LMFDB - Embedded newform 7105.2.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7105,2,Mod(1,7105)] 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("7105.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7105 = 5 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7105.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.7337106361$
Analytic rank:	$0$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1015)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	7105.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{8} -3.00000 q^{9} +1.73205 q^{10} -5.46410 q^{11} +2.00000 q^{13} -5.00000 q^{16} +3.46410 q^{17} -5.19615 q^{18} -2.00000 q^{19} +1.00000 q^{20} -9.46410 q^{22} +6.92820 q^{23} +1.00000 q^{25} +3.46410 q^{26} +1.00000 q^{29} +8.92820 q^{31} -5.19615 q^{32} +6.00000 q^{34} -3.00000 q^{36} -3.46410 q^{38} -1.73205 q^{40} -5.46410 q^{41} +3.46410 q^{43} -5.46410 q^{44} -3.00000 q^{45} +12.0000 q^{46} +4.00000 q^{47} +1.73205 q^{50} +2.00000 q^{52} +4.92820 q^{53} -5.46410 q^{55} +1.73205 q^{58} +8.00000 q^{59} +1.46410 q^{61} +15.4641 q^{62} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} +3.46410 q^{68} -8.00000 q^{71} +5.19615 q^{72} -14.3923 q^{73} -2.00000 q^{76} -1.46410 q^{79} -5.00000 q^{80} +9.00000 q^{81} -9.46410 q^{82} +1.07180 q^{83} +3.46410 q^{85} +6.00000 q^{86} +9.46410 q^{88} +17.4641 q^{89} -5.19615 q^{90} +6.92820 q^{92} +6.92820 q^{94} -2.00000 q^{95} +11.4641 q^{97} +16.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{13} - 10 q^{16} - 4 q^{19} + 2 q^{20} - 12 q^{22} + 2 q^{25} + 2 q^{29} + 4 q^{31} + 12 q^{34} - 6 q^{36} - 4 q^{41} - 4 q^{44} - 6 q^{45} + 24 q^{46}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^13 - 10 * q^16 - 4 * q^19 + 2 * q^20 - 12 * q^22 + 2 * q^25 + 2 * q^29 + 4 * q^31 + 12 * q^34 - 6 * q^36 - 4 * q^41 - 4 * q^44 - 6 * q^45 + 24 * q^46 + 8 * q^47 + 4 * q^52 - 4 * q^53 - 4 * q^55 + 16 * q^59 - 4 * q^61 + 24 * q^62 + 2 * q^64 + 4 * q^65 - 8 * q^67 - 16 * q^71 - 8 * q^73 - 4 * q^76 + 4 * q^79 - 10 * q^80 + 18 * q^81 - 12 * q^82 + 16 * q^83 + 12 * q^86 + 12 * q^88 + 28 * q^89 - 4 * q^95 + 16 * q^97 + 12 * 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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.73205	−0.612372
$9$	−3.00000	−1.00000
$10$	1.73205	0.547723
$11$	−5.46410	−1.64749	−0.823744	−	0.566961i	$-0.808119\pi$
$11$	−5.46410	−1.64749	−0.823744	+	0.566961i	$0.808119\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	−5.00000	−1.25000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	−5.19615	−1.22474
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−9.46410	−2.01775
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.46410	0.679366
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	−3.00000	−0.500000
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−3.46410	−0.561951
$39$	0	0
$40$	−1.73205	−0.273861
$41$	−5.46410	−0.853349	−0.426675	−	0.904405i	$-0.640315\pi$
$41$	−5.46410	−0.853349	−0.426675	+	0.904405i	$0.640315\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	−5.46410	−0.823744
$45$	−3.00000	−0.447214
$46$	12.0000	1.76930
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	0	0
$50$	1.73205	0.244949
$51$	0	0
$52$	2.00000	0.277350
$53$	4.92820	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$53$	4.92820	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$54$	0	0
$55$	−5.46410	−0.736779
$56$	0	0
$57$	0	0
$58$	1.73205	0.227429
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	1.46410	0.187459	0.0937295	−	0.995598i	$-0.470121\pi$
$61$	1.46410	0.187459	0.0937295	+	0.995598i	$0.470121\pi$
$62$	15.4641	1.96394
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.46410	0.420084
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	5.19615	0.612372
$73$	−14.3923	−1.68449	−0.842246	−	0.539093i	$-0.818767\pi$
$73$	−14.3923	−1.68449	−0.842246	+	0.539093i	$0.818767\pi$
$74$	0	0
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	−1.46410	−0.164724	−0.0823622	−	0.996602i	$-0.526246\pi$
$79$	−1.46410	−0.164724	−0.0823622	+	0.996602i	$0.526246\pi$
$80$	−5.00000	−0.559017
$81$	9.00000	1.00000
$82$	−9.46410	−1.04514
$83$	1.07180	0.117645	0.0588225	−	0.998268i	$-0.481265\pi$
$83$	1.07180	0.117645	0.0588225	+	0.998268i	$0.481265\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	6.00000	0.646997
$87$	0	0
$88$	9.46410	1.00888
$89$	17.4641	1.85119	0.925596	−	0.378514i	$-0.123565\pi$
$89$	17.4641	1.85119	0.925596	+	0.378514i	$0.123565\pi$
$90$	−5.19615	−0.547723
$91$	0	0
$92$	6.92820	0.722315
$93$	0	0
$94$	6.92820	0.714590
$95$	−2.00000	−0.205196
$96$	0	0
$97$	11.4641	1.16400	0.582002	−	0.813188i	$-0.302270\pi$
$97$	11.4641	1.16400	0.582002	+	0.813188i	$0.302270\pi$
$98$	0	0
$99$	16.3923	1.64749
$100$	1.00000	0.100000
$101$	13.4641	1.33973	0.669864	−	0.742484i	$-0.266352\pi$
$101$	13.4641	1.33973	0.669864	+	0.742484i	$0.266352\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−3.46410	−0.339683
$105$	0	0
$106$	8.53590	0.829080
$107$	10.9282	1.05647	0.528235	−	0.849098i	$-0.322854\pi$
$107$	10.9282	1.05647	0.528235	+	0.849098i	$0.322854\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−9.46410	−0.902367
$111$	0	0
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	6.92820	0.646058
$116$	1.00000	0.0928477
$117$	−6.00000	−0.554700
$118$	13.8564	1.27559
$119$	0	0
$120$	0	0
$121$	18.8564	1.71422
$122$	2.53590	0.229589
$123$	0	0
$124$	8.92820	0.801776
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.53590	0.757438	0.378719	−	0.925512i	$-0.376365\pi$
$127$	8.53590	0.757438	0.378719	+	0.925512i	$0.376365\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	3.46410	0.303822
$131$	4.92820	0.430579	0.215290	−	0.976550i	$-0.430930\pi$
$131$	4.92820	0.430579	0.215290	+	0.976550i	$0.430930\pi$
$132$	0	0
$133$	0	0
$134$	−6.92820	−0.598506
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−18.9282	−1.61715	−0.808573	−	0.588396i	$-0.799760\pi$
$137$	−18.9282	−1.61715	−0.808573	+	0.588396i	$0.799760\pi$
$138$	0	0
$139$	−17.8564	−1.51456	−0.757280	−	0.653090i	$-0.773472\pi$
$139$	−17.8564	−1.51456	−0.757280	+	0.653090i	$0.773472\pi$
$140$	0	0
$141$	0	0
$142$	−13.8564	−1.16280
$143$	−10.9282	−0.913862
$144$	15.0000	1.25000
$145$	1.00000	0.0830455
$146$	−24.9282	−2.06307
$147$	0	0
$148$	0	0
$149$	23.8564	1.95439	0.977196	−	0.212337i	$-0.0681075\pi$
$149$	23.8564	1.95439	0.977196	+	0.212337i	$0.0681075\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	3.46410	0.280976
$153$	−10.3923	−0.840168
$154$	0	0
$155$	8.92820	0.717131
$156$	0	0
$157$	11.4641	0.914935	0.457467	−	0.889226i	$-0.348757\pi$
$157$	11.4641	0.914935	0.457467	+	0.889226i	$0.348757\pi$
$158$	−2.53590	−0.201745
$159$	0	0
$160$	−5.19615	−0.410792
$161$	0	0
$162$	15.5885	1.22474
$163$	17.3205	1.35665	0.678323	−	0.734763i	$-0.262707\pi$
$163$	17.3205	1.35665	0.678323	+	0.734763i	$0.262707\pi$
$164$	−5.46410	−0.426675
$165$	0	0
$166$	1.85641	0.144085
$167$	−5.85641	−0.453182	−0.226591	−	0.973990i	$-0.572758\pi$
$167$	−5.85641	−0.453182	−0.226591	+	0.973990i	$0.572758\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	6.00000	0.458831
$172$	3.46410	0.264135
$173$	0.928203	0.0705700	0.0352850	−	0.999377i	$-0.488766\pi$
$173$	0.928203	0.0705700	0.0352850	+	0.999377i	$0.488766\pi$
$174$	0	0
$175$	0	0
$176$	27.3205	2.05936
$177$	0	0
$178$	30.2487	2.26724
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	−3.00000	−0.223607
$181$	12.9282	0.960946	0.480473	−	0.877010i	$-0.340465\pi$
$181$	12.9282	0.960946	0.480473	+	0.877010i	$0.340465\pi$
$182$	0	0
$183$	0	0
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	−18.9282	−1.38417
$188$	4.00000	0.291730
$189$	0	0
$190$	−3.46410	−0.251312
$191$	8.39230	0.607246	0.303623	−	0.952792i	$-0.401804\pi$
$191$	8.39230	0.607246	0.303623	+	0.952792i	$0.401804\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	19.8564	1.42561
$195$	0	0
$196$	0	0
$197$	−18.7846	−1.33835	−0.669174	−	0.743106i	$-0.733352\pi$
$197$	−18.7846	−1.33835	−0.669174	+	0.743106i	$0.733352\pi$
$198$	28.3923	2.01775
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−1.73205	−0.122474
$201$	0	0
$202$	23.3205	1.64083
$203$	0	0
$204$	0	0
$205$	−5.46410	−0.381629
$206$	27.7128	1.93084
$207$	−20.7846	−1.44463
$208$	−10.0000	−0.693375
$209$	10.9282	0.755920
$210$	0	0
$211$	−0.392305	−0.0270074	−0.0135037	−	0.999909i	$-0.504298\pi$
$211$	−0.392305	−0.0270074	−0.0135037	+	0.999909i	$0.504298\pi$
$212$	4.92820	0.338470
$213$	0	0
$214$	18.9282	1.29391
$215$	3.46410	0.236250
$216$	0	0
$217$	0	0
$218$	17.3205	1.17309
$219$	0	0
$220$	−5.46410	−0.368390
$221$	6.92820	0.466041
$222$	0	0
$223$	−5.07180	−0.339633	−0.169816	−	0.985476i	$-0.554317\pi$
$223$	−5.07180	−0.339633	−0.169816	+	0.985476i	$0.554317\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	−27.7128	−1.84343
$227$	−9.85641	−0.654193	−0.327096	−	0.944991i	$-0.606070\pi$
$227$	−9.85641	−0.654193	−0.327096	+	0.944991i	$0.606070\pi$
$228$	0	0
$229$	−26.2487	−1.73456	−0.867282	−	0.497817i	$-0.834135\pi$
$229$	−26.2487	−1.73456	−0.867282	+	0.497817i	$0.834135\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	−1.73205	−0.113715
$233$	−3.85641	−0.252642	−0.126321	−	0.991989i	$-0.540317\pi$
$233$	−3.85641	−0.252642	−0.126321	+	0.991989i	$0.540317\pi$
$234$	−10.3923	−0.679366
$235$	4.00000	0.260931
$236$	8.00000	0.520756
$237$	0	0
$238$	0	0
$239$	−24.7846	−1.60318	−0.801592	−	0.597872i	$-0.796013\pi$
$239$	−24.7846	−1.60318	−0.801592	+	0.597872i	$0.796013\pi$
$240$	0	0
$241$	−0.143594	−0.00924967	−0.00462484	−	0.999989i	$-0.501472\pi$
$241$	−0.143594	−0.00924967	−0.00462484	+	0.999989i	$0.501472\pi$
$242$	32.6603	2.09948
$243$	0	0
$244$	1.46410	0.0937295
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	−15.4641	−0.981971
$249$	0	0
$250$	1.73205	0.109545
$251$	12.9282	0.816021	0.408010	−	0.912977i	$-0.366223\pi$
$251$	12.9282	0.816021	0.408010	+	0.912977i	$0.366223\pi$
$252$	0	0
$253$	−37.8564	−2.38001
$254$	14.7846	0.927669
$255$	0	0
$256$	19.0000	1.18750
$257$	−19.8564	−1.23861	−0.619304	−	0.785151i	$-0.712585\pi$
$257$	−19.8564	−1.23861	−0.619304	+	0.785151i	$0.712585\pi$
$258$	0	0
$259$	0	0
$260$	2.00000	0.124035
$261$	−3.00000	−0.185695
$262$	8.53590	0.527350
$263$	−13.3205	−0.821378	−0.410689	−	0.911776i	$-0.634712\pi$
$263$	−13.3205	−0.821378	−0.410689	+	0.911776i	$0.634712\pi$
$264$	0	0
$265$	4.92820	0.302737
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	28.3923	1.73111	0.865555	−	0.500814i	$-0.166966\pi$
$269$	28.3923	1.73111	0.865555	+	0.500814i	$0.166966\pi$
$270$	0	0
$271$	−0.143594	−0.00872269	−0.00436134	−	0.999990i	$-0.501388\pi$
$271$	−0.143594	−0.00872269	−0.00436134	+	0.999990i	$0.501388\pi$
$272$	−17.3205	−1.05021
$273$	0	0
$274$	−32.7846	−1.98059
$275$	−5.46410	−0.329498
$276$	0	0
$277$	−27.8564	−1.67373	−0.836865	−	0.547410i	$-0.815614\pi$
$277$	−27.8564	−1.67373	−0.836865	+	0.547410i	$0.815614\pi$
$278$	−30.9282	−1.85495
$279$	−26.7846	−1.60355
$280$	0	0
$281$	27.8564	1.66177	0.830887	−	0.556441i	$-0.187834\pi$
$281$	27.8564	1.66177	0.830887	+	0.556441i	$0.187834\pi$
$282$	0	0
$283$	6.92820	0.411839	0.205919	−	0.978569i	$-0.433982\pi$
$283$	6.92820	0.411839	0.205919	+	0.978569i	$0.433982\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−18.9282	−1.11925
$287$	0	0
$288$	15.5885	0.918559
$289$	−5.00000	−0.294118
$290$	1.73205	0.101710
$291$	0	0
$292$	−14.3923	−0.842246
$293$	−11.4641	−0.669740	−0.334870	−	0.942264i	$-0.608692\pi$
$293$	−11.4641	−0.669740	−0.334870	+	0.942264i	$0.608692\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	0	0
$298$	41.3205	2.39363
$299$	13.8564	0.801337
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	10.0000	0.573539
$305$	1.46410	0.0838342
$306$	−18.0000	−1.02899
$307$	6.92820	0.395413	0.197707	−	0.980261i	$-0.436651\pi$
$307$	6.92820	0.395413	0.197707	+	0.980261i	$0.436651\pi$
$308$	0	0
$309$	0	0
$310$	15.4641	0.878302
$311$	30.7846	1.74564	0.872818	−	0.488047i	$-0.162290\pi$
$311$	30.7846	1.74564	0.872818	+	0.488047i	$0.162290\pi$
$312$	0	0
$313$	3.07180	0.173628	0.0868141	−	0.996225i	$-0.472331\pi$
$313$	3.07180	0.173628	0.0868141	+	0.996225i	$0.472331\pi$
$314$	19.8564	1.12056
$315$	0	0
$316$	−1.46410	−0.0823622
$317$	−22.9282	−1.28778	−0.643888	−	0.765120i	$-0.722680\pi$
$317$	−22.9282	−1.28778	−0.643888	+	0.765120i	$0.722680\pi$
$318$	0	0
$319$	−5.46410	−0.305931
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	−6.92820	−0.385496
$324$	9.00000	0.500000
$325$	2.00000	0.110940
$326$	30.0000	1.66155
$327$	0	0
$328$	9.46410	0.522568
$329$	0	0
$330$	0	0
$331$	32.3923	1.78044	0.890221	−	0.455529i	$-0.150550\pi$
$331$	32.3923	1.78044	0.890221	+	0.455529i	$0.150550\pi$
$332$	1.07180	0.0588225
$333$	0	0
$334$	−10.1436	−0.555033
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−34.9282	−1.90266	−0.951330	−	0.308173i	$-0.900282\pi$
$337$	−34.9282	−1.90266	−0.951330	+	0.308173i	$0.900282\pi$
$338$	−15.5885	−0.847900
$339$	0	0
$340$	3.46410	0.187867
$341$	−48.7846	−2.64183
$342$	10.3923	0.561951
$343$	0	0
$344$	−6.00000	−0.323498
$345$	0	0
$346$	1.60770	0.0864302
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	16.9282	0.906146	0.453073	−	0.891473i	$-0.350328\pi$
$349$	16.9282	0.906146	0.453073	+	0.891473i	$0.350328\pi$
$350$	0	0
$351$	0	0
$352$	28.3923	1.51331
$353$	35.8564	1.90844	0.954222	−	0.299099i	$-0.0966862\pi$
$353$	35.8564	1.90844	0.954222	+	0.299099i	$0.0966862\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	17.4641	0.925596
$357$	0	0
$358$	−12.0000	−0.634220
$359$	16.3923	0.865153	0.432576	−	0.901597i	$-0.357605\pi$
$359$	16.3923	0.865153	0.432576	+	0.901597i	$0.357605\pi$
$360$	5.19615	0.273861
$361$	−15.0000	−0.789474
$362$	22.3923	1.17691
$363$	0	0
$364$	0	0
$365$	−14.3923	−0.753328
$366$	0	0
$367$	18.9282	0.988044	0.494022	−	0.869449i	$-0.335526\pi$
$367$	18.9282	0.988044	0.494022	+	0.869449i	$0.335526\pi$
$368$	−34.6410	−1.80579
$369$	16.3923	0.853349
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.9282	0.876509	0.438255	−	0.898851i	$-0.355597\pi$
$373$	16.9282	0.876509	0.438255	+	0.898851i	$0.355597\pi$
$374$	−32.7846	−1.69525
$375$	0	0
$376$	−6.92820	−0.357295
$377$	2.00000	0.103005
$378$	0	0
$379$	5.46410	0.280672	0.140336	−	0.990104i	$-0.455182\pi$
$379$	5.46410	0.280672	0.140336	+	0.990104i	$0.455182\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	14.5359	0.743721
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	27.7128	1.41055
$387$	−10.3923	−0.528271
$388$	11.4641	0.582002
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	0	0
$394$	−32.5359	−1.63913
$395$	−1.46410	−0.0736669
$396$	16.3923	0.823744
$397$	0.928203	0.0465852	0.0232926	−	0.999729i	$-0.492585\pi$
$397$	0.928203	0.0465852	0.0232926	+	0.999729i	$0.492585\pi$
$398$	−6.92820	−0.347279
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	17.8564	0.889491
$404$	13.4641	0.669864
$405$	9.00000	0.447214
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.2487	−0.704553	−0.352277	−	0.935896i	$-0.614592\pi$
$409$	−14.2487	−0.704553	−0.352277	+	0.935896i	$0.614592\pi$
$410$	−9.46410	−0.467399
$411$	0	0
$412$	16.0000	0.788263
$413$	0	0
$414$	−36.0000	−1.76930
$415$	1.07180	0.0526124
$416$	−10.3923	−0.509525
$417$	0	0
$418$	18.9282	0.925809
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	18.7846	0.915506	0.457753	−	0.889079i	$-0.348654\pi$
$421$	18.7846	0.915506	0.457753	+	0.889079i	$0.348654\pi$
$422$	−0.679492	−0.0330771
$423$	−12.0000	−0.583460
$424$	−8.53590	−0.414540
$425$	3.46410	0.168034
$426$	0	0
$427$	0	0
$428$	10.9282	0.528235
$429$	0	0
$430$	6.00000	0.289346
$431$	5.85641	0.282093	0.141047	−	0.990003i	$-0.454953\pi$
$431$	5.85641	0.282093	0.141047	+	0.990003i	$0.454953\pi$
$432$	0	0
$433$	−1.32051	−0.0634596	−0.0317298	−	0.999496i	$-0.510102\pi$
$433$	−1.32051	−0.0634596	−0.0317298	+	0.999496i	$0.510102\pi$
$434$	0	0
$435$	0	0
$436$	10.0000	0.478913
$437$	−13.8564	−0.662842
$438$	0	0
$439$	−33.8564	−1.61588	−0.807939	−	0.589266i	$-0.799417\pi$
$439$	−33.8564	−1.61588	−0.807939	+	0.589266i	$0.799417\pi$
$440$	9.46410	0.451183
$441$	0	0
$442$	12.0000	0.570782
$443$	−5.32051	−0.252785	−0.126392	−	0.991980i	$-0.540340\pi$
$443$	−5.32051	−0.252785	−0.126392	+	0.991980i	$0.540340\pi$
$444$	0	0
$445$	17.4641	0.827878
$446$	−8.78461	−0.415963
$447$	0	0
$448$	0	0
$449$	7.85641	0.370767	0.185383	−	0.982666i	$-0.440647\pi$
$449$	7.85641	0.370767	0.185383	+	0.982666i	$0.440647\pi$
$450$	−5.19615	−0.244949
$451$	29.8564	1.40588
$452$	−16.0000	−0.752577
$453$	0	0
$454$	−17.0718	−0.801219
$455$	0	0
$456$	0	0
$457$	18.7846	0.878707	0.439353	−	0.898314i	$-0.355208\pi$
$457$	18.7846	0.878707	0.439353	+	0.898314i	$0.355208\pi$
$458$	−45.4641	−2.12440
$459$	0	0
$460$	6.92820	0.323029
$461$	−15.3205	−0.713547	−0.356774	−	0.934191i	$-0.616123\pi$
$461$	−15.3205	−0.713547	−0.356774	+	0.934191i	$0.616123\pi$
$462$	0	0
$463$	−20.7846	−0.965943	−0.482971	−	0.875636i	$-0.660442\pi$
$463$	−20.7846	−0.965943	−0.482971	+	0.875636i	$0.660442\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	−6.67949	−0.309421
$467$	13.8564	0.641198	0.320599	−	0.947215i	$-0.396116\pi$
$467$	13.8564	0.641198	0.320599	+	0.947215i	$0.396116\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	6.92820	0.319574
$471$	0	0
$472$	−13.8564	−0.637793
$473$	−18.9282	−0.870320
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	−14.7846	−0.676941
$478$	−42.9282	−1.96349
$479$	−23.8564	−1.09003	−0.545014	−	0.838427i	$-0.683475\pi$
$479$	−23.8564	−1.09003	−0.545014	+	0.838427i	$0.683475\pi$
$480$	0	0
$481$	0	0
$482$	−0.248711	−0.0113285
$483$	0	0
$484$	18.8564	0.857109
$485$	11.4641	0.520558
$486$	0	0
$487$	−20.7846	−0.941841	−0.470920	−	0.882176i	$-0.656078\pi$
$487$	−20.7846	−0.941841	−0.470920	+	0.882176i	$0.656078\pi$
$488$	−2.53590	−0.114795
$489$	0	0
$490$	0	0
$491$	−14.5359	−0.655996	−0.327998	−	0.944678i	$-0.606374\pi$
$491$	−14.5359	−0.655996	−0.327998	+	0.944678i	$0.606374\pi$
$492$	0	0
$493$	3.46410	0.156015
$494$	−6.92820	−0.311715
$495$	16.3923	0.736779
$496$	−44.6410	−2.00444
$497$	0	0
$498$	0	0
$499$	22.9282	1.02641	0.513204	−	0.858267i	$-0.328459\pi$
$499$	22.9282	1.02641	0.513204	+	0.858267i	$0.328459\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	22.3923	0.999417
$503$	29.8564	1.33123	0.665616	−	0.746295i	$-0.268169\pi$
$503$	29.8564	1.33123	0.665616	+	0.746295i	$0.268169\pi$
$504$	0	0
$505$	13.4641	0.599145
$506$	−65.5692	−2.91491
$507$	0	0
$508$	8.53590	0.378719
$509$	−7.85641	−0.348229	−0.174115	−	0.984725i	$-0.555706\pi$
$509$	−7.85641	−0.348229	−0.174115	+	0.984725i	$0.555706\pi$
$510$	0	0
$511$	0	0
$512$	8.66025	0.382733
$513$	0	0
$514$	−34.3923	−1.51698
$515$	16.0000	0.705044
$516$	0	0
$517$	−21.8564	−0.961244
$518$	0	0
$519$	0	0
$520$	−3.46410	−0.151911
$521$	14.7846	0.647726	0.323863	−	0.946104i	$-0.395018\pi$
$521$	14.7846	0.647726	0.323863	+	0.946104i	$0.395018\pi$
$522$	−5.19615	−0.227429
$523$	3.21539	0.140599	0.0702996	−	0.997526i	$-0.477604\pi$
$523$	3.21539	0.140599	0.0702996	+	0.997526i	$0.477604\pi$
$524$	4.92820	0.215290
$525$	0	0
$526$	−23.0718	−1.00598
$527$	30.9282	1.34725
$528$	0	0
$529$	25.0000	1.08696
$530$	8.53590	0.370776
$531$	−24.0000	−1.04151
$532$	0	0
$533$	−10.9282	−0.473353
$534$	0	0
$535$	10.9282	0.472467
$536$	6.92820	0.299253
$537$	0	0
$538$	49.1769	2.12017
$539$	0	0
$540$	0	0
$541$	−32.6410	−1.40335	−0.701673	−	0.712499i	$-0.747563\pi$
$541$	−32.6410	−1.40335	−0.701673	+	0.712499i	$0.747563\pi$
$542$	−0.248711	−0.0106831
$543$	0	0
$544$	−18.0000	−0.771744
$545$	10.0000	0.428353
$546$	0	0
$547$	−0.784610	−0.0335475	−0.0167737	−	0.999859i	$-0.505339\pi$
$547$	−0.784610	−0.0335475	−0.0167737	+	0.999859i	$0.505339\pi$
$548$	−18.9282	−0.808573
$549$	−4.39230	−0.187459
$550$	−9.46410	−0.403551
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	0	0
$554$	−48.2487	−2.04989
$555$	0	0
$556$	−17.8564	−0.757280
$557$	13.7128	0.581031	0.290515	−	0.956870i	$-0.406173\pi$
$557$	13.7128	0.581031	0.290515	+	0.956870i	$0.406173\pi$
$558$	−46.3923	−1.96394
$559$	6.92820	0.293032
$560$	0	0
$561$	0	0
$562$	48.2487	2.03525
$563$	−1.85641	−0.0782382	−0.0391191	−	0.999235i	$-0.512455\pi$
$563$	−1.85641	−0.0782382	−0.0391191	+	0.999235i	$0.512455\pi$
$564$	0	0
$565$	−16.0000	−0.673125
$566$	12.0000	0.504398
$567$	0	0
$568$	13.8564	0.581402
$569$	−34.7846	−1.45825	−0.729123	−	0.684382i	$-0.760072\pi$
$569$	−34.7846	−1.45825	−0.729123	+	0.684382i	$0.760072\pi$
$570$	0	0
$571$	1.85641	0.0776882	0.0388441	−	0.999245i	$-0.487632\pi$
$571$	1.85641	0.0776882	0.0388441	+	0.999245i	$0.487632\pi$
$572$	−10.9282	−0.456931
$573$	0	0
$574$	0	0
$575$	6.92820	0.288926
$576$	−3.00000	−0.125000
$577$	31.4641	1.30987	0.654934	−	0.755686i	$-0.272696\pi$
$577$	31.4641	1.30987	0.654934	+	0.755686i	$0.272696\pi$
$578$	−8.66025	−0.360219
$579$	0	0
$580$	1.00000	0.0415227
$581$	0	0
$582$	0	0
$583$	−26.9282	−1.11525
$584$	24.9282	1.03154
$585$	−6.00000	−0.248069
$586$	−19.8564	−0.820261
$587$	6.92820	0.285958	0.142979	−	0.989726i	$-0.454332\pi$
$587$	6.92820	0.285958	0.142979	+	0.989726i	$0.454332\pi$
$588$	0	0
$589$	−17.8564	−0.735760
$590$	13.8564	0.570459
$591$	0	0
$592$	0	0
$593$	12.9282	0.530898	0.265449	−	0.964125i	$-0.414480\pi$
$593$	12.9282	0.530898	0.265449	+	0.964125i	$0.414480\pi$
$594$	0	0
$595$	0	0
$596$	23.8564	0.977196
$597$	0	0
$598$	24.0000	0.981433
$599$	28.3923	1.16008	0.580039	−	0.814589i	$-0.303037\pi$
$599$	28.3923	1.16008	0.580039	+	0.814589i	$0.303037\pi$
$600$	0	0
$601$	−6.53590	−0.266605	−0.133302	−	0.991075i	$-0.542558\pi$
$601$	−6.53590	−0.266605	−0.133302	+	0.991075i	$0.542558\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	18.8564	0.766622
$606$	0	0
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	10.3923	0.421464
$609$	0	0
$610$	2.53590	0.102676
$611$	8.00000	0.323645
$612$	−10.3923	−0.420084
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	18.9282	0.762021	0.381010	−	0.924571i	$-0.375576\pi$
$617$	18.9282	0.762021	0.381010	+	0.924571i	$0.375576\pi$
$618$	0	0
$619$	−27.0718	−1.08811	−0.544054	−	0.839050i	$-0.683111\pi$
$619$	−27.0718	−1.08811	−0.544054	+	0.839050i	$0.683111\pi$
$620$	8.92820	0.358565
$621$	0	0
$622$	53.3205	2.13796
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	5.32051	0.212650
$627$	0	0
$628$	11.4641	0.457467
$629$	0	0
$630$	0	0
$631$	18.9282	0.753520	0.376760	−	0.926311i	$-0.377038\pi$
$631$	18.9282	0.753520	0.376760	+	0.926311i	$0.377038\pi$
$632$	2.53590	0.100873
$633$	0	0
$634$	−39.7128	−1.57720
$635$	8.53590	0.338737
$636$	0	0
$637$	0	0
$638$	−9.46410	−0.374687
$639$	24.0000	0.949425
$640$	12.1244	0.479257
$641$	−14.7846	−0.583957	−0.291978	−	0.956425i	$-0.594314\pi$
$641$	−14.7846	−0.583957	−0.291978	+	0.956425i	$0.594314\pi$
$642$	0	0
$643$	−38.9282	−1.53518	−0.767589	−	0.640942i	$-0.778544\pi$
$643$	−38.9282	−1.53518	−0.767589	+	0.640942i	$0.778544\pi$
$644$	0	0
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−8.78461	−0.345359	−0.172679	−	0.984978i	$-0.555242\pi$
$647$	−8.78461	−0.345359	−0.172679	+	0.984978i	$0.555242\pi$
$648$	−15.5885	−0.612372
$649$	−43.7128	−1.71588
$650$	3.46410	0.135873
$651$	0	0
$652$	17.3205	0.678323
$653$	−27.7128	−1.08449	−0.542243	−	0.840222i	$-0.682425\pi$
$653$	−27.7128	−1.08449	−0.542243	+	0.840222i	$0.682425\pi$
$654$	0	0
$655$	4.92820	0.192561
$656$	27.3205	1.06669
$657$	43.1769	1.68449
$658$	0	0
$659$	34.5359	1.34533	0.672664	−	0.739948i	$-0.265150\pi$
$659$	34.5359	1.34533	0.672664	+	0.739948i	$0.265150\pi$
$660$	0	0
$661$	−12.9282	−0.502849	−0.251424	−	0.967877i	$-0.580899\pi$
$661$	−12.9282	−0.502849	−0.251424	+	0.967877i	$0.580899\pi$
$662$	56.1051	2.18059
$663$	0	0
$664$	−1.85641	−0.0720425
$665$	0	0
$666$	0	0
$667$	6.92820	0.268261
$668$	−5.85641	−0.226591
$669$	0	0
$670$	−6.92820	−0.267660
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−42.7846	−1.64923	−0.824613	−	0.565698i	$-0.808607\pi$
$673$	−42.7846	−1.64923	−0.824613	+	0.565698i	$0.808607\pi$
$674$	−60.4974	−2.33027
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−32.2487	−1.23942	−0.619709	−	0.784831i	$-0.712750\pi$
$677$	−32.2487	−1.23942	−0.619709	+	0.784831i	$0.712750\pi$
$678$	0	0
$679$	0	0
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−84.4974	−3.23557
$683$	18.9282	0.724268	0.362134	−	0.932126i	$-0.382048\pi$
$683$	18.9282	0.724268	0.362134	+	0.932126i	$0.382048\pi$
$684$	6.00000	0.229416
$685$	−18.9282	−0.723209
$686$	0	0
$687$	0	0
$688$	−17.3205	−0.660338
$689$	9.85641	0.375499
$690$	0	0
$691$	−23.7128	−0.902078	−0.451039	−	0.892504i	$-0.648946\pi$
$691$	−23.7128	−0.902078	−0.451039	+	0.892504i	$0.648946\pi$
$692$	0.928203	0.0352850
$693$	0	0
$694$	6.92820	0.262991
$695$	−17.8564	−0.677332
$696$	0	0
$697$	−18.9282	−0.716957
$698$	29.3205	1.10980
$699$	0	0
$700$	0	0
$701$	−35.8564	−1.35428	−0.677139	−	0.735855i	$-0.736780\pi$
$701$	−35.8564	−1.35428	−0.677139	+	0.735855i	$0.736780\pi$
$702$	0	0
$703$	0	0
$704$	−5.46410	−0.205936
$705$	0	0
$706$	62.1051	2.33736
$707$	0	0
$708$	0	0
$709$	−25.7128	−0.965665	−0.482832	−	0.875713i	$-0.660392\pi$
$709$	−25.7128	−0.965665	−0.482832	+	0.875713i	$0.660392\pi$
$710$	−13.8564	−0.520022
$711$	4.39230	0.164724
$712$	−30.2487	−1.13362
$713$	61.8564	2.31654
$714$	0	0
$715$	−10.9282	−0.408692
$716$	−6.92820	−0.258919
$717$	0	0
$718$	28.3923	1.05959
$719$	−15.7128	−0.585989	−0.292995	−	0.956114i	$-0.594652\pi$
$719$	−15.7128	−0.585989	−0.292995	+	0.956114i	$0.594652\pi$
$720$	15.0000	0.559017
$721$	0	0
$722$	−25.9808	−0.966904
$723$	0	0
$724$	12.9282	0.480473
$725$	1.00000	0.0371391
$726$	0	0
$727$	17.8564	0.662257	0.331129	−	0.943586i	$-0.392571\pi$
$727$	17.8564	0.662257	0.331129	+	0.943586i	$0.392571\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−24.9282	−0.922634
$731$	12.0000	0.443836
$732$	0	0
$733$	35.4641	1.30990	0.654948	−	0.755674i	$-0.272690\pi$
$733$	35.4641	1.30990	0.654948	+	0.755674i	$0.272690\pi$
$734$	32.7846	1.21010
$735$	0	0
$736$	−36.0000	−1.32698
$737$	21.8564	0.805091
$738$	28.3923	1.04514
$739$	31.3205	1.15214	0.576072	−	0.817399i	$-0.304585\pi$
$739$	31.3205	1.15214	0.576072	+	0.817399i	$0.304585\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.5359	1.48712	0.743559	−	0.668670i	$-0.233136\pi$
$743$	40.5359	1.48712	0.743559	+	0.668670i	$0.233136\pi$
$744$	0	0
$745$	23.8564	0.874031
$746$	29.3205	1.07350
$747$	−3.21539	−0.117645
$748$	−18.9282	−0.692084
$749$	0	0
$750$	0	0
$751$	16.3923	0.598164	0.299082	−	0.954227i	$-0.403320\pi$
$751$	16.3923	0.598164	0.299082	+	0.954227i	$0.403320\pi$
$752$	−20.0000	−0.729325
$753$	0	0
$754$	3.46410	0.126155
$755$	0	0
$756$	0	0
$757$	−27.7128	−1.00724	−0.503620	−	0.863925i	$-0.667999\pi$
$757$	−27.7128	−1.00724	−0.503620	+	0.863925i	$0.667999\pi$
$758$	9.46410	0.343752
$759$	0	0
$760$	3.46410	0.125656
$761$	50.7846	1.84094	0.920470	−	0.390812i	$-0.127806\pi$
$761$	50.7846	1.84094	0.920470	+	0.390812i	$0.127806\pi$
$762$	0	0
$763$	0	0
$764$	8.39230	0.303623
$765$	−10.3923	−0.375735
$766$	27.7128	1.00130
$767$	16.0000	0.577727
$768$	0	0
$769$	−16.6795	−0.601478	−0.300739	−	0.953707i	$-0.597233\pi$
$769$	−16.6795	−0.601478	−0.300739	+	0.953707i	$0.597233\pi$
$770$	0	0
$771$	0	0
$772$	16.0000	0.575853
$773$	8.53590	0.307015	0.153507	−	0.988147i	$-0.450943\pi$
$773$	8.53590	0.307015	0.153507	+	0.988147i	$0.450943\pi$
$774$	−18.0000	−0.646997
$775$	8.92820	0.320711
$776$	−19.8564	−0.712803
$777$	0	0
$778$	3.46410	0.124194
$779$	10.9282	0.391544
$780$	0	0
$781$	43.7128	1.56417
$782$	41.5692	1.48651
$783$	0	0
$784$	0	0
$785$	11.4641	0.409171
$786$	0	0
$787$	28.7846	1.02606	0.513030	−	0.858371i	$-0.328523\pi$
$787$	28.7846	1.02606	0.513030	+	0.858371i	$0.328523\pi$
$788$	−18.7846	−0.669174
$789$	0	0
$790$	−2.53590	−0.0902232
$791$	0	0
$792$	−28.3923	−1.00888
$793$	2.92820	0.103984
$794$	1.60770	0.0570550
$795$	0	0
$796$	−4.00000	−0.141776
$797$	45.3205	1.60533	0.802667	−	0.596427i	$-0.203413\pi$
$797$	45.3205	1.60533	0.802667	+	0.596427i	$0.203413\pi$
$798$	0	0
$799$	13.8564	0.490204
$800$	−5.19615	−0.183712
$801$	−52.3923	−1.85119
$802$	−10.3923	−0.366965
$803$	78.6410	2.77518
$804$	0	0
$805$	0	0
$806$	30.9282	1.08940
$807$	0	0
$808$	−23.3205	−0.820413
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	15.5885	0.547723
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17.3205	0.606711
$816$	0	0
$817$	−6.92820	−0.242387
$818$	−24.6795	−0.862898
$819$	0	0
$820$	−5.46410	−0.190815
$821$	−35.5692	−1.24137	−0.620687	−	0.784058i	$-0.713146\pi$
$821$	−35.5692	−1.24137	−0.620687	+	0.784058i	$0.713146\pi$
$822$	0	0
$823$	27.1769	0.947328	0.473664	−	0.880706i	$-0.342931\pi$
$823$	27.1769	0.947328	0.473664	+	0.880706i	$0.342931\pi$
$824$	−27.7128	−0.965422
$825$	0	0
$826$	0	0
$827$	36.5359	1.27048	0.635239	−	0.772316i	$-0.280902\pi$
$827$	36.5359	1.27048	0.635239	+	0.772316i	$0.280902\pi$
$828$	−20.7846	−0.722315
$829$	−32.3923	−1.12503	−0.562516	−	0.826787i	$-0.690166\pi$
$829$	−32.3923	−1.12503	−0.562516	+	0.826787i	$0.690166\pi$
$830$	1.85641	0.0644368
$831$	0	0
$832$	2.00000	0.0693375
$833$	0	0
$834$	0	0
$835$	−5.85641	−0.202669
$836$	10.9282	0.377960
$837$	0	0
$838$	−41.5692	−1.43598
$839$	26.7846	0.924707	0.462354	−	0.886696i	$-0.347005\pi$
$839$	26.7846	0.924707	0.462354	+	0.886696i	$0.347005\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	32.5359	1.12126
$843$	0	0
$844$	−0.392305	−0.0135037
$845$	−9.00000	−0.309609
$846$	−20.7846	−0.714590
$847$	0	0
$848$	−24.6410	−0.846176
$849$	0	0
$850$	6.00000	0.205798
$851$	0	0
$852$	0	0
$853$	−57.3205	−1.96262	−0.981308	−	0.192442i	$-0.938359\pi$
$853$	−57.3205	−1.96262	−0.981308	+	0.192442i	$0.938359\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	−18.9282	−0.646953
$857$	−17.7128	−0.605058	−0.302529	−	0.953140i	$-0.597831\pi$
$857$	−17.7128	−0.605058	−0.302529	+	0.953140i	$0.597831\pi$
$858$	0	0
$859$	9.21539	0.314425	0.157213	−	0.987565i	$-0.449749\pi$
$859$	9.21539	0.314425	0.157213	+	0.987565i	$0.449749\pi$
$860$	3.46410	0.118125
$861$	0	0
$862$	10.1436	0.345492
$863$	43.7128	1.48800	0.744001	−	0.668179i	$-0.232926\pi$
$863$	43.7128	1.48800	0.744001	+	0.668179i	$0.232926\pi$
$864$	0	0
$865$	0.928203	0.0315599
$866$	−2.28719	−0.0777218
$867$	0	0
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−17.3205	−0.586546
$873$	−34.3923	−1.16400
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	−0.928203	−0.0313432	−0.0156716	−	0.999877i	$-0.504989\pi$
$877$	−0.928203	−0.0313432	−0.0156716	+	0.999877i	$0.504989\pi$
$878$	−58.6410	−1.97904
$879$	0	0
$880$	27.3205	0.920974
$881$	−25.1769	−0.848232	−0.424116	−	0.905608i	$-0.639415\pi$
$881$	−25.1769	−0.848232	−0.424116	+	0.905608i	$0.639415\pi$
$882$	0	0
$883$	−24.7846	−0.834069	−0.417034	−	0.908891i	$-0.636930\pi$
$883$	−24.7846	−0.834069	−0.417034	+	0.908891i	$0.636930\pi$
$884$	6.92820	0.233021
$885$	0	0
$886$	−9.21539	−0.309597
$887$	−20.7846	−0.697879	−0.348939	−	0.937145i	$-0.613458\pi$
$887$	−20.7846	−0.697879	−0.348939	+	0.937145i	$0.613458\pi$
$888$	0	0
$889$	0	0
$890$	30.2487	1.01394
$891$	−49.1769	−1.64749
$892$	−5.07180	−0.169816
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−6.92820	−0.231584
$896$	0	0
$897$	0	0
$898$	13.6077	0.454095
$899$	8.92820	0.297772
$900$	−3.00000	−0.100000
$901$	17.0718	0.568744
$902$	51.7128	1.72185
$903$	0	0
$904$	27.7128	0.921714
$905$	12.9282	0.429748
$906$	0	0
$907$	−21.6077	−0.717472	−0.358736	−	0.933439i	$-0.616792\pi$
$907$	−21.6077	−0.717472	−0.358736	+	0.933439i	$0.616792\pi$
$908$	−9.85641	−0.327096
$909$	−40.3923	−1.33973
$910$	0	0
$911$	47.3205	1.56780	0.783899	−	0.620888i	$-0.213228\pi$
$911$	47.3205	1.56780	0.783899	+	0.620888i	$0.213228\pi$
$912$	0	0
$913$	−5.85641	−0.193819
$914$	32.5359	1.07619
$915$	0	0
$916$	−26.2487	−0.867282
$917$	0	0
$918$	0	0
$919$	10.9282	0.360488	0.180244	−	0.983622i	$-0.442311\pi$
$919$	10.9282	0.360488	0.180244	+	0.983622i	$0.442311\pi$
$920$	−12.0000	−0.395628
$921$	0	0
$922$	−26.5359	−0.873913
$923$	−16.0000	−0.526646
$924$	0	0
$925$	0	0
$926$	−36.0000	−1.18303
$927$	−48.0000	−1.57653
$928$	−5.19615	−0.170572
$929$	45.7128	1.49979	0.749894	−	0.661558i	$-0.230104\pi$
$929$	45.7128	1.49979	0.749894	+	0.661558i	$0.230104\pi$
$930$	0	0
$931$	0	0
$932$	−3.85641	−0.126321
$933$	0	0
$934$	24.0000	0.785304
$935$	−18.9282	−0.619018
$936$	10.3923	0.339683
$937$	40.6410	1.32768	0.663842	−	0.747873i	$-0.268925\pi$
$937$	40.6410	1.32768	0.663842	+	0.747873i	$0.268925\pi$
$938$	0	0
$939$	0	0
$940$	4.00000	0.130466
$941$	32.9282	1.07343	0.536714	−	0.843764i	$-0.319665\pi$
$941$	32.9282	1.07343	0.536714	+	0.843764i	$0.319665\pi$
$942$	0	0
$943$	−37.8564	−1.23277
$944$	−40.0000	−1.30189
$945$	0	0
$946$	−32.7846	−1.06592
$947$	33.6077	1.09210	0.546052	−	0.837751i	$-0.316130\pi$
$947$	33.6077	1.09210	0.546052	+	0.837751i	$0.316130\pi$
$948$	0	0
$949$	−28.7846	−0.934388
$950$	−3.46410	−0.112390
$951$	0	0
$952$	0	0
$953$	1.21539	0.0393704	0.0196852	−	0.999806i	$-0.493734\pi$
$953$	1.21539	0.0393704	0.0196852	+	0.999806i	$0.493734\pi$
$954$	−25.6077	−0.829080
$955$	8.39230	0.271569
$956$	−24.7846	−0.801592
$957$	0	0
$958$	−41.3205	−1.33501
$959$	0	0
$960$	0	0
$961$	48.7128	1.57138
$962$	0	0
$963$	−32.7846	−1.05647
$964$	−0.143594	−0.00462484
$965$	16.0000	0.515058
$966$	0	0
$967$	38.3923	1.23461	0.617307	−	0.786723i	$-0.288224\pi$
$967$	38.3923	1.23461	0.617307	+	0.786723i	$0.288224\pi$
$968$	−32.6603	−1.04974
$969$	0	0
$970$	19.8564	0.637551
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	0	0
$974$	−36.0000	−1.15351
$975$	0	0
$976$	−7.32051	−0.234324
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−95.4256	−3.04982
$980$	0	0
$981$	−30.0000	−0.957826
$982$	−25.1769	−0.803428
$983$	−12.7846	−0.407766	−0.203883	−	0.978995i	$-0.565356\pi$
$983$	−12.7846	−0.407766	−0.203883	+	0.978995i	$0.565356\pi$
$984$	0	0
$985$	−18.7846	−0.598527
$986$	6.00000	0.191079
$987$	0	0
$988$	−4.00000	−0.127257
$989$	24.0000	0.763156
$990$	28.3923	0.902367
$991$	35.7128	1.13445	0.567227	−	0.823561i	$-0.308016\pi$
$991$	35.7128	1.13445	0.567227	+	0.823561i	$0.308016\pi$
$992$	−46.3923	−1.47296
$993$	0	0
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	−50.1051	−1.58685	−0.793423	−	0.608671i	$-0.791703\pi$
$997$	−50.1051	−1.58685	−0.793423	+	0.608671i	$0.791703\pi$
$998$	39.7128	1.25709
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.g.1.2		2
7.6	odd	2		1015.2.a.d.1.2	✓	2
21.20	even	2		9135.2.a.r.1.1		2
35.34	odd	2		5075.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1015.2.a.d.1.2	✓	2	7.6	odd	2
5075.2.a.n.1.1		2	35.34	odd	2
7105.2.a.g.1.2		2	1.1	even	1	trivial
9135.2.a.r.1.1		2	21.20	even	2

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:	\( N \)	\(=\)	\( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7105.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{8} -3.00000 q^{9} +1.73205 q^{10} -5.46410 q^{11} +2.00000 q^{13} -5.00000 q^{16} +3.46410 q^{17} -5.19615 q^{18} -2.00000 q^{19} +1.00000 q^{20} -9.46410 q^{22} +6.92820 q^{23} +1.00000 q^{25} +3.46410 q^{26} +1.00000 q^{29} +8.92820 q^{31} -5.19615 q^{32} +6.00000 q^{34} -3.00000 q^{36} -3.46410 q^{38} -1.73205 q^{40} -5.46410 q^{41} +3.46410 q^{43} -5.46410 q^{44} -3.00000 q^{45} +12.0000 q^{46} +4.00000 q^{47} +1.73205 q^{50} +2.00000 q^{52} +4.92820 q^{53} -5.46410 q^{55} +1.73205 q^{58} +8.00000 q^{59} +1.46410 q^{61} +15.4641 q^{62} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} +3.46410 q^{68} -8.00000 q^{71} +5.19615 q^{72} -14.3923 q^{73} -2.00000 q^{76} -1.46410 q^{79} -5.00000 q^{80} +9.00000 q^{81} -9.46410 q^{82} +1.07180 q^{83} +3.46410 q^{85} +6.00000 q^{86} +9.46410 q^{88} +17.4641 q^{89} -5.19615 q^{90} +6.92820 q^{92} +6.92820 q^{94} -2.00000 q^{95} +11.4641 q^{97} +16.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{13} - 10 q^{16} - 4 q^{19} + 2 q^{20} - 12 q^{22} + 2 q^{25} + 2 q^{29} + 4 q^{31} + 12 q^{34} - 6 q^{36} - 4 q^{41} - 4 q^{44} - 6 q^{45} + 24 q^{46}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^13 - 10 * q^16 - 4 * q^19 + 2 * q^20 - 12 * q^22 + 2 * q^25 + 2 * q^29 + 4 * q^31 + 12 * q^34 - 6 * q^36 - 4 * q^41 - 4 * q^44 - 6 * q^45 + 24 * q^46 + 8 * q^47 + 4 * q^52 - 4 * q^53 - 4 * q^55 + 16 * q^59 - 4 * q^61 + 24 * q^62 + 2 * q^64 + 4 * q^65 - 8 * q^67 - 16 * q^71 - 8 * q^73 - 4 * q^76 + 4 * q^79 - 10 * q^80 + 18 * q^81 - 12 * q^82 + 16 * q^83 + 12 * q^86 + 12 * q^88 + 28 * q^89 - 4 * q^95 + 16 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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