$-0.797813\pi$
$173$	−21.1752	−1.60992	−0.804959	+	0.593331i	$0.797813\pi$
$174$	0	0
$175$	−12.5352	−0.947574
$176$	0	0
$177$	−22.7018	−1.70637
$178$	0	0
$179$	−16.5913	−1.24009	−0.620045	−	0.784566i	$-0.712886\pi$
$179$	−16.5913	−1.24009	−0.620045	+	0.784566i	$0.712886\pi$
$180$	0	0
$181$	−11.7859	−0.876042	−0.438021	−	0.898965i	$-0.644321\pi$
$181$	−11.7859	−0.876042	−0.438021	+	0.898965i	$0.644321\pi$
$182$	0	0
$183$	−3.32834	−0.246038
$184$	0	0
$185$	−1.84753	−0.135833
$186$	0	0
$187$	2.65428	0.194100
$188$	0	0
$189$	−16.5280	−1.20224
$190$	0	0
$191$	7.96248	0.576145	0.288072	−	0.957609i	$-0.406986\pi$
$191$	7.96248	0.576145	0.288072	+	0.957609i	$0.406986\pi$
$192$	0	0
$193$	25.2229	1.81559	0.907793	−	0.419419i	$-0.137766\pi$
$193$	25.2229	1.81559	0.907793	+	0.419419i	$0.137766\pi$
$194$	0	0
$195$	−21.1037	−1.51127
$196$	0	0
$197$	12.8833	0.917895	0.458948	−	0.888463i	$-0.348227\pi$
$197$	12.8833	0.917895	0.458948	+	0.888463i	$0.348227\pi$
$198$	0	0
$199$	−0.339615	−0.0240747	−0.0120373	−	0.999928i	$-0.503832\pi$
$199$	−0.339615	−0.0240747	−0.0120373	+	0.999928i	$0.503832\pi$
$200$	0	0
$201$	−20.6792	−1.45860
$202$	0	0
$203$	−7.92129	−0.555966
$204$	0	0
$205$	−5.53465	−0.386557
$206$	0	0
$207$	−4.06896	−0.282812
$208$	0	0
$209$	6.49859	0.449517
$210$	0	0
$211$	6.36434	0.438139	0.219069	−	0.975709i	$-0.429698\pi$
$211$	6.36434	0.438139	0.219069	+	0.975709i	$0.429698\pi$
$212$	0	0
$213$	−5.29075	−0.362516
$214$	0	0
$215$	10.3106	0.703174
$216$	0	0
$217$	8.43364	0.572512
$218$	0	0
$219$	5.30857	0.358720
$220$	0	0
$221$	6.34060	0.426515
$222$	0	0
$223$	13.0629	0.874755	0.437378	−	0.899278i	$-0.355907\pi$
$223$	13.0629	0.874755	0.437378	+	0.899278i	$0.355907\pi$
$224$	0	0
$225$	−3.58610	−0.239073
$226$	0	0
$227$	20.3705	1.35204	0.676018	−	0.736885i	$-0.263704\pi$
$227$	20.3705	1.35204	0.676018	+	0.736885i	$0.263704\pi$
$228$	0	0
$229$	17.9245	1.18448	0.592241	−	0.805761i	$-0.298243\pi$
$229$	17.9245	1.18448	0.592241	+	0.805761i	$0.298243\pi$
$230$	0	0
$231$	28.0076	1.84276
$232$	0	0
$233$	−15.8926	−1.04116	−0.520580	−	0.853813i	$-0.674284\pi$
$233$	−15.8926	−1.04116	−0.520580	+	0.853813i	$0.674284\pi$
$234$	0	0
$235$	−13.9571	−0.910463
$236$	0	0
$237$	1.71747	0.111562
$238$	0	0
$239$	−16.3537	−1.05783	−0.528916	−	0.848674i	$-0.677402\pi$
$239$	−16.3537	−1.05783	−0.528916	+	0.848674i	$0.677402\pi$
$240$	0	0
$241$	29.6820	1.91199	0.955994	−	0.293386i	$-0.0947820\pi$
$241$	29.6820	1.91199	0.955994	+	0.293386i	$0.0947820\pi$
$242$	0	0
$243$	−13.7844	−0.884272
$244$	0	0
$245$	−28.6309	−1.82916
$246$	0	0
$247$	15.5240	0.987765
$248$	0	0
$249$	−17.3134	−1.09719
$250$	0	0
$251$	23.7425	1.49861	0.749305	−	0.662225i	$-0.230388\pi$
$251$	23.7425	1.49861	0.749305	+	0.662225i	$0.230388\pi$
$252$	0	0
$253$	−7.53342	−0.473622
$254$	0	0
$255$	−3.32834	−0.208429
$256$	0	0
$257$	−18.9548	−1.18237	−0.591183	−	0.806537i	$-0.701339\pi$
$257$	−18.9548	−1.18237	−0.591183	+	0.806537i	$0.701339\pi$
$258$	0	0
$259$	5.85723	0.363951
$260$	0	0
$261$	−2.26614	−0.140270
$262$	0	0
$263$	−9.54650	−0.588662	−0.294331	−	0.955703i	$-0.595097\pi$
$263$	−9.54650	−0.588662	−0.294331	+	0.955703i	$0.595097\pi$
$264$	0	0
$265$	−11.6539	−0.715892
$266$	0	0
$267$	−3.06025	−0.187284
$268$	0	0
$269$	20.3467	1.24056	0.620282	−	0.784379i	$-0.287018\pi$
$269$	20.3467	1.24056	0.620282	+	0.784379i	$0.287018\pi$
$270$	0	0
$271$	28.8748	1.75402	0.877011	−	0.480471i	$-0.159534\pi$
$271$	28.8748	1.75402	0.877011	+	0.480471i	$0.159534\pi$
$272$	0	0
$273$	66.9050	4.04927
$274$	0	0
$275$	−6.63944	−0.400373
$276$	0	0
$277$	−10.4151	−0.625780	−0.312890	−	0.949789i	$-0.601297\pi$
$277$	−10.4151	−0.625780	−0.312890	+	0.949789i	$0.601297\pi$
$278$	0	0
$279$	2.41271	0.144445
$280$	0	0
$281$	30.2032	1.80177	0.900885	−	0.434057i	$-0.142918\pi$
$281$	30.2032	1.80177	0.900885	+	0.434057i	$0.142918\pi$
$282$	0	0
$283$	26.6458	1.58393	0.791964	−	0.610568i	$-0.209059\pi$
$283$	26.6458	1.58393	0.791964	+	0.610568i	$0.209059\pi$
$284$	0	0
$285$	−8.14892	−0.482700
$286$	0	0
$287$	17.5465	1.03574
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−15.0663	−0.883202
$292$	0	0
$293$	−3.52580	−0.205979	−0.102990	−	0.994682i	$-0.532841\pi$
$293$	−3.52580	−0.205979	−0.102990	+	0.994682i	$0.532841\pi$
$294$	0	0
$295$	17.0423	0.992240
$296$	0	0
$297$	−8.75428	−0.507975
$298$	0	0
$299$	−17.9959	−1.04073
$300$	0	0
$301$	−32.6875	−1.88408
$302$	0	0
$303$	−18.2730	−1.04976
$304$	0	0
$305$	2.49859	0.143069
$306$	0	0
$307$	−1.34483	−0.0767534	−0.0383767	−	0.999263i	$-0.512219\pi$
$307$	−1.34483	−0.0767534	−0.0383767	+	0.999263i	$0.512219\pi$
$308$	0	0
$309$	15.0188	0.854391
$310$	0	0
$311$	−32.5803	−1.84746	−0.923730	−	0.383044i	$-0.874876\pi$
$311$	−32.5803	−1.84746	−0.923730	+	0.383044i	$0.874876\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	−11.3562	−0.639852
$316$	0	0
$317$	−25.8814	−1.45364	−0.726822	−	0.686826i	$-0.759004\pi$
$317$	−25.8814	−1.45364	−0.726822	+	0.686826i	$0.759004\pi$
$318$	0	0
$319$	−4.19561	−0.234909
$320$	0	0
$321$	−11.0903	−0.619001
$322$	0	0
$323$	2.44834	0.136229
$324$	0	0
$325$	−15.8604	−0.879777
$326$	0	0
$327$	−17.5840	−0.972399
$328$	0	0
$329$	44.2483	2.43949
$330$	0	0
$331$	−1.26171	−0.0693499	−0.0346749	−	0.999399i	$-0.511040\pi$
$331$	−1.26171	−0.0693499	−0.0346749	+	0.999399i	$0.511040\pi$
$332$	0	0
$333$	1.67565	0.0918250
$334$	0	0
$335$	15.5240	0.848164
$336$	0	0
$337$	17.8120	0.970279	0.485140	−	0.874437i	$-0.338769\pi$
$337$	17.8120	0.970279	0.485140	+	0.874437i	$0.338769\pi$
$338$	0	0
$339$	−16.8924	−0.917471
$340$	0	0
$341$	4.46698	0.241901
$342$	0	0
$343$	55.6896	3.00696
$344$	0	0
$345$	9.44653	0.508584
$346$	0	0
$347$	4.34210	0.233096	0.116548	−	0.993185i	$-0.462817\pi$
$347$	4.34210	0.233096	0.116548	+	0.993185i	$0.462817\pi$
$348$	0	0
$349$	−12.4908	−0.668615	−0.334307	−	0.942464i	$-0.608502\pi$
$349$	−12.4908	−0.668615	−0.334307	+	0.942464i	$0.608502\pi$
$350$	0	0
$351$	−20.9124	−1.11622
$352$	0	0
$353$	10.2782	0.547055	0.273527	−	0.961864i	$-0.411810\pi$
$353$	10.2782	0.547055	0.273527	+	0.961864i	$0.411810\pi$
$354$	0	0
$355$	3.97178	0.210800
$356$	0	0
$357$	10.5518	0.558462
$358$	0	0
$359$	−7.45826	−0.393632	−0.196816	−	0.980440i	$-0.563060\pi$
$359$	−7.45826	−0.393632	−0.196816	+	0.980440i	$0.563060\pi$
$360$	0	0
$361$	−13.0056	−0.684506
$362$	0	0
$363$	−8.32726	−0.437068
$364$	0	0
$365$	−3.98516	−0.208593
$366$	0	0
$367$	−6.92083	−0.361264	−0.180632	−	0.983551i	$-0.557814\pi$
$367$	−6.92083	−0.361264	−0.180632	+	0.983551i	$0.557814\pi$
$368$	0	0
$369$	5.01974	0.261317
$370$	0	0
$371$	36.9463	1.91815
$372$	0	0
$373$	−22.0882	−1.14369	−0.571843	−	0.820363i	$-0.693771\pi$
$373$	−22.0882	−1.14369	−0.571843	+	0.820363i	$0.693771\pi$
$374$	0	0
$375$	24.9672	1.28930
$376$	0	0
$377$	−10.0225	−0.516187
$378$	0	0
$379$	−35.4430	−1.82058	−0.910291	−	0.413968i	$-0.864143\pi$
$379$	−35.4430	−1.82058	−0.910291	+	0.413968i	$0.864143\pi$
$380$	0	0
$381$	28.4288	1.45645
$382$	0	0
$383$	−28.5493	−1.45880	−0.729401	−	0.684087i	$-0.760201\pi$
$383$	−28.5493	−1.45880	−0.729401	+	0.684087i	$0.760201\pi$
$384$	0	0
$385$	−21.0254	−1.07155
$386$	0	0
$387$	−9.35132	−0.475354
$388$	0	0
$389$	−19.0218	−0.964443	−0.482222	−	0.876049i	$-0.660170\pi$
$389$	−19.0218	−0.964443	−0.482222	+	0.876049i	$0.660170\pi$
$390$	0	0
$391$	−2.83821	−0.143534
$392$	0	0
$393$	−12.4336	−0.627194
$394$	0	0
$395$	−1.28931	−0.0648724
$396$	0	0
$397$	−6.52486	−0.327473	−0.163737	−	0.986504i	$-0.552355\pi$
$397$	−6.52486	−0.327473	−0.163737	+	0.986504i	$0.552355\pi$
$398$	0	0
$399$	25.8345	1.29334
$400$	0	0
$401$	−5.41096	−0.270210	−0.135105	−	0.990831i	$-0.543137\pi$
$401$	−5.41096	−0.270210	−0.135105	+	0.990831i	$0.543137\pi$
$402$	0	0
$403$	10.6708	0.531550
$404$	0	0
$405$	17.7758	0.883289
$406$	0	0
$407$	3.10236	0.153778
$408$	0	0
$409$	28.4307	1.40581	0.702904	−	0.711285i	$-0.251886\pi$
$409$	28.4307	1.40581	0.702904	+	0.711285i	$0.251886\pi$
$410$	0	0
$411$	−3.29817	−0.162687
$412$	0	0
$413$	−54.0291	−2.65860
$414$	0	0
$415$	12.9972	0.638007
$416$	0	0
$417$	−6.93223	−0.339473
$418$	0	0
$419$	−14.4224	−0.704581	−0.352290	−	0.935891i	$-0.614597\pi$
$419$	−14.4224	−0.704581	−0.352290	+	0.935891i	$0.614597\pi$
$420$	0	0
$421$	−21.2645	−1.03637	−0.518183	−	0.855270i	$-0.673391\pi$
$421$	−21.2645	−1.03637	−0.518183	+	0.855270i	$0.673391\pi$
$422$	0	0
$423$	12.6586	0.615484
$424$	0	0
$425$	−2.50141	−0.121336
$426$	0	0
$427$	−7.92129	−0.383338
$428$	0	0
$429$	35.4370	1.71092
$430$	0	0
$431$	18.4752	0.889917	0.444958	−	0.895551i	$-0.353218\pi$
$431$	18.4752	0.889917	0.444958	+	0.895551i	$0.353218\pi$
$432$	0	0
$433$	−27.1580	−1.30513	−0.652564	−	0.757733i	$-0.726307\pi$
$433$	−27.1580	−1.30513	−0.652564	+	0.757733i	$0.726307\pi$
$434$	0	0
$435$	5.26109	0.252250
$436$	0	0
$437$	−6.94891	−0.332411
$438$	0	0
$439$	−2.29647	−0.109605	−0.0548023	−	0.998497i	$-0.517453\pi$
$439$	−2.29647	−0.109605	−0.0548023	+	0.998497i	$0.517453\pi$
$440$	0	0
$441$	25.9672	1.23654
$442$	0	0
$443$	−14.9912	−0.712254	−0.356127	−	0.934438i	$-0.615903\pi$
$443$	−14.9912	−0.712254	−0.356127	+	0.934438i	$0.615903\pi$
$444$	0	0
$445$	2.29734	0.108904
$446$	0	0
$447$	−5.68944	−0.269101
$448$	0	0
$449$	−20.6567	−0.974849	−0.487425	−	0.873165i	$-0.662064\pi$
$449$	−20.6567	−0.974849	−0.487425	+	0.873165i	$0.662064\pi$
$450$	0	0
$451$	9.29372	0.437624
$452$	0	0
$453$	−6.08486	−0.285892
$454$	0	0
$455$	−50.2257	−2.35462
$456$	0	0
$457$	−26.4206	−1.23590	−0.617952	−	0.786216i	$-0.712037\pi$
$457$	−26.4206	−1.23590	−0.617952	+	0.786216i	$0.712037\pi$
$458$	0	0
$459$	−3.29817	−0.153945
$460$	0	0
$461$	39.2364	1.82742	0.913711	−	0.406365i	$-0.133204\pi$
$461$	39.2364	1.82742	0.913711	+	0.406365i	$0.133204\pi$
$462$	0	0
$463$	7.30774	0.339620	0.169810	−	0.985477i	$-0.445685\pi$
$463$	7.30774	0.339620	0.169810	+	0.985477i	$0.445685\pi$
$464$	0	0
$465$	−5.60137	−0.259757
$466$	0	0
$467$	−24.3923	−1.12874	−0.564370	−	0.825522i	$-0.690881\pi$
$467$	−24.3923	−1.12874	−0.564370	+	0.825522i	$0.690881\pi$
$468$	0	0
$469$	−49.2156	−2.27256
$470$	0	0
$471$	−2.13554	−0.0984005
$472$	0	0
$473$	−17.3134	−0.796069
$474$	0	0
$475$	−6.12430	−0.281002
$476$	0	0
$477$	10.5697	0.483951
$478$	0	0
$479$	−6.68012	−0.305223	−0.152611	−	0.988286i	$-0.548768\pi$
$479$	−6.68012	−0.305223	−0.152611	+	0.988286i	$0.548768\pi$
$480$	0	0
$481$	7.41096	0.337911
$482$	0	0
$483$	−29.9483	−1.36270
$484$	0	0
$485$	11.3103	0.513575
$486$	0	0
$487$	−17.2314	−0.780829	−0.390414	−	0.920639i	$-0.627668\pi$
$487$	−17.2314	−0.780829	−0.390414	+	0.920639i	$0.627668\pi$
$488$	0	0
$489$	32.6917	1.47837
$490$	0	0
$491$	−8.73548	−0.394227	−0.197113	−	0.980381i	$-0.563157\pi$
$491$	−8.73548	−0.394227	−0.197113	+	0.980381i	$0.563157\pi$
$492$	0	0
$493$	−1.58069	−0.0711909
$494$	0	0
$495$	−6.01498	−0.270353
$496$	0	0
$497$	−12.5917	−0.564816
$498$	0	0
$499$	5.23137	0.234188	0.117094	−	0.993121i	$-0.462642\pi$
$499$	5.23137	0.234188	0.117094	+	0.993121i	$0.462642\pi$
$500$	0	0
$501$	33.4817	1.49585
$502$	0	0
$503$	−2.39969	−0.106997	−0.0534984	−	0.998568i	$-0.517037\pi$
$503$	−2.39969	−0.106997	−0.0534984	+	0.998568i	$0.517037\pi$
$504$	0	0
$505$	13.7176	0.610426
$506$	0	0
$507$	57.2796	2.54387
$508$	0	0
$509$	−6.72283	−0.297984	−0.148992	−	0.988838i	$-0.547603\pi$
$509$	−6.72283	−0.297984	−0.148992	+	0.988838i	$0.547603\pi$
$510$	0	0
$511$	12.6341	0.558901
$512$	0	0
$513$	−8.07505	−0.356522
$514$	0	0
$515$	−11.2747	−0.496821
$516$	0	0
$517$	23.4367	1.03074
$518$	0	0
$519$	−44.5868	−1.95714
$520$	0	0
$521$	−23.2754	−1.01971	−0.509857	−	0.860259i	$-0.670302\pi$
$521$	−23.2754	−1.01971	−0.509857	+	0.860259i	$0.670302\pi$
$522$	0	0
$523$	3.46255	0.151407	0.0757034	−	0.997130i	$-0.475880\pi$
$523$	3.46255	0.151407	0.0757034	+	0.997130i	$0.475880\pi$
$524$	0	0
$525$	−26.3944	−1.15195
$526$	0	0
$527$	1.68293	0.0733097
$528$	0	0
$529$	−14.9446	−0.649764
$530$	0	0
$531$	−15.4568	−0.670766
$532$	0	0
$533$	22.2010	0.961632
$534$	0	0
$535$	8.32553	0.359944
$536$	0	0
$537$	−34.9349	−1.50755
$538$	0	0
$539$	48.0767	2.07081
$540$	0	0
$541$	−9.41268	−0.404683	−0.202341	−	0.979315i	$-0.564855\pi$
$541$	−9.41268	−0.404683	−0.202341	+	0.979315i	$0.564855\pi$
$542$	0	0
$543$	−24.8167	−1.06499
$544$	0	0
$545$	13.2004	0.565442
$546$	0	0
$547$	−8.92340	−0.381537	−0.190768	−	0.981635i	$-0.561098\pi$
$547$	−8.92340	−0.381537	−0.190768	+	0.981635i	$0.561098\pi$
$548$	0	0
$549$	−2.26614	−0.0967164
$550$	0	0
$551$	−3.87008	−0.164871
$552$	0	0
$553$	4.08751	0.173818
$554$	0	0
$555$	−3.89020	−0.165130
$556$	0	0
$557$	−8.36902	−0.354607	−0.177303	−	0.984156i	$-0.556737\pi$
$557$	−8.36902	−0.354607	−0.177303	+	0.984156i	$0.556737\pi$
$558$	0	0
$559$	−41.3585	−1.74928
$560$	0	0
$561$	5.58891	0.235964
$562$	0	0
$563$	−12.1127	−0.510488	−0.255244	−	0.966877i	$-0.582156\pi$
$563$	−12.1127	−0.510488	−0.255244	+	0.966877i	$0.582156\pi$
$564$	0	0
$565$	12.6812	0.533502
$566$	0	0
$567$	−56.3548	−2.36668
$568$	0	0
$569$	−18.8542	−0.790411	−0.395206	−	0.918593i	$-0.629327\pi$
$569$	−18.8542	−0.790411	−0.395206	+	0.918593i	$0.629327\pi$
$570$	0	0
$571$	3.35775	0.140517	0.0702587	−	0.997529i	$-0.477618\pi$
$571$	3.35775	0.140517	0.0702587	+	0.997529i	$0.477618\pi$
$572$	0	0
$573$	16.7659	0.700408
$574$	0	0
$575$	7.09951	0.296070
$576$	0	0
$577$	−24.7442	−1.03011	−0.515057	−	0.857156i	$-0.672229\pi$
$577$	−24.7442	−1.03011	−0.515057	+	0.857156i	$0.672229\pi$
$578$	0	0
$579$	53.1099	2.20717
$580$	0	0
$581$	−41.2050	−1.70947
$582$	0	0
$583$	19.5690	0.810467
$584$	0	0
$585$	−14.3687	−0.594072
$586$	0	0
$587$	25.2517	1.04225	0.521124	−	0.853481i	$-0.325513\pi$
$587$	25.2517	1.04225	0.521124	+	0.853481i	$0.325513\pi$
$588$	0	0
$589$	4.12039	0.169778
$590$	0	0
$591$	27.1273	1.11587
$592$	0	0
$593$	22.1581	0.909924	0.454962	−	0.890511i	$-0.349653\pi$
$593$	22.1581	0.909924	0.454962	+	0.890511i	$0.349653\pi$
$594$	0	0
$595$	−7.92129	−0.324741
$596$	0	0
$597$	−0.715101	−0.0292671
$598$	0	0
$599$	34.4007	1.40558	0.702788	−	0.711399i	$-0.251938\pi$
$599$	34.4007	1.40558	0.702788	+	0.711399i	$0.251938\pi$
$600$	0	0
$601$	25.2908	1.03163	0.515817	−	0.856699i	$-0.327488\pi$
$601$	25.2908	1.03163	0.515817	+	0.856699i	$0.327488\pi$
$602$	0	0
$603$	−14.0797	−0.573369
$604$	0	0
$605$	6.25129	0.254151
$606$	0	0
$607$	14.2041	0.576526	0.288263	−	0.957551i	$-0.406922\pi$
$607$	14.2041	0.576526	0.288263	+	0.957551i	$0.406922\pi$
$608$	0	0
$609$	−16.6792	−0.675877
$610$	0	0
$611$	55.9859	2.26495
$612$	0	0
$613$	33.8370	1.36666	0.683332	−	0.730108i	$-0.260530\pi$
$613$	33.8370	1.36666	0.683332	+	0.730108i	$0.260530\pi$
$614$	0	0
$615$	−11.6539	−0.469929
$616$	0	0
$617$	−33.3261	−1.34166	−0.670830	−	0.741611i	$-0.734062\pi$
$617$	−33.3261	−1.34166	−0.670830	+	0.741611i	$0.734062\pi$
$618$	0	0
$619$	17.9900	0.723080	0.361540	−	0.932357i	$-0.382251\pi$
$619$	17.9900	0.723080	0.361540	+	0.932357i	$0.382251\pi$
$620$	0	0
$621$	9.36090	0.375640
$622$	0	0
$623$	−7.28325	−0.291797
$624$	0	0
$625$	−6.23595	−0.249438
$626$	0	0
$627$	13.6836	0.546469
$628$	0	0
$629$	1.16881	0.0466035
$630$	0	0
$631$	9.75904	0.388501	0.194251	−	0.980952i	$-0.437773\pi$
$631$	9.75904	0.388501	0.194251	+	0.980952i	$0.437773\pi$
$632$	0	0
$633$	13.4009	0.532637
$634$	0	0
$635$	−21.3416	−0.846915
$636$	0	0
$637$	114.846	4.55038
$638$	0	0
$639$	−3.60226	−0.142503
$640$	0	0
$641$	−12.1525	−0.479994	−0.239997	−	0.970774i	$-0.577146\pi$
$641$	−12.1525	−0.479994	−0.239997	+	0.970774i	$0.577146\pi$
$642$	0	0
$643$	−20.1310	−0.793890	−0.396945	−	0.917842i	$-0.629930\pi$
$643$	−20.1310	−0.793890	−0.396945	+	0.917842i	$0.629930\pi$
$644$	0	0
$645$	21.7101	0.854835
$646$	0	0
$647$	−8.21254	−0.322868	−0.161434	−	0.986883i	$-0.551612\pi$
$647$	−8.21254	−0.322868	−0.161434	+	0.986883i	$0.551612\pi$
$648$	0	0
$649$	−28.6172	−1.12332
$650$	0	0
$651$	17.7580	0.695992
$652$	0	0
$653$	−39.5293	−1.54690	−0.773451	−	0.633856i	$-0.781471\pi$
$653$	−39.5293	−1.54690	−0.773451	+	0.633856i	$0.781471\pi$
$654$	0	0
$655$	9.33396	0.364708
$656$	0	0
$657$	3.61440	0.141011
$658$	0	0
$659$	−15.5399	−0.605348	−0.302674	−	0.953094i	$-0.597879\pi$
$659$	−15.5399	−0.605348	−0.302674	+	0.953094i	$0.597879\pi$
$660$	0	0
$661$	46.1373	1.79453	0.897267	−	0.441489i	$-0.145550\pi$
$661$	46.1373	1.79453	0.897267	+	0.441489i	$0.145550\pi$
$662$	0	0
$663$	13.3509	0.518506
$664$	0	0
$665$	−19.3940	−0.752068
$666$	0	0
$667$	4.48634	0.173712
$668$	0	0
$669$	27.5055	1.06342
$670$	0	0
$671$	−4.19561	−0.161970
$672$	0	0
$673$	−32.0095	−1.23388	−0.616938	−	0.787012i	$-0.711627\pi$
$673$	−32.0095	−1.23388	−0.616938	+	0.787012i	$0.711627\pi$
$674$	0	0
$675$	8.25006	0.317545
$676$	0	0
$677$	13.7551	0.528650	0.264325	−	0.964434i	$-0.414851\pi$
$677$	13.7551	0.528650	0.264325	+	0.964434i	$0.414851\pi$
$678$	0	0
$679$	−35.8571	−1.37607
$680$	0	0
$681$	42.8925	1.64364
$682$	0	0
$683$	−23.7998	−0.910674	−0.455337	−	0.890319i	$-0.650481\pi$
$683$	−23.7998	−0.910674	−0.455337	+	0.890319i	$0.650481\pi$
$684$	0	0
$685$	2.47594	0.0946010
$686$	0	0
$687$	37.7421	1.43995
$688$	0	0
$689$	46.7469	1.78091
$690$	0	0
$691$	−37.8178	−1.43866	−0.719328	−	0.694671i	$-0.755550\pi$
$691$	−37.8178	−1.43866	−0.719328	+	0.694671i	$0.755550\pi$
$692$	0	0
$693$	19.0693	0.724382
$694$	0	0
$695$	5.20404	0.197401
$696$	0	0
$697$	3.50141	0.132625
$698$	0	0
$699$	−33.4638	−1.26572
$700$	0	0
$701$	4.14594	0.156590	0.0782950	−	0.996930i	$-0.475052\pi$
$701$	4.14594	0.156590	0.0782950	+	0.996930i	$0.475052\pi$
$702$	0	0
$703$	2.86165	0.107929
$704$	0	0
$705$	−29.3884	−1.10683
$706$	0	0
$707$	−43.4889	−1.63557
$708$	0	0
$709$	−47.2230	−1.77350	−0.886748	−	0.462252i	$-0.847041\pi$
$709$	−47.2230	−1.77350	−0.886748	+	0.462252i	$0.847041\pi$
$710$	0	0
$711$	1.16936	0.0438545
$712$	0	0
$713$	−4.77652	−0.178882
$714$	0	0
$715$	−26.6027	−0.994884
$716$	0	0
$717$	−34.4347	−1.28599
$718$	0	0
$719$	−19.5647	−0.729642	−0.364821	−	0.931078i	$-0.618870\pi$
$719$	−19.5647	−0.729642	−0.364821	+	0.931078i	$0.618870\pi$
$720$	0	0
$721$	35.7441	1.33118
$722$	0	0
$723$	62.4991	2.32437
$724$	0	0
$725$	3.95396	0.146846
$726$	0	0
$727$	−2.57406	−0.0954668	−0.0477334	−	0.998860i	$-0.515200\pi$
$727$	−2.57406	−0.0954668	−0.0477334	+	0.998860i	$0.515200\pi$
$728$	0	0
$729$	4.71199	0.174518
$730$	0	0
$731$	−6.52280	−0.241255
$732$	0	0
$733$	−8.78689	−0.324551	−0.162276	−	0.986745i	$-0.551883\pi$
$733$	−8.78689	−0.324551	−0.162276	+	0.986745i	$0.551883\pi$
$734$	0	0
$735$	−60.2858	−2.22368
$736$	0	0
$737$	−26.0676	−0.960214
$738$	0	0
$739$	21.3574	0.785643	0.392822	−	0.919615i	$-0.371499\pi$
$739$	21.3574	0.785643	0.392822	+	0.919615i	$0.371499\pi$
$740$	0	0
$741$	32.6875	1.20081
$742$	0	0
$743$	15.6886	0.575557	0.287779	−	0.957697i	$-0.407083\pi$
$743$	15.6886	0.575557	0.287779	+	0.957697i	$0.407083\pi$
$744$	0	0
$745$	4.27108	0.156480
$746$	0	0
$747$	−11.7880	−0.431300
$748$	0	0
$749$	−26.3944	−0.964431
$750$	0	0
$751$	−19.4799	−0.710832	−0.355416	−	0.934708i	$-0.615661\pi$
$751$	−19.4799	−0.710832	−0.355416	+	0.934708i	$0.615661\pi$
$752$	0	0
$753$	49.9926	1.82183
$754$	0	0
$755$	4.56792	0.166244
$756$	0	0
$757$	−18.0194	−0.654927	−0.327463	−	0.944864i	$-0.606194\pi$
$757$	−18.0194	−0.654927	−0.327463	+	0.944864i	$0.606194\pi$
$758$	0	0
$759$	−15.8625	−0.575773
$760$	0	0
$761$	31.8671	1.15518	0.577592	−	0.816326i	$-0.303993\pi$
$761$	31.8671	1.15518	0.577592	+	0.816326i	$0.303993\pi$
$762$	0	0
$763$	−41.8491	−1.51504
$764$	0	0
$765$	−2.26614	−0.0819324
$766$	0	0
$767$	−68.3613	−2.46838
$768$	0	0
$769$	−2.60475	−0.0939296	−0.0469648	−	0.998897i	$-0.514955\pi$
$769$	−2.60475	−0.0939296	−0.0469648	+	0.998897i	$0.514955\pi$
$770$	0	0
$771$	−39.9116	−1.43738
$772$	0	0
$773$	9.40703	0.338347	0.169174	−	0.985586i	$-0.445890\pi$
$773$	9.40703	0.338347	0.169174	+	0.985586i	$0.445890\pi$
$774$	0	0
$775$	−4.20970	−0.151217
$776$	0	0
$777$	12.3331	0.442448
$778$	0	0
$779$	8.57264	0.307147
$780$	0	0
$781$	−6.66936	−0.238649
$782$	0	0
$783$	5.21340	0.186312
$784$	0	0
$785$	1.60316	0.0572191
$786$	0	0
$787$	−3.38310	−0.120594	−0.0602972	−	0.998180i	$-0.519205\pi$
$787$	−3.38310	−0.120594	−0.0602972	+	0.998180i	$0.519205\pi$
$788$	0	0
$789$	−20.1013	−0.715625
$790$	0	0
$791$	−40.2032	−1.42946
$792$	0	0
$793$	−10.0225	−0.355911
$794$	0	0
$795$	−24.5386	−0.870295
$796$	0	0
$797$	20.3351	0.720307	0.360153	−	0.932893i	$-0.382724\pi$
$797$	20.3351	0.720307	0.360153	+	0.932893i	$0.382724\pi$
$798$	0	0
$799$	8.82975	0.312374
$800$	0	0
$801$	−2.08361	−0.0736206
$802$	0	0
$803$	6.69183	0.236149
$804$	0	0
$805$	22.4823	0.792397
$806$	0	0
$807$	42.8425	1.50813
$808$	0	0
$809$	−44.1507	−1.55226	−0.776128	−	0.630576i	$-0.782819\pi$
$809$	−44.1507	−1.55226	−0.776128	+	0.630576i	$0.782819\pi$
$810$	0	0
$811$	7.73677	0.271675	0.135837	−	0.990731i	$-0.456628\pi$
$811$	7.73677	0.271675	0.135837	+	0.990731i	$0.456628\pi$
$812$	0	0
$813$	60.7994	2.13233
$814$	0	0
$815$	−24.5417	−0.859660
$816$	0	0
$817$	−15.9700	−0.558721
$818$	0	0
$819$	45.5530	1.59175
$820$	0	0
$821$	48.5660	1.69496	0.847482	−	0.530824i	$-0.178118\pi$
$821$	48.5660	1.69496	0.847482	+	0.530824i	$0.178118\pi$
$822$	0	0
$823$	37.3368	1.30148	0.650740	−	0.759301i	$-0.274459\pi$
$823$	37.3368	1.30148	0.650740	+	0.759301i	$0.274459\pi$
$824$	0	0
$825$	−13.9801	−0.486726
$826$	0	0
$827$	−36.3380	−1.26360	−0.631798	−	0.775133i	$-0.717683\pi$
$827$	−36.3380	−1.26360	−0.631798	+	0.775133i	$0.717683\pi$
$828$	0	0
$829$	−10.6053	−0.368337	−0.184169	−	0.982895i	$-0.558959\pi$
$829$	−10.6053	−0.368337	−0.184169	+	0.982895i	$0.558959\pi$
$830$	0	0
$831$	−21.9302	−0.760749
$832$	0	0
$833$	18.1129	0.627574
$834$	0	0
$835$	−25.1348	−0.869824
$836$	0	0
$837$	−5.55060	−0.191857
$838$	0	0
$839$	43.2821	1.49426	0.747132	−	0.664676i	$-0.231430\pi$
$839$	43.2821	1.49426	0.747132	+	0.664676i	$0.231430\pi$
$840$	0	0
$841$	−26.5014	−0.913842
$842$	0	0
$843$	63.5964	2.19038
$844$	0	0
$845$	−42.9999	−1.47924
$846$	0	0
$847$	−19.8185	−0.680971
$848$	0	0
$849$	56.1059	1.92555
$850$	0	0
$851$	−3.31733	−0.113717
$852$	0	0
$853$	−39.7672	−1.36160	−0.680802	−	0.732467i	$-0.738369\pi$
$853$	−39.7672	−1.36160	−0.680802	+	0.732467i	$0.738369\pi$
$854$	0	0
$855$	−5.54828	−0.189747
$856$	0	0
$857$	14.9672	0.511271	0.255636	−	0.966773i	$-0.417715\pi$
$857$	14.9672	0.511271	0.255636	+	0.966773i	$0.417715\pi$
$858$	0	0
$859$	47.5063	1.62090	0.810448	−	0.585811i	$-0.199224\pi$
$859$	47.5063	1.62090	0.810448	+	0.585811i	$0.199224\pi$
$860$	0	0
$861$	36.9463	1.25913
$862$	0	0
$863$	−20.9221	−0.712198	−0.356099	−	0.934448i	$-0.615893\pi$
$863$	−20.9221	−0.712198	−0.356099	+	0.934448i	$0.615893\pi$
$864$	0	0
$865$	33.4715	1.13806
$866$	0	0
$867$	2.10562	0.0715106
$868$	0	0
$869$	2.16500	0.0734426
$870$	0	0
$871$	−62.2708	−2.10997
$872$	0	0
$873$	−10.2581	−0.347183
$874$	0	0
$875$	59.4208	2.00879
$876$	0	0
$877$	−29.6926	−1.00265	−0.501324	−	0.865260i	$-0.667153\pi$
$877$	−29.6926	−1.00265	−0.501324	+	0.865260i	$0.667153\pi$
$878$	0	0
$879$	−7.42398	−0.250405
$880$	0	0
$881$	−16.1976	−0.545710	−0.272855	−	0.962055i	$-0.587968\pi$
$881$	−16.1976	−0.545710	−0.272855	+	0.962055i	$0.587968\pi$
$882$	0	0
$883$	−37.4853	−1.26148	−0.630741	−	0.775993i	$-0.717249\pi$
$883$	−37.4853	−1.26148	−0.630741	+	0.775993i	$0.717249\pi$
$884$	0	0
$885$	35.8846	1.20625
$886$	0	0
$887$	8.05161	0.270347	0.135173	−	0.990822i	$-0.456841\pi$
$887$	8.05161	0.270347	0.135173	+	0.990822i	$0.456841\pi$
$888$	0	0
$889$	67.6592	2.26922
$890$	0	0
$891$	−29.8490	−0.999979
$892$	0	0
$893$	21.6182	0.723427
$894$	0	0
$895$	26.2257	0.876630
$896$	0	0
$897$	−37.8926	−1.26520
$898$	0	0
$899$	−2.66020	−0.0887227
$900$	0	0
$901$	7.37263	0.245618
$902$	0	0
$903$	−68.8275	−2.29044
$904$	0	0
$905$	18.6300	0.619281
$906$	0	0
$907$	−33.3609	−1.10773	−0.553865	−	0.832606i	$-0.686848\pi$
$907$	−33.3609	−1.10773	−0.553865	+	0.832606i	$0.686848\pi$
$908$	0	0
$909$	−12.4414	−0.412655
$910$	0	0
$911$	−25.7866	−0.854347	−0.427174	−	0.904170i	$-0.640491\pi$
$911$	−25.7866	−0.854347	−0.427174	+	0.904170i	$0.640491\pi$
$912$	0	0
$913$	−21.8247	−0.722293
$914$	0	0
$915$	5.26109	0.173926
$916$	0	0
$917$	−29.5915	−0.977196
$918$	0	0
$919$	25.8720	0.853440	0.426720	−	0.904384i	$-0.359669\pi$
$919$	25.8720	0.853440	0.426720	+	0.904384i	$0.359669\pi$
$920$	0	0
$921$	−2.83170	−0.0933076
$922$	0	0
$923$	−15.9319	−0.524404
$924$	0	0
$925$	−2.92367	−0.0961297
$926$	0	0
$927$	10.2257	0.335857
$928$	0	0
$929$	40.6849	1.33483	0.667413	−	0.744687i	$-0.267401\pi$
$929$	40.6849	1.33483	0.667413	+	0.744687i	$0.267401\pi$
$930$	0	0
$931$	44.3465	1.45340
$932$	0	0
$933$	−68.6018	−2.24592
$934$	0	0
$935$	−4.19561	−0.137211
$936$	0	0
$937$	0.426795	0.0139428	0.00697140	−	0.999976i	$-0.497781\pi$
$937$	0.426795	0.0139428	0.00697140	+	0.999976i	$0.497781\pi$
$938$	0	0
$939$	12.6337	0.412286
$940$	0	0
$941$	21.4487	0.699208	0.349604	−	0.936898i	$-0.386316\pi$
$941$	21.4487	0.699208	0.349604	+	0.936898i	$0.386316\pi$
$942$	0	0
$943$	−9.93772	−0.323617
$944$	0	0
$945$	26.1258	0.849871
$946$	0	0
$947$	45.9708	1.49385	0.746925	−	0.664908i	$-0.231529\pi$
$947$	45.9708	1.49385	0.746925	+	0.664908i	$0.231529\pi$
$948$	0	0
$949$	15.9856	0.518913
$950$	0	0
$951$	−54.4964	−1.76717
$952$	0	0
$953$	−28.3714	−0.919038	−0.459519	−	0.888168i	$-0.651978\pi$
$953$	−28.3714	−0.919038	−0.459519	+	0.888168i	$0.651978\pi$
$954$	0	0
$955$	−12.5862	−0.407281
$956$	0	0
$957$	−8.83436	−0.285574
$958$	0	0
$959$	−7.84948	−0.253473
$960$	0	0
$961$	−28.1677	−0.908637
$962$	0	0
$963$	−7.55097	−0.243327
$964$	0	0
$965$	−39.8697	−1.28345
$966$	0	0
$967$	29.4541	0.947180	0.473590	−	0.880745i	$-0.342958\pi$
$967$	29.4541	0.947180	0.473590	+	0.880745i	$0.342958\pi$
$968$	0	0
$969$	5.15528	0.165611
$970$	0	0
$971$	38.8816	1.24777	0.623885	−	0.781516i	$-0.285554\pi$
$971$	38.8816	1.24777	0.623885	+	0.781516i	$0.285554\pi$
$972$	0	0
$973$	−16.4984	−0.528914
$974$	0	0
$975$	−33.3960	−1.06953
$976$	0	0
$977$	51.5465	1.64912	0.824559	−	0.565775i	$-0.191423\pi$
$977$	51.5465	1.64912	0.824559	+	0.565775i	$0.191423\pi$
$978$	0	0
$979$	−3.85766	−0.123291
$980$	0	0
$981$	−11.9723	−0.382246
$982$	0	0
$983$	−39.7261	−1.26707	−0.633533	−	0.773716i	$-0.718396\pi$
$983$	−39.7261	−1.26707	−0.633533	+	0.773716i	$0.718396\pi$
$984$	0	0
$985$	−20.3645	−0.648868
$986$	0	0
$987$	93.1701	2.96564
$988$	0	0
$989$	18.5131	0.588682
$990$	0	0
$991$	8.01408	0.254576	0.127288	−	0.991866i	$-0.459373\pi$
$991$	8.01408	0.254576	0.127288	+	0.991866i	$0.459373\pi$
$992$	0	0
$993$	−2.65668	−0.0843073
$994$	0	0
$995$	0.536828	0.0170186
$996$	0	0
$997$	18.6589	0.590934	0.295467	−	0.955353i	$-0.404525\pi$
$997$	18.6589	0.590934	0.295467	+	0.955353i	$0.404525\pi$
$998$	0	0
$999$	−3.85494	−0.121965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4352.2.a.bf.1.7		8
4.3	odd	2		4352.2.a.bb.1.2		8
8.3	odd	2		4352.2.a.bb.1.7		8
8.5	even	2	inner	4352.2.a.bf.1.2		8
16.3	odd	4		544.2.c.b.273.2		8
16.5	even	4		136.2.c.b.69.2	yes	8
16.11	odd	4		544.2.c.b.273.7		8
16.13	even	4		136.2.c.b.69.1	✓	8
48.5	odd	4		1224.2.f.c.613.7		8
48.11	even	4		4896.2.f.d.2449.6		8
48.29	odd	4		1224.2.f.c.613.8		8
48.35	even	4		4896.2.f.d.2449.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.c.b.69.1	✓	8	16.13	even	4
136.2.c.b.69.2	yes	8	16.5	even	4
544.2.c.b.273.2		8	16.3	odd	4
544.2.c.b.273.7		8	16.11	odd	4
1224.2.f.c.613.7		8	48.5	odd	4
1224.2.f.c.613.8		8	48.29	odd	4
4352.2.a.bb.1.2		8	4.3	odd	2
4352.2.a.bb.1.7		8	8.3	odd	2
4352.2.a.bf.1.2		8	8.5	even	2	inner
4352.2.a.bf.1.7		8	1.1	even	1	trivial
4896.2.f.d.2449.3		8	48.35	even	4
4896.2.f.d.2449.6		8	48.11	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4352 = 2^{8} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4352.a (trivial)

\(f(q)\)	\(=\)	\(q+2.10562 q^{3} -1.58069 q^{5} +5.01127 q^{7} +1.43364 q^{9} +2.65428 q^{11} +6.34060 q^{13} -3.32834 q^{15} +1.00000 q^{17} +2.44834 q^{19} +10.5518 q^{21} -2.83821 q^{23} -2.50141 q^{25} -3.29817 q^{27} -1.58069 q^{29} +1.68293 q^{31} +5.58891 q^{33} -7.92129 q^{35} +1.16881 q^{37} +13.3509 q^{39} +3.50141 q^{41} -6.52280 q^{43} -2.26614 q^{45} +8.82975 q^{47} +18.1129 q^{49} +2.10562 q^{51} +7.37263 q^{53} -4.19561 q^{55} +5.15528 q^{57} -10.7815 q^{59} -1.58069 q^{61} +7.18434 q^{63} -10.0225 q^{65} -9.82097 q^{67} -5.97619 q^{69} -2.51268 q^{71} +2.52114 q^{73} -5.26701 q^{75} +13.3013 q^{77} +0.815662 q^{79} -11.2456 q^{81} -8.22246 q^{83} -1.58069 q^{85} -3.32834 q^{87} -1.45337 q^{89} +31.7745 q^{91} +3.54362 q^{93} -3.87008 q^{95} -7.15528 q^{97} +3.80527 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{7} + 8 q^{9} + 12 q^{15} + 8 q^{17} + 16 q^{23} + 8 q^{25} + 24 q^{31} - 8 q^{33} + 12 q^{39} + 4 q^{47} + 8 q^{49} + 12 q^{55} - 8 q^{57} + 40 q^{63} - 24 q^{65} + 36 q^{71} - 8 q^{73} + 24 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) \) 8 * q + 12 * q^7 + 8 * q^9 + 12 * q^15 + 8 * q^17 + 16 * q^23 + 8 * q^25 + 24 * q^31 - 8 * q^33 + 12 * q^39 + 4 * q^47 + 8 * q^49 + 12 * q^55 - 8 * q^57 + 40 * q^63 - 24 * q^65 + 36 * q^71 - 8 * q^73 + 24 * q^79 + 40 * q^81 + 12 * q^87 - 8 * q^89 + 32 * q^95 - 8 * q^97

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