LMFDB - Embedded newform 9702.2.a.df.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, 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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.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:	$77.4708600410$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1078)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{8} -2.82843 q^{10} -1.00000 q^{11} +4.24264 q^{13} +1.00000 q^{16} +2.82843 q^{17} +1.41421 q^{19} -2.82843 q^{20} -1.00000 q^{22} -6.00000 q^{23} +3.00000 q^{25} +4.24264 q^{26} -8.00000 q^{29} -1.41421 q^{31} +1.00000 q^{32} +2.82843 q^{34} -6.00000 q^{37} +1.41421 q^{38} -2.82843 q^{40} -8.48528 q^{41} +10.0000 q^{43} -1.00000 q^{44} -6.00000 q^{46} +7.07107 q^{47} +3.00000 q^{50} +4.24264 q^{52} -6.00000 q^{53} +2.82843 q^{55} -8.00000 q^{58} +14.1421 q^{59} -4.24264 q^{61} -1.41421 q^{62} +1.00000 q^{64} -12.0000 q^{65} -4.00000 q^{67} +2.82843 q^{68} +8.48528 q^{73} -6.00000 q^{74} +1.41421 q^{76} -2.82843 q^{80} -8.48528 q^{82} +7.07107 q^{83} -8.00000 q^{85} +10.0000 q^{86} -1.00000 q^{88} -18.3848 q^{89} -6.00000 q^{92} +7.07107 q^{94} -4.00000 q^{95} -1.41421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 2 q^{16} - 2 q^{22} - 12 q^{23} + 6 q^{25} - 16 q^{29} + 2 q^{32} - 12 q^{37} + 20 q^{43} - 2 q^{44} - 12 q^{46} + 6 q^{50} - 12 q^{53} - 16 q^{58} + 2 q^{64} - 24 q^{65} - 8 q^{67} - 12 q^{74} - 16 q^{85} + 20 q^{86} - 2 q^{88} - 12 q^{92} - 8 q^{95}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 + 2 * q^16 - 2 * q^22 - 12 * q^23 + 6 * q^25 - 16 * q^29 + 2 * q^32 - 12 * q^37 + 20 * q^43 - 2 * q^44 - 12 * q^46 + 6 * q^50 - 12 * q^53 - 16 * q^58 + 2 * q^64 - 24 * q^65 - 8 * q^67 - 12 * q^74 - 16 * q^85 + 20 * q^86 - 2 * q^88 - 12 * q^92 - 8 * q^95

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.82843	−0.894427
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	1.41421	0.324443	0.162221	−	0.986754i	$-0.448134\pi$
$19$	1.41421	0.324443	0.162221	+	0.986754i	$0.448134\pi$
$20$	−2.82843	−0.632456
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	4.24264	0.832050
$27$	0	0
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−1.41421	−0.254000	−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421	−0.254000	−0.127000	+	0.991903i	$0.540535\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.82843	0.485071
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	1.41421	0.229416
$39$	0	0
$40$	−2.82843	−0.447214
$41$	−8.48528	−1.32518	−0.662589	−	0.748983i	$-0.730542\pi$
$41$	−8.48528	−1.32518	−0.662589	+	0.748983i	$0.730542\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−6.00000	−0.884652
$47$	7.07107	1.03142	0.515711	−	0.856763i	$-0.327528\pi$
$47$	7.07107	1.03142	0.515711	+	0.856763i	$0.327528\pi$
$48$	0	0
$49$	0	0
$50$	3.00000	0.424264
$51$	0	0
$52$	4.24264	0.588348
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	0	0
$58$	−8.00000	−1.05045
$59$	14.1421	1.84115	0.920575	−	0.390567i	$-0.127721\pi$
$59$	14.1421	1.84115	0.920575	+	0.390567i	$0.127721\pi$
$60$	0	0
$61$	−4.24264	−0.543214	−0.271607	−	0.962408i	$-0.587555\pi$
$61$	−4.24264	−0.543214	−0.271607	+	0.962408i	$0.587555\pi$
$62$	−1.41421	−0.179605
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.82843	0.342997
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	1.41421	0.162221
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.82843	−0.316228
$81$	0	0
$82$	−8.48528	−0.937043
$83$	7.07107	0.776151	0.388075	−	0.921628i	$-0.373140\pi$
$83$	7.07107	0.776151	0.388075	+	0.921628i	$0.373140\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	10.0000	1.07833
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−18.3848	−1.94878	−0.974391	−	0.224860i	$-0.927808\pi$
$89$	−18.3848	−1.94878	−0.974391	+	0.224860i	$0.927808\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	7.07107	0.729325
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−1.41421	−0.143592	−0.0717958	−	0.997419i	$-0.522873\pi$
$97$	−1.41421	−0.143592	−0.0717958	+	0.997419i	$0.522873\pi$
$98$	0	0
$99$	0	0
$100$	3.00000	0.300000
$101$	1.41421	0.140720	0.0703598	−	0.997522i	$-0.477585\pi$
$101$	1.41421	0.140720	0.0703598	+	0.997522i	$0.477585\pi$
$102$	0	0
$103$	4.24264	0.418040	0.209020	−	0.977911i	$-0.432973\pi$
$103$	4.24264	0.418040	0.209020	+	0.977911i	$0.432973\pi$
$104$	4.24264	0.416025
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	2.82843	0.269680
$111$	0	0
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	16.9706	1.58251
$116$	−8.00000	−0.742781
$117$	0	0
$118$	14.1421	1.30189
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.24264	−0.384111
$123$	0	0
$124$	−1.41421	−0.127000
$125$	5.65685	0.505964
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−12.0000	−1.05247
$131$	−7.07107	−0.617802	−0.308901	−	0.951094i	$-0.599961\pi$
$131$	−7.07107	−0.617802	−0.308901	+	0.951094i	$0.599961\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.82843	0.242536
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−21.2132	−1.79928	−0.899640	−	0.436632i	$-0.856171\pi$
$139$	−21.2132	−1.79928	−0.899640	+	0.436632i	$0.856171\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.24264	−0.354787
$144$	0	0
$145$	22.6274	1.87910
$146$	8.48528	0.702247
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	1.41421	0.114708
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	−2.82843	−0.223607
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−8.48528	−0.662589
$165$	0	0
$166$	7.07107	0.548821
$167$	14.1421	1.09435	0.547176	−	0.837018i	$-0.315703\pi$
$167$	14.1421	1.09435	0.547176	+	0.837018i	$0.315703\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	−8.00000	−0.613572
$171$	0	0
$172$	10.0000	0.762493
$173$	15.5563	1.18273	0.591364	−	0.806405i	$-0.298590\pi$
$173$	15.5563	1.18273	0.591364	+	0.806405i	$0.298590\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−18.3848	−1.37800
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−8.48528	−0.630706	−0.315353	−	0.948974i	$-0.602123\pi$
$181$	−8.48528	−0.630706	−0.315353	+	0.948974i	$0.602123\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	16.9706	1.24770
$186$	0	0
$187$	−2.82843	−0.206835
$188$	7.07107	0.515711
$189$	0	0
$190$	−4.00000	−0.290191
$191$	22.0000	1.59186	0.795932	−	0.605386i	$-0.206981\pi$
$191$	22.0000	1.59186	0.795932	+	0.605386i	$0.206981\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−1.41421	−0.101535
$195$	0	0
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	4.24264	0.300753	0.150376	−	0.988629i	$-0.451951\pi$
$199$	4.24264	0.300753	0.150376	+	0.988629i	$0.451951\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	1.41421	0.0995037
$203$	0	0
$204$	0	0
$205$	24.0000	1.67623
$206$	4.24264	0.295599
$207$	0	0
$208$	4.24264	0.294174
$209$	−1.41421	−0.0978232
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−18.0000	−1.23045
$215$	−28.2843	−1.92897
$216$	0	0
$217$	0	0
$218$	14.0000	0.948200
$219$	0	0
$220$	2.82843	0.190693
$221$	12.0000	0.807207
$222$	0	0
$223$	−12.7279	−0.852325	−0.426162	−	0.904647i	$-0.640135\pi$
$223$	−12.7279	−0.852325	−0.426162	+	0.904647i	$0.640135\pi$
$224$	0	0
$225$	0	0
$226$	−8.00000	−0.532152
$227$	−26.8701	−1.78343	−0.891714	−	0.452599i	$-0.850497\pi$
$227$	−26.8701	−1.78343	−0.891714	+	0.452599i	$0.850497\pi$
$228$	0	0
$229$	−25.4558	−1.68217	−0.841085	−	0.540903i	$-0.818082\pi$
$229$	−25.4558	−1.68217	−0.841085	+	0.540903i	$0.818082\pi$
$230$	16.9706	1.11901
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	−20.0000	−1.30466
$236$	14.1421	0.920575
$237$	0	0
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−2.82843	−0.182195	−0.0910975	−	0.995842i	$-0.529037\pi$
$241$	−2.82843	−0.182195	−0.0910975	+	0.995842i	$0.529037\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−4.24264	−0.271607
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	−1.41421	−0.0898027
$249$	0	0
$250$	5.65685	0.357771
$251$	−25.4558	−1.60676	−0.803379	−	0.595468i	$-0.796967\pi$
$251$	−25.4558	−1.60676	−0.803379	+	0.595468i	$0.796967\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.3848	−1.14681	−0.573405	−	0.819272i	$-0.694378\pi$
$257$	−18.3848	−1.14681	−0.573405	+	0.819272i	$0.694378\pi$
$258$	0	0
$259$	0	0
$260$	−12.0000	−0.744208
$261$	0	0
$262$	−7.07107	−0.436852
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	16.9706	1.04249
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−25.4558	−1.55207	−0.776035	−	0.630690i	$-0.782772\pi$
$269$	−25.4558	−1.55207	−0.776035	+	0.630690i	$0.782772\pi$
$270$	0	0
$271$	−8.48528	−0.515444	−0.257722	−	0.966219i	$-0.582972\pi$
$271$	−8.48528	−0.515444	−0.257722	+	0.966219i	$0.582972\pi$
$272$	2.82843	0.171499
$273$	0	0
$274$	−10.0000	−0.604122
$275$	−3.00000	−0.180907
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	−21.2132	−1.27228
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−4.24264	−0.252199	−0.126099	−	0.992018i	$-0.540246\pi$
$283$	−4.24264	−0.252199	−0.126099	+	0.992018i	$0.540246\pi$
$284$	0	0
$285$	0	0
$286$	−4.24264	−0.250873
$287$	0	0
$288$	0	0
$289$	−9.00000	−0.529412
$290$	22.6274	1.32873
$291$	0	0
$292$	8.48528	0.496564
$293$	15.5563	0.908812	0.454406	−	0.890795i	$-0.349852\pi$
$293$	15.5563	0.908812	0.454406	+	0.890795i	$0.349852\pi$
$294$	0	0
$295$	−40.0000	−2.32889
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−25.4558	−1.47215
$300$	0	0
$301$	0	0
$302$	−20.0000	−1.15087
$303$	0	0
$304$	1.41421	0.0811107
$305$	12.0000	0.687118
$306$	0	0
$307$	−24.0416	−1.37213	−0.686064	−	0.727541i	$-0.740663\pi$
$307$	−24.0416	−1.37213	−0.686064	+	0.727541i	$0.740663\pi$
$308$	0	0
$309$	0	0
$310$	4.00000	0.227185
$311$	24.0416	1.36328	0.681638	−	0.731690i	$-0.261268\pi$
$311$	24.0416	1.36328	0.681638	+	0.731690i	$0.261268\pi$
$312$	0	0
$313$	−12.7279	−0.719425	−0.359712	−	0.933063i	$-0.617125\pi$
$313$	−12.7279	−0.719425	−0.359712	+	0.933063i	$0.617125\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	−2.82843	−0.158114
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	12.7279	0.706018
$326$	−16.0000	−0.886158
$327$	0	0
$328$	−8.48528	−0.468521
$329$	0	0
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	7.07107	0.388075
$333$	0	0
$334$	14.1421	0.773823
$335$	11.3137	0.618134
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	−8.00000	−0.433861
$341$	1.41421	0.0765840
$342$	0	0
$343$	0	0
$344$	10.0000	0.539164
$345$	0	0
$346$	15.5563	0.836315
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	−9.89949	−0.529908	−0.264954	−	0.964261i	$-0.585357\pi$
$349$	−9.89949	−0.529908	−0.264954	+	0.964261i	$0.585357\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	29.6985	1.58069	0.790345	−	0.612661i	$-0.209901\pi$
$353$	29.6985	1.58069	0.790345	+	0.612661i	$0.209901\pi$
$354$	0	0
$355$	0	0
$356$	−18.3848	−0.974391
$357$	0	0
$358$	12.0000	0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	−8.48528	−0.445976
$363$	0	0
$364$	0	0
$365$	−24.0000	−1.25622
$366$	0	0
$367$	24.0416	1.25496	0.627481	−	0.778632i	$-0.284086\pi$
$367$	24.0416	1.25496	0.627481	+	0.778632i	$0.284086\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	16.9706	0.882258
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−2.82843	−0.146254
$375$	0	0
$376$	7.07107	0.364662
$377$	−33.9411	−1.74806
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	22.0000	1.12562
$383$	−35.3553	−1.80657	−0.903287	−	0.429037i	$-0.858853\pi$
$383$	−35.3553	−1.80657	−0.903287	+	0.429037i	$0.858853\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	−1.41421	−0.0717958
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−16.9706	−0.858238
$392$	0	0
$393$	0	0
$394$	−10.0000	−0.503793
$395$	0	0
$396$	0	0
$397$	−11.3137	−0.567819	−0.283909	−	0.958851i	$-0.591631\pi$
$397$	−11.3137	−0.567819	−0.283909	+	0.958851i	$0.591631\pi$
$398$	4.24264	0.212664
$399$	0	0
$400$	3.00000	0.150000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	1.41421	0.0703598
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−28.2843	−1.39857	−0.699284	−	0.714844i	$-0.746498\pi$
$409$	−28.2843	−1.39857	−0.699284	+	0.714844i	$0.746498\pi$
$410$	24.0000	1.18528
$411$	0	0
$412$	4.24264	0.209020
$413$	0	0
$414$	0	0
$415$	−20.0000	−0.981761
$416$	4.24264	0.208013
$417$	0	0
$418$	−1.41421	−0.0691714
$419$	22.6274	1.10542	0.552711	−	0.833373i	$-0.313593\pi$
$419$	22.6274	1.10542	0.552711	+	0.833373i	$0.313593\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−14.0000	−0.681509
$423$	0	0
$424$	−6.00000	−0.291386
$425$	8.48528	0.411597
$426$	0	0
$427$	0	0
$428$	−18.0000	−0.870063
$429$	0	0
$430$	−28.2843	−1.36399
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	24.0416	1.15537	0.577684	−	0.816261i	$-0.303957\pi$
$433$	24.0416	1.15537	0.577684	+	0.816261i	$0.303957\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	−8.48528	−0.405906
$438$	0	0
$439$	−31.1127	−1.48493	−0.742464	−	0.669886i	$-0.766343\pi$
$439$	−31.1127	−1.48493	−0.742464	+	0.669886i	$0.766343\pi$
$440$	2.82843	0.134840
$441$	0	0
$442$	12.0000	0.570782
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	52.0000	2.46504
$446$	−12.7279	−0.602685
$447$	0	0
$448$	0	0
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	0	0
$451$	8.48528	0.399556
$452$	−8.00000	−0.376288
$453$	0	0
$454$	−26.8701	−1.26107
$455$	0	0
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	−25.4558	−1.18947
$459$	0	0
$460$	16.9706	0.791257
$461$	−18.3848	−0.856264	−0.428132	−	0.903716i	$-0.640828\pi$
$461$	−18.3848	−0.856264	−0.428132	+	0.903716i	$0.640828\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−2.00000	−0.0926482
$467$	33.9411	1.57061	0.785304	−	0.619110i	$-0.212507\pi$
$467$	33.9411	1.57061	0.785304	+	0.619110i	$0.212507\pi$
$468$	0	0
$469$	0	0
$470$	−20.0000	−0.922531
$471$	0	0
$472$	14.1421	0.650945
$473$	−10.0000	−0.459800
$474$	0	0
$475$	4.24264	0.194666
$476$	0	0
$477$	0	0
$478$	4.00000	0.182956
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	−25.4558	−1.16069
$482$	−2.82843	−0.128831
$483$	0	0
$484$	1.00000	0.0454545
$485$	4.00000	0.181631
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	−4.24264	−0.192055
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−22.6274	−1.01909
$494$	6.00000	0.269953
$495$	0	0
$496$	−1.41421	−0.0635001
$497$	0	0
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	5.65685	0.252982
$501$	0	0
$502$	−25.4558	−1.13615
$503$	14.1421	0.630567	0.315283	−	0.948998i	$-0.397900\pi$
$503$	14.1421	0.630567	0.315283	+	0.948998i	$0.397900\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	6.00000	0.266733
$507$	0	0
$508$	8.00000	0.354943
$509$	31.1127	1.37905	0.689523	−	0.724264i	$-0.257820\pi$
$509$	31.1127	1.37905	0.689523	+	0.724264i	$0.257820\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.3848	−0.810918
$515$	−12.0000	−0.528783
$516$	0	0
$517$	−7.07107	−0.310985
$518$	0	0
$519$	0	0
$520$	−12.0000	−0.526235
$521$	41.0122	1.79678	0.898388	−	0.439202i	$-0.144739\pi$
$521$	41.0122	1.79678	0.898388	+	0.439202i	$0.144739\pi$
$522$	0	0
$523$	−18.3848	−0.803910	−0.401955	−	0.915659i	$-0.631669\pi$
$523$	−18.3848	−0.803910	−0.401955	+	0.915659i	$0.631669\pi$
$524$	−7.07107	−0.308901
$525$	0	0
$526$	−12.0000	−0.523225
$527$	−4.00000	−0.174243
$528$	0	0
$529$	13.0000	0.565217
$530$	16.9706	0.737154
$531$	0	0
$532$	0	0
$533$	−36.0000	−1.55933
$534$	0	0
$535$	50.9117	2.20110
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−25.4558	−1.09748
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−8.48528	−0.364474
$543$	0	0
$544$	2.82843	0.121268
$545$	−39.5980	−1.69619
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−10.0000	−0.427179
$549$	0	0
$550$	−3.00000	−0.127920
$551$	−11.3137	−0.481980
$552$	0	0
$553$	0	0
$554$	4.00000	0.169944
$555$	0	0
$556$	−21.2132	−0.899640
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	42.4264	1.79445
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	9.89949	0.417214	0.208607	−	0.978000i	$-0.433107\pi$
$563$	9.89949	0.417214	0.208607	+	0.978000i	$0.433107\pi$
$564$	0	0
$565$	22.6274	0.951943
$566$	−4.24264	−0.178331
$567$	0	0
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−4.24264	−0.177394
$573$	0	0
$574$	0	0
$575$	−18.0000	−0.750652
$576$	0	0
$577$	12.7279	0.529870	0.264935	−	0.964266i	$-0.414649\pi$
$577$	12.7279	0.529870	0.264935	+	0.964266i	$0.414649\pi$
$578$	−9.00000	−0.374351
$579$	0	0
$580$	22.6274	0.939552
$581$	0	0
$582$	0	0
$583$	6.00000	0.248495
$584$	8.48528	0.351123
$585$	0	0
$586$	15.5563	0.642627
$587$	19.7990	0.817192	0.408596	−	0.912715i	$-0.366019\pi$
$587$	19.7990	0.817192	0.408596	+	0.912715i	$0.366019\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	−40.0000	−1.64677
$591$	0	0
$592$	−6.00000	−0.246598
$593$	−2.82843	−0.116150	−0.0580748	−	0.998312i	$-0.518496\pi$
$593$	−2.82843	−0.116150	−0.0580748	+	0.998312i	$0.518496\pi$
$594$	0	0
$595$	0	0
$596$	−4.00000	−0.163846
$597$	0	0
$598$	−25.4558	−1.04097
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	−20.0000	−0.813788
$605$	−2.82843	−0.114992
$606$	0	0
$607$	−33.9411	−1.37763	−0.688814	−	0.724938i	$-0.741868\pi$
$607$	−33.9411	−1.37763	−0.688814	+	0.724938i	$0.741868\pi$
$608$	1.41421	0.0573539
$609$	0	0
$610$	12.0000	0.485866
$611$	30.0000	1.21367
$612$	0	0
$613$	44.0000	1.77714	0.888572	−	0.458738i	$-0.151698\pi$
$613$	44.0000	1.77714	0.888572	+	0.458738i	$0.151698\pi$
$614$	−24.0416	−0.970241
$615$	0	0
$616$	0	0
$617$	4.00000	0.161034	0.0805170	−	0.996753i	$-0.474343\pi$
$617$	4.00000	0.161034	0.0805170	+	0.996753i	$0.474343\pi$
$618$	0	0
$619$	28.2843	1.13684	0.568420	−	0.822738i	$-0.307555\pi$
$619$	28.2843	1.13684	0.568420	+	0.822738i	$0.307555\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	24.0416	0.963982
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	−12.7279	−0.508710
$627$	0	0
$628$	0	0
$629$	−16.9706	−0.676661
$630$	0	0
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	0	0
$633$	0	0
$634$	30.0000	1.19145
$635$	−22.6274	−0.897942
$636$	0	0
$637$	0	0
$638$	8.00000	0.316723
$639$	0	0
$640$	−2.82843	−0.111803
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	0	0
$643$	25.4558	1.00388	0.501940	−	0.864902i	$-0.332620\pi$
$643$	25.4558	1.00388	0.501940	+	0.864902i	$0.332620\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	7.07107	0.277992	0.138996	−	0.990293i	$-0.455612\pi$
$647$	7.07107	0.277992	0.138996	+	0.990293i	$0.455612\pi$
$648$	0	0
$649$	−14.1421	−0.555127
$650$	12.7279	0.499230
$651$	0	0
$652$	−16.0000	−0.626608
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	20.0000	0.781465
$656$	−8.48528	−0.331295
$657$	0	0
$658$	0	0
$659$	−46.0000	−1.79191	−0.895953	−	0.444149i	$-0.853506\pi$
$659$	−46.0000	−1.79191	−0.895953	+	0.444149i	$0.853506\pi$
$660$	0	0
$661$	−8.48528	−0.330039	−0.165020	−	0.986290i	$-0.552769\pi$
$661$	−8.48528	−0.330039	−0.165020	+	0.986290i	$0.552769\pi$
$662$	−32.0000	−1.24372
$663$	0	0
$664$	7.07107	0.274411
$665$	0	0
$666$	0	0
$667$	48.0000	1.85857
$668$	14.1421	0.547176
$669$	0	0
$670$	11.3137	0.437087
$671$	4.24264	0.163785
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	5.00000	0.192308
$677$	1.41421	0.0543526	0.0271763	−	0.999631i	$-0.491348\pi$
$677$	1.41421	0.0543526	0.0271763	+	0.999631i	$0.491348\pi$
$678$	0	0
$679$	0	0
$680$	−8.00000	−0.306786
$681$	0	0
$682$	1.41421	0.0541530
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	28.2843	1.08069
$686$	0	0
$687$	0	0
$688$	10.0000	0.381246
$689$	−25.4558	−0.969790
$690$	0	0
$691$	8.48528	0.322795	0.161398	−	0.986889i	$-0.448400\pi$
$691$	8.48528	0.322795	0.161398	+	0.986889i	$0.448400\pi$
$692$	15.5563	0.591364
$693$	0	0
$694$	−10.0000	−0.379595
$695$	60.0000	2.27593
$696$	0	0
$697$	−24.0000	−0.909065
$698$	−9.89949	−0.374701
$699$	0	0
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	−8.48528	−0.320028
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	29.6985	1.11772
$707$	0	0
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	−18.3848	−0.688999
$713$	8.48528	0.317776
$714$	0	0
$715$	12.0000	0.448775
$716$	12.0000	0.448461
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−4.24264	−0.158224	−0.0791119	−	0.996866i	$-0.525208\pi$
$719$	−4.24264	−0.158224	−0.0791119	+	0.996866i	$0.525208\pi$
$720$	0	0
$721$	0	0
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−8.48528	−0.315353
$725$	−24.0000	−0.891338
$726$	0	0
$727$	15.5563	0.576953	0.288477	−	0.957487i	$-0.406851\pi$
$727$	15.5563	0.576953	0.288477	+	0.957487i	$0.406851\pi$
$728$	0	0
$729$	0	0
$730$	−24.0000	−0.888280
$731$	28.2843	1.04613
$732$	0	0
$733$	12.7279	0.470117	0.235058	−	0.971981i	$-0.424472\pi$
$733$	12.7279	0.470117	0.235058	+	0.971981i	$0.424472\pi$
$734$	24.0416	0.887393
$735$	0	0
$736$	−6.00000	−0.221163
$737$	4.00000	0.147342
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	16.9706	0.623850
$741$	0	0
$742$	0	0
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	11.3137	0.414502
$746$	26.0000	0.951928
$747$	0	0
$748$	−2.82843	−0.103418
$749$	0	0
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	7.07107	0.257855
$753$	0	0
$754$	−33.9411	−1.23606
$755$	56.5685	2.05874
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−22.6274	−0.820243	−0.410122	−	0.912031i	$-0.634514\pi$
$761$	−22.6274	−0.820243	−0.410122	+	0.912031i	$0.634514\pi$
$762$	0	0
$763$	0	0
$764$	22.0000	0.795932
$765$	0	0
$766$	−35.3553	−1.27744
$767$	60.0000	2.16647
$768$	0	0
$769$	16.9706	0.611974	0.305987	−	0.952036i	$-0.401014\pi$
$769$	16.9706	0.611974	0.305987	+	0.952036i	$0.401014\pi$
$770$	0	0
$771$	0	0
$772$	−14.0000	−0.503871
$773$	22.6274	0.813852	0.406926	−	0.913461i	$-0.366601\pi$
$773$	22.6274	0.813852	0.406926	+	0.913461i	$0.366601\pi$
$774$	0	0
$775$	−4.24264	−0.152400
$776$	−1.41421	−0.0507673
$777$	0	0
$778$	18.0000	0.645331
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	−16.9706	−0.606866
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15.5563	0.554524	0.277262	−	0.960794i	$-0.410573\pi$
$787$	15.5563	0.554524	0.277262	+	0.960794i	$0.410573\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−18.0000	−0.639199
$794$	−11.3137	−0.401508
$795$	0	0
$796$	4.24264	0.150376
$797$	39.5980	1.40263	0.701316	−	0.712850i	$-0.252596\pi$
$797$	39.5980	1.40263	0.701316	+	0.712850i	$0.252596\pi$
$798$	0	0
$799$	20.0000	0.707549
$800$	3.00000	0.106066
$801$	0	0
$802$	−18.0000	−0.635602
$803$	−8.48528	−0.299439
$804$	0	0
$805$	0	0
$806$	−6.00000	−0.211341
$807$	0	0
$808$	1.41421	0.0497519
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	18.3848	0.645577	0.322788	−	0.946471i	$-0.395380\pi$
$811$	18.3848	0.645577	0.322788	+	0.946471i	$0.395380\pi$
$812$	0	0
$813$	0	0
$814$	6.00000	0.210300
$815$	45.2548	1.58521
$816$	0	0
$817$	14.1421	0.494771
$818$	−28.2843	−0.988936
$819$	0	0
$820$	24.0000	0.838116
$821$	20.0000	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$822$	0	0
$823$	−42.0000	−1.46403	−0.732014	−	0.681290i	$-0.761419\pi$
$823$	−42.0000	−1.46403	−0.732014	+	0.681290i	$0.761419\pi$
$824$	4.24264	0.147799
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	25.4558	0.884118	0.442059	−	0.896986i	$-0.354248\pi$
$829$	25.4558	0.884118	0.442059	+	0.896986i	$0.354248\pi$
$830$	−20.0000	−0.694210
$831$	0	0
$832$	4.24264	0.147087
$833$	0	0
$834$	0	0
$835$	−40.0000	−1.38426
$836$	−1.41421	−0.0489116
$837$	0	0
$838$	22.6274	0.781651
$839$	15.5563	0.537065	0.268532	−	0.963271i	$-0.413461\pi$
$839$	15.5563	0.537065	0.268532	+	0.963271i	$0.413461\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	2.00000	0.0689246
$843$	0	0
$844$	−14.0000	−0.481900
$845$	−14.1421	−0.486504
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	0	0
$850$	8.48528	0.291043
$851$	36.0000	1.23406
$852$	0	0
$853$	−38.1838	−1.30739	−0.653694	−	0.756759i	$-0.726781\pi$
$853$	−38.1838	−1.30739	−0.653694	+	0.756759i	$0.726781\pi$
$854$	0	0
$855$	0	0
$856$	−18.0000	−0.615227
$857$	−22.6274	−0.772938	−0.386469	−	0.922302i	$-0.626305\pi$
$857$	−22.6274	−0.772938	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	−19.7990	−0.675533	−0.337766	−	0.941230i	$-0.609671\pi$
$859$	−19.7990	−0.675533	−0.337766	+	0.941230i	$0.609671\pi$
$860$	−28.2843	−0.964486
$861$	0	0
$862$	12.0000	0.408722
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	0	0
$865$	−44.0000	−1.49604
$866$	24.0416	0.816968
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.9706	−0.575026
$872$	14.0000	0.474100
$873$	0	0
$874$	−8.48528	−0.287019
$875$	0	0
$876$	0	0
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	−31.1127	−1.05000
$879$	0	0
$880$	2.82843	0.0953463
$881$	−43.8406	−1.47703	−0.738514	−	0.674238i	$-0.764472\pi$
$881$	−43.8406	−1.47703	−0.738514	+	0.674238i	$0.764472\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−36.0000	−1.20944
$887$	−5.65685	−0.189939	−0.0949693	−	0.995480i	$-0.530275\pi$
$887$	−5.65685	−0.189939	−0.0949693	+	0.995480i	$0.530275\pi$
$888$	0	0
$889$	0	0
$890$	52.0000	1.74304
$891$	0	0
$892$	−12.7279	−0.426162
$893$	10.0000	0.334637
$894$	0	0
$895$	−33.9411	−1.13453
$896$	0	0
$897$	0	0
$898$	−8.00000	−0.266963
$899$	11.3137	0.377333
$900$	0	0
$901$	−16.9706	−0.565371
$902$	8.48528	0.282529
$903$	0	0
$904$	−8.00000	−0.266076
$905$	24.0000	0.797787
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−26.8701	−0.891714
$909$	0	0
$910$	0	0
$911$	10.0000	0.331315	0.165657	−	0.986183i	$-0.447025\pi$
$911$	10.0000	0.331315	0.165657	+	0.986183i	$0.447025\pi$
$912$	0	0
$913$	−7.07107	−0.234018
$914$	38.0000	1.25693
$915$	0	0
$916$	−25.4558	−0.841085
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	16.9706	0.559503
$921$	0	0
$922$	−18.3848	−0.605470
$923$	0	0
$924$	0	0
$925$	−18.0000	−0.591836
$926$	32.0000	1.05159
$927$	0	0
$928$	−8.00000	−0.262613
$929$	−1.41421	−0.0463988	−0.0231994	−	0.999731i	$-0.507385\pi$
$929$	−1.41421	−0.0463988	−0.0231994	+	0.999731i	$0.507385\pi$
$930$	0	0
$931$	0	0
$932$	−2.00000	−0.0655122
$933$	0	0
$934$	33.9411	1.11059
$935$	8.00000	0.261628
$936$	0	0
$937$	−19.7990	−0.646805	−0.323402	−	0.946262i	$-0.604827\pi$
$937$	−19.7990	−0.646805	−0.323402	+	0.946262i	$0.604827\pi$
$938$	0	0
$939$	0	0
$940$	−20.0000	−0.652328
$941$	−38.1838	−1.24476	−0.622378	−	0.782717i	$-0.713833\pi$
$941$	−38.1838	−1.24476	−0.622378	+	0.782717i	$0.713833\pi$
$942$	0	0
$943$	50.9117	1.65791
$944$	14.1421	0.460287
$945$	0	0
$946$	−10.0000	−0.325128
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	36.0000	1.16861
$950$	4.24264	0.137649
$951$	0	0
$952$	0	0
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	0	0
$955$	−62.2254	−2.01357
$956$	4.00000	0.129369
$957$	0	0
$958$	11.3137	0.365529
$959$	0	0
$960$	0	0
$961$	−29.0000	−0.935484
$962$	−25.4558	−0.820729
$963$	0	0
$964$	−2.82843	−0.0910975
$965$	39.5980	1.27470
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	4.00000	0.128432
$971$	11.3137	0.363074	0.181537	−	0.983384i	$-0.441893\pi$
$971$	11.3137	0.363074	0.181537	+	0.983384i	$0.441893\pi$
$972$	0	0
$973$	0	0
$974$	22.0000	0.704925
$975$	0	0
$976$	−4.24264	−0.135804
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	0	0
$979$	18.3848	0.587580
$980$	0	0
$981$	0	0
$982$	12.0000	0.382935
$983$	21.2132	0.676596	0.338298	−	0.941039i	$-0.390149\pi$
$983$	21.2132	0.676596	0.338298	+	0.941039i	$0.390149\pi$
$984$	0	0
$985$	28.2843	0.901212
$986$	−22.6274	−0.720604
$987$	0	0
$988$	6.00000	0.190885
$989$	−60.0000	−1.90789
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	−1.41421	−0.0449013
$993$	0	0
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	57.9828	1.83633	0.918166	−	0.396196i	$-0.129670\pi$
$997$	57.9828	1.83633	0.918166	+	0.396196i	$0.129670\pi$
$998$	16.0000	0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.df.1.1		2
3.2	odd	2		1078.2.a.p.1.2	yes	2
7.6	odd	2	inner	9702.2.a.df.1.2		2
12.11	even	2		8624.2.a.bl.1.2		2
21.2	odd	6		1078.2.e.t.67.1		4
21.5	even	6		1078.2.e.t.67.2		4
21.11	odd	6		1078.2.e.t.177.1		4
21.17	even	6		1078.2.e.t.177.2		4
21.20	even	2		1078.2.a.p.1.1	✓	2
84.83	odd	2		8624.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.a.p.1.1	✓	2	21.20	even	2
1078.2.a.p.1.2	yes	2	3.2	odd	2
1078.2.e.t.67.1		4	21.2	odd	6
1078.2.e.t.67.2		4	21.5	even	6
1078.2.e.t.177.1		4	21.11	odd	6
1078.2.e.t.177.2		4	21.17	even	6
8624.2.a.bl.1.1		2	84.83	odd	2
8624.2.a.bl.1.2		2	12.11	even	2
9702.2.a.df.1.1		2	1.1	even	1	trivial
9702.2.a.df.1.2		2	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{8} -2.82843 q^{10} -1.00000 q^{11} +4.24264 q^{13} +1.00000 q^{16} +2.82843 q^{17} +1.41421 q^{19} -2.82843 q^{20} -1.00000 q^{22} -6.00000 q^{23} +3.00000 q^{25} +4.24264 q^{26} -8.00000 q^{29} -1.41421 q^{31} +1.00000 q^{32} +2.82843 q^{34} -6.00000 q^{37} +1.41421 q^{38} -2.82843 q^{40} -8.48528 q^{41} +10.0000 q^{43} -1.00000 q^{44} -6.00000 q^{46} +7.07107 q^{47} +3.00000 q^{50} +4.24264 q^{52} -6.00000 q^{53} +2.82843 q^{55} -8.00000 q^{58} +14.1421 q^{59} -4.24264 q^{61} -1.41421 q^{62} +1.00000 q^{64} -12.0000 q^{65} -4.00000 q^{67} +2.82843 q^{68} +8.48528 q^{73} -6.00000 q^{74} +1.41421 q^{76} -2.82843 q^{80} -8.48528 q^{82} +7.07107 q^{83} -8.00000 q^{85} +10.0000 q^{86} -1.00000 q^{88} -18.3848 q^{89} -6.00000 q^{92} +7.07107 q^{94} -4.00000 q^{95} -1.41421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 2 q^{16} - 2 q^{22} - 12 q^{23} + 6 q^{25} - 16 q^{29} + 2 q^{32} - 12 q^{37} + 20 q^{43} - 2 q^{44} - 12 q^{46} + 6 q^{50} - 12 q^{53} - 16 q^{58} + 2 q^{64} - 24 q^{65} - 8 q^{67} - 12 q^{74} - 16 q^{85} + 20 q^{86} - 2 q^{88} - 12 q^{92} - 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 + 2 * q^16 - 2 * q^22 - 12 * q^23 + 6 * q^25 - 16 * q^29 + 2 * q^32 - 12 * q^37 + 20 * q^43 - 2 * q^44 - 12 * q^46 + 6 * q^50 - 12 * q^53 - 16 * q^58 + 2 * q^64 - 24 * q^65 - 8 * q^67 - 12 * q^74 - 16 * q^85 + 20 * q^86 - 2 * q^88 - 12 * q^92 - 8 * q^95

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