LMFDB - Embedded newform 2816.2.c.w.1409.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2816,2,Mod(1409,2816)] 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("2816.1409"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,4,0,-6,0,0,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4858732092$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1409.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	2816.1409
Dual form			2816.2.c.w.1409.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} +3.56155i q^{5} -3.12311 q^{7} +0.561553 q^{9} -1.00000i q^{11} +5.12311i q^{13} -5.56155 q^{15} +2.00000 q^{17} +4.00000i q^{19} -4.87689i q^{21} -2.43845 q^{23} -7.68466 q^{25} +5.56155i q^{27} +5.12311i q^{29} -5.56155 q^{31} +1.56155 q^{33} -11.1231i q^{35} -7.56155i q^{37} -8.00000 q^{39} +1.12311 q^{41} -7.12311i q^{43} +2.00000i q^{45} +8.00000 q^{47} +2.75379 q^{49} +3.12311i q^{51} +12.2462i q^{53} +3.56155 q^{55} -6.24621 q^{57} +7.80776i q^{59} -1.12311i q^{61} -1.75379 q^{63} -18.2462 q^{65} -9.56155i q^{67} -3.80776i q^{69} +8.68466 q^{71} -5.12311 q^{73} -12.0000i q^{75} +3.12311i q^{77} -11.1231 q^{79} -7.00000 q^{81} -0.876894i q^{83} +7.12311i q^{85} -8.00000 q^{87} -2.68466 q^{89} -16.0000i q^{91} -8.68466i q^{93} -14.2462 q^{95} +15.5616 q^{97} -0.561553i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} - 6 q^{9} - 14 q^{15} + 8 q^{17} - 18 q^{23} - 6 q^{25} - 14 q^{31} - 2 q^{33} - 32 q^{39} - 12 q^{41} + 32 q^{47} + 44 q^{49} + 6 q^{55} + 8 q^{57} - 40 q^{63} - 40 q^{65} + 10 q^{71} - 4 q^{73}+ \cdots + 54 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 6 * q^9 - 14 * q^15 + 8 * q^17 - 18 * q^23 - 6 * q^25 - 14 * q^31 - 2 * q^33 - 32 * q^39 - 12 * q^41 + 32 * q^47 + 44 * q^49 + 6 * q^55 + 8 * q^57 - 40 * q^63 - 40 * q^65 + 10 * q^71 - 4 * q^73 - 28 * q^79 - 28 * q^81 - 32 * q^87 + 14 * q^89 - 24 * q^95 + 54 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	3.56155i	1.59277i	0.604787	+	0.796387i	$0.293258\pi$
$5$	3.56155i	1.59277i	−0.604787	+	0.796387i	$0.706742\pi$
$6$	0	0
$7$	−3.12311	−1.18042	−0.590211	−	0.807249i	$-0.700956\pi$
$7$	−3.12311	−1.18042	−0.590211	+	0.807249i	$0.700956\pi$
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	5.12311i	1.42089i	0.703751	+	0.710447i	$0.251507\pi$
$13$	5.12311i	1.42089i	−0.703751	+	0.710447i	$0.748493\pi$
$14$	0	0
$15$	−5.56155	−1.43599
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	− 4.87689i	− 1.06423i
$22$	0	0
$23$	−2.43845	−0.508451	−0.254226	−	0.967145i	$-0.581821\pi$
$23$	−2.43845	−0.508451	−0.254226	+	0.967145i	$0.581821\pi$
$24$	0	0
$25$	−7.68466	−1.53693
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	5.12311i	0.951337i	0.879625	+	0.475668i	$0.157794\pi$
$29$	5.12311i	0.951337i	−0.879625	+	0.475668i	$0.842206\pi$
$30$	0	0
$31$	−5.56155	−0.998884	−0.499442	−	0.866347i	$-0.666462\pi$
$31$	−5.56155	−0.998884	−0.499442	+	0.866347i	$0.666462\pi$
$32$	0	0
$33$	1.56155	0.271831
$34$	0	0
$35$	− 11.1231i	− 1.88015i
$36$	0	0
$37$	− 7.56155i	− 1.24311i	−0.783370	−	0.621556i	$-0.786501\pi$
$37$	− 7.56155i	− 1.24311i	0.783370	−	0.621556i	$-0.213499\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	1.12311	0.175400	0.0876998	−	0.996147i	$-0.472048\pi$
$41$	1.12311	0.175400	0.0876998	+	0.996147i	$0.472048\pi$
$42$	0	0
$43$	− 7.12311i	− 1.08626i	−0.839648	−	0.543132i	$-0.817238\pi$
$43$	− 7.12311i	− 1.08626i	0.839648	−	0.543132i	$-0.182762\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	3.12311i	0.437322i
$52$	0	0
$53$	12.2462i	1.68215i	0.540921	+	0.841073i	$0.318076\pi$
$53$	12.2462i	1.68215i	−0.540921	+	0.841073i	$0.681924\pi$
$54$	0	0
$55$	3.56155	0.480240
$56$	0	0
$57$	−6.24621	−0.827331
$58$	0	0
$59$	7.80776i	1.01648i	0.861214	+	0.508242i	$0.169705\pi$
$59$	7.80776i	1.01648i	−0.861214	+	0.508242i	$0.830295\pi$
$60$	0	0
$61$	− 1.12311i	− 0.143799i	−0.997412	−	0.0718995i	$-0.977094\pi$
$61$	− 1.12311i	− 0.143799i	0.997412	−	0.0718995i	$-0.0229061\pi$
$62$	0	0
$63$	−1.75379	−0.220957
$64$	0	0
$65$	−18.2462	−2.26316
$66$	0	0
$67$	− 9.56155i	− 1.16813i	−0.811707	−	0.584065i	$-0.801461\pi$
$67$	− 9.56155i	− 1.16813i	0.811707	−	0.584065i	$-0.198539\pi$
$68$	0	0
$69$	− 3.80776i	− 0.458401i
$70$	0	0
$71$	8.68466	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$71$	8.68466	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$72$	0	0
$73$	−5.12311	−0.599614	−0.299807	−	0.954000i	$-0.596922\pi$
$73$	−5.12311	−0.599614	−0.299807	+	0.954000i	$0.596922\pi$
$74$	0	0
$75$	− 12.0000i	− 1.38564i
$76$	0	0
$77$	3.12311i	0.355911i
$78$	0	0
$79$	−11.1231	−1.25145	−0.625724	−	0.780045i	$-0.715196\pi$
$79$	−11.1231	−1.25145	−0.625724	+	0.780045i	$0.715196\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 0.876894i	− 0.0962517i	−0.998841	−	0.0481258i	$-0.984675\pi$
$83$	− 0.876894i	− 0.0962517i	0.998841	−	0.0481258i	$-0.0153248\pi$
$84$	0	0
$85$	7.12311i	0.772609i
$86$	0	0
$87$	−8.00000	−0.857690
$88$	0	0
$89$	−2.68466	−0.284573	−0.142287	−	0.989825i	$-0.545445\pi$
$89$	−2.68466	−0.284573	−0.142287	+	0.989825i	$0.545445\pi$
$90$	0	0
$91$	− 16.0000i	− 1.67726i
$92$	0	0
$93$	− 8.68466i	− 0.900557i
$94$	0	0
$95$	−14.2462	−1.46163
$96$	0	0
$97$	15.5616	1.58004	0.790018	−	0.613083i	$-0.210071\pi$
$97$	15.5616	1.58004	0.790018	+	0.613083i	$0.210071\pi$
$98$	0	0
$99$	− 0.561553i	− 0.0564382i
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	17.3693	1.69507
$106$	0	0
$107$	− 13.3693i	− 1.29246i	−0.763142	−	0.646230i	$-0.776345\pi$
$107$	− 13.3693i	− 1.29246i	0.763142	−	0.646230i	$-0.223655\pi$
$108$	0	0
$109$	− 12.2462i	− 1.17297i	−0.809959	−	0.586487i	$-0.800510\pi$
$109$	− 12.2462i	− 1.17297i	0.809959	−	0.586487i	$-0.199490\pi$
$110$	0	0
$111$	11.8078	1.12074
$112$	0	0
$113$	−0.438447	−0.0412456	−0.0206228	−	0.999787i	$-0.506565\pi$
$113$	−0.438447	−0.0412456	−0.0206228	+	0.999787i	$0.506565\pi$
$114$	0	0
$115$	− 8.68466i	− 0.809849i
$116$	0	0
$117$	2.87689i	0.265969i
$118$	0	0
$119$	−6.24621	−0.572589
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	1.75379i	0.158134i
$124$	0	0
$125$	− 9.56155i	− 0.855211i
$126$	0	0
$127$	6.24621	0.554262	0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621	0.554262	0.277131	+	0.960832i	$0.410616\pi$
$128$	0	0
$129$	11.1231	0.979335
$130$	0	0
$131$	− 13.3693i	− 1.16808i	−0.811724	−	0.584041i	$-0.801471\pi$
$131$	− 13.3693i	− 1.16808i	0.811724	−	0.584041i	$-0.198529\pi$
$132$	0	0
$133$	− 12.4924i	− 1.08323i
$134$	0	0
$135$	−19.8078	−1.70478
$136$	0	0
$137$	8.43845	0.720945	0.360473	−	0.932770i	$-0.382615\pi$
$137$	8.43845	0.720945	0.360473	+	0.932770i	$0.382615\pi$
$138$	0	0
$139$	15.1231i	1.28273i	0.767238	+	0.641363i	$0.221631\pi$
$139$	15.1231i	1.28273i	−0.767238	+	0.641363i	$0.778369\pi$
$140$	0	0
$141$	12.4924i	1.05205i
$142$	0	0
$143$	5.12311	0.428416
$144$	0	0
$145$	−18.2462	−1.51527
$146$	0	0
$147$	4.30019i	0.354673i
$148$	0	0
$149$	4.24621i	0.347863i	0.984758	+	0.173932i	$0.0556472\pi$
$149$	4.24621i	0.347863i	−0.984758	+	0.173932i	$0.944353\pi$
$150$	0	0
$151$	9.36932	0.762464	0.381232	−	0.924479i	$-0.375500\pi$
$151$	9.36932	0.762464	0.381232	+	0.924479i	$0.375500\pi$
$152$	0	0
$153$	1.12311	0.0907977
$154$	0	0
$155$	− 19.8078i	− 1.59100i
$156$	0	0
$157$	4.43845i	0.354227i	0.984190	+	0.177113i	$0.0566759\pi$
$157$	4.43845i	0.354227i	−0.984190	+	0.177113i	$0.943324\pi$
$158$	0	0
$159$	−19.1231	−1.51656
$160$	0	0
$161$	7.61553	0.600188
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	5.56155i	0.432966i
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−13.2462	−1.01894
$170$	0	0
$171$	2.24621i	0.171772i
$172$	0	0
$173$	− 12.2462i	− 0.931062i	−0.885032	−	0.465531i	$-0.845863\pi$
$173$	− 12.2462i	− 0.931062i	0.885032	−	0.465531i	$-0.154137\pi$
$174$	0	0
$175$	24.0000	1.81423
$176$	0	0
$177$	−12.1922	−0.916425
$178$	0	0
$179$	6.43845i	0.481232i	0.970620	+	0.240616i	$0.0773495\pi$
$179$	6.43845i	0.481232i	−0.970620	+	0.240616i	$0.922651\pi$
$180$	0	0
$181$	− 1.31534i	− 0.0977686i	−0.998804	−	0.0488843i	$-0.984433\pi$
$181$	− 1.31534i	− 0.0977686i	0.998804	−	0.0488843i	$-0.0155666\pi$
$182$	0	0
$183$	1.75379	0.129644
$184$	0	0
$185$	26.9309	1.98000
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	0	0
$189$	− 17.3693i	− 1.26343i
$190$	0	0
$191$	−10.4384	−0.755300	−0.377650	−	0.925949i	$-0.623268\pi$
$191$	−10.4384	−0.755300	−0.377650	+	0.925949i	$0.623268\pi$
$192$	0	0
$193$	−9.12311	−0.656696	−0.328348	−	0.944557i	$-0.606492\pi$
$193$	−9.12311	−0.656696	−0.328348	+	0.944557i	$0.606492\pi$
$194$	0	0
$195$	− 28.4924i	− 2.04038i
$196$	0	0
$197$	− 14.4924i	− 1.03254i	−0.856425	−	0.516271i	$-0.827320\pi$
$197$	− 14.4924i	− 1.03254i	0.856425	−	0.516271i	$-0.172680\pi$
$198$	0	0
$199$	12.4924	0.885564	0.442782	−	0.896629i	$-0.353991\pi$
$199$	12.4924	0.885564	0.442782	+	0.896629i	$0.353991\pi$
$200$	0	0
$201$	14.9309	1.05314
$202$	0	0
$203$	− 16.0000i	− 1.12298i
$204$	0	0
$205$	4.00000i	0.279372i
$206$	0	0
$207$	−1.36932	−0.0951741
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	− 8.49242i	− 0.584642i	−0.956320	−	0.292321i	$-0.905572\pi$
$211$	− 8.49242i	− 0.584642i	0.956320	−	0.292321i	$-0.0944276\pi$
$212$	0	0
$213$	13.5616i	0.929222i
$214$	0	0
$215$	25.3693	1.73017
$216$	0	0
$217$	17.3693	1.17911
$218$	0	0
$219$	− 8.00000i	− 0.540590i
$220$	0	0
$221$	10.2462i	0.689235i
$222$	0	0
$223$	−11.8078	−0.790706	−0.395353	−	0.918529i	$-0.629378\pi$
$223$	−11.8078	−0.790706	−0.395353	+	0.918529i	$0.629378\pi$
$224$	0	0
$225$	−4.31534	−0.287689
$226$	0	0
$227$	− 23.1231i	− 1.53473i	−0.641208	−	0.767367i	$-0.721566\pi$
$227$	− 23.1231i	− 1.53473i	0.641208	−	0.767367i	$-0.278434\pi$
$228$	0	0
$229$	14.6847i	0.970390i	0.874406	+	0.485195i	$0.161251\pi$
$229$	14.6847i	0.970390i	−0.874406	+	0.485195i	$0.838749\pi$
$230$	0	0
$231$	−4.87689	−0.320876
$232$	0	0
$233$	7.36932	0.482780	0.241390	−	0.970428i	$-0.422397\pi$
$233$	7.36932	0.482780	0.241390	+	0.970428i	$0.422397\pi$
$234$	0	0
$235$	28.4924i	1.85864i
$236$	0	0
$237$	− 17.3693i	− 1.12826i
$238$	0	0
$239$	−4.87689	−0.315460	−0.157730	−	0.987482i	$-0.550418\pi$
$239$	−4.87689	−0.315460	−0.157730	+	0.987482i	$0.550418\pi$
$240$	0	0
$241$	29.1231	1.87598	0.937992	−	0.346657i	$-0.112683\pi$
$241$	29.1231	1.87598	0.937992	+	0.346657i	$0.112683\pi$
$242$	0	0
$243$	5.75379i	0.369106i
$244$	0	0
$245$	9.80776i	0.626595i
$246$	0	0
$247$	−20.4924	−1.30390
$248$	0	0
$249$	1.36932	0.0867769
$250$	0	0
$251$	1.56155i	0.0985643i	0.998785	+	0.0492822i	$0.0156934\pi$
$251$	1.56155i	0.0985643i	−0.998785	+	0.0492822i	$0.984307\pi$
$252$	0	0
$253$	2.43845i	0.153304i
$254$	0	0
$255$	−11.1231	−0.696556
$256$	0	0
$257$	11.7538	0.733181	0.366591	−	0.930382i	$-0.380525\pi$
$257$	11.7538	0.733181	0.366591	+	0.930382i	$0.380525\pi$
$258$	0	0
$259$	23.6155i	1.46740i
$260$	0	0
$261$	2.87689i	0.178075i
$262$	0	0
$263$	−19.1231	−1.17918	−0.589591	−	0.807702i	$-0.700711\pi$
$263$	−19.1231	−1.17918	−0.589591	+	0.807702i	$0.700711\pi$
$264$	0	0
$265$	−43.6155	−2.67928
$266$	0	0
$267$	− 4.19224i	− 0.256561i
$268$	0	0
$269$	20.7386i	1.26446i	0.774782	+	0.632228i	$0.217860\pi$
$269$	20.7386i	1.26446i	−0.774782	+	0.632228i	$0.782140\pi$
$270$	0	0
$271$	−28.4924	−1.73079	−0.865396	−	0.501089i	$-0.832933\pi$
$271$	−28.4924	−1.73079	−0.865396	+	0.501089i	$0.832933\pi$
$272$	0	0
$273$	24.9848	1.51215
$274$	0	0
$275$	7.68466i	0.463402i
$276$	0	0
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	0	0
$279$	−3.12311	−0.186975
$280$	0	0
$281$	−16.2462	−0.969168	−0.484584	−	0.874745i	$-0.661029\pi$
$281$	−16.2462	−0.969168	−0.484584	+	0.874745i	$0.661029\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	− 22.2462i	− 1.31775i
$286$	0	0
$287$	−3.50758	−0.207046
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	24.3002i	1.42450i
$292$	0	0
$293$	− 3.36932i	− 0.196838i	−0.995145	−	0.0984188i	$-0.968622\pi$
$293$	− 3.36932i	− 0.196838i	0.995145	−	0.0984188i	$-0.0313785\pi$
$294$	0	0
$295$	−27.8078	−1.61903
$296$	0	0
$297$	5.56155	0.322714
$298$	0	0
$299$	− 12.4924i	− 0.722455i
$300$	0	0
$301$	22.2462i	1.28225i
$302$	0	0
$303$	3.12311	0.179418
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	32.4924i	1.85444i	0.374516	+	0.927220i	$0.377809\pi$
$307$	32.4924i	1.85444i	−0.374516	+	0.927220i	$0.622191\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.75379	−0.553087	−0.276543	−	0.961001i	$-0.589189\pi$
$311$	−9.75379	−0.553087	−0.276543	+	0.961001i	$0.589189\pi$
$312$	0	0
$313$	9.80776	0.554368	0.277184	−	0.960817i	$-0.410599\pi$
$313$	9.80776	0.554368	0.277184	+	0.960817i	$0.410599\pi$
$314$	0	0
$315$	− 6.24621i	− 0.351934i
$316$	0	0
$317$	14.1922i	0.797115i	0.917143	+	0.398558i	$0.130489\pi$
$317$	14.1922i	0.797115i	−0.917143	+	0.398558i	$0.869511\pi$
$318$	0	0
$319$	5.12311	0.286839
$320$	0	0
$321$	20.8769	1.16523
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	− 39.3693i	− 2.18382i
$326$	0	0
$327$	19.1231	1.05751
$328$	0	0
$329$	−24.9848	−1.37746
$330$	0	0
$331$	34.9309i	1.91997i	0.280044	+	0.959987i	$0.409651\pi$
$331$	34.9309i	1.91997i	−0.280044	+	0.959987i	$0.590349\pi$
$332$	0	0
$333$	− 4.24621i	− 0.232691i
$334$	0	0
$335$	34.0540	1.86057
$336$	0	0
$337$	−16.7386	−0.911811	−0.455906	−	0.890028i	$-0.650685\pi$
$337$	−16.7386	−0.911811	−0.455906	+	0.890028i	$0.650685\pi$
$338$	0	0
$339$	− 0.684658i	− 0.0371855i
$340$	0	0
$341$	5.56155i	0.301175i
$342$	0	0
$343$	13.2614	0.716046
$344$	0	0
$345$	13.5616	0.730129
$346$	0	0
$347$	22.7386i	1.22067i	0.792142	+	0.610337i	$0.208966\pi$
$347$	22.7386i	1.22067i	−0.792142	+	0.610337i	$0.791034\pi$
$348$	0	0
$349$	32.2462i	1.72610i	0.505118	+	0.863050i	$0.331449\pi$
$349$	32.2462i	1.72610i	−0.505118	+	0.863050i	$0.668551\pi$
$350$	0	0
$351$	−28.4924	−1.52081
$352$	0	0
$353$	−24.0540	−1.28026	−0.640132	−	0.768265i	$-0.721120\pi$
$353$	−24.0540	−1.28026	−0.640132	+	0.768265i	$0.721120\pi$
$354$	0	0
$355$	30.9309i	1.64164i
$356$	0	0
$357$	− 9.75379i	− 0.516225i
$358$	0	0
$359$	−4.49242	−0.237101	−0.118550	−	0.992948i	$-0.537825\pi$
$359$	−4.49242	−0.237101	−0.118550	+	0.992948i	$0.537825\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 1.56155i	− 0.0819603i
$364$	0	0
$365$	− 18.2462i	− 0.955050i
$366$	0	0
$367$	22.9309	1.19698	0.598491	−	0.801130i	$-0.295767\pi$
$367$	22.9309	1.19698	0.598491	+	0.801130i	$0.295767\pi$
$368$	0	0
$369$	0.630683	0.0328321
$370$	0	0
$371$	− 38.2462i	− 1.98564i
$372$	0	0
$373$	− 8.24621i	− 0.426973i	−0.976946	−	0.213486i	$-0.931518\pi$
$373$	− 8.24621i	− 0.426973i	0.976946	−	0.213486i	$-0.0684819\pi$
$374$	0	0
$375$	14.9309	0.771027
$376$	0	0
$377$	−26.2462	−1.35175
$378$	0	0
$379$	0.192236i	0.00987450i	0.999988	+	0.00493725i	$0.00157158\pi$
$379$	0.192236i	0.00987450i	−0.999988	+	0.00493725i	$0.998428\pi$
$380$	0	0
$381$	9.75379i	0.499702i
$382$	0	0
$383$	−2.05398	−0.104953	−0.0524766	−	0.998622i	$-0.516712\pi$
$383$	−2.05398	−0.104953	−0.0524766	+	0.998622i	$0.516712\pi$
$384$	0	0
$385$	−11.1231	−0.566886
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	3.56155i	0.180578i	0.995916	+	0.0902889i	$0.0287791\pi$
$389$	3.56155i	0.180578i	−0.995916	+	0.0902889i	$0.971221\pi$
$390$	0	0
$391$	−4.87689	−0.246635
$392$	0	0
$393$	20.8769	1.05310
$394$	0	0
$395$	− 39.6155i	− 1.99327i
$396$	0	0
$397$	− 10.4924i	− 0.526600i	−0.964714	−	0.263300i	$-0.915189\pi$
$397$	− 10.4924i	− 0.526600i	0.964714	−	0.263300i	$-0.0848108\pi$
$398$	0	0
$399$	19.5076	0.976600
$400$	0	0
$401$	30.4924	1.52272	0.761359	−	0.648330i	$-0.224532\pi$
$401$	30.4924	1.52272	0.761359	+	0.648330i	$0.224532\pi$
$402$	0	0
$403$	− 28.4924i	− 1.41931i
$404$	0	0
$405$	− 24.9309i	− 1.23882i
$406$	0	0
$407$	−7.56155	−0.374812
$408$	0	0
$409$	−22.4924	−1.11218	−0.556089	−	0.831123i	$-0.687699\pi$
$409$	−22.4924	−1.11218	−0.556089	+	0.831123i	$0.687699\pi$
$410$	0	0
$411$	13.1771i	0.649977i
$412$	0	0
$413$	− 24.3845i	− 1.19988i
$414$	0	0
$415$	3.12311	0.153307
$416$	0	0
$417$	−23.6155	−1.15646
$418$	0	0
$419$	32.4924i	1.58736i	0.608336	+	0.793679i	$0.291837\pi$
$419$	32.4924i	1.58736i	−0.608336	+	0.793679i	$0.708163\pi$
$420$	0	0
$421$	2.49242i	0.121473i	0.998154	+	0.0607366i	$0.0193450\pi$
$421$	2.49242i	0.121473i	−0.998154	+	0.0607366i	$0.980655\pi$
$422$	0	0
$423$	4.49242	0.218429
$424$	0	0
$425$	−15.3693	−0.745521
$426$	0	0
$427$	3.50758i	0.169744i
$428$	0	0
$429$	8.00000i	0.386244i
$430$	0	0
$431$	27.1231	1.30647	0.653237	−	0.757153i	$-0.273411\pi$
$431$	27.1231	1.30647	0.653237	+	0.757153i	$0.273411\pi$
$432$	0	0
$433$	−22.6847	−1.09016	−0.545078	−	0.838386i	$-0.683500\pi$
$433$	−22.6847	−1.09016	−0.545078	+	0.838386i	$0.683500\pi$
$434$	0	0
$435$	− 28.4924i	− 1.36611i
$436$	0	0
$437$	− 9.75379i	− 0.466587i
$438$	0	0
$439$	−4.49242	−0.214412	−0.107206	−	0.994237i	$-0.534190\pi$
$439$	−4.49242	−0.214412	−0.107206	+	0.994237i	$0.534190\pi$
$440$	0	0
$441$	1.54640	0.0736380
$442$	0	0
$443$	11.3153i	0.537608i	0.963195	+	0.268804i	$0.0866284\pi$
$443$	11.3153i	0.537608i	−0.963195	+	0.268804i	$0.913372\pi$
$444$	0	0
$445$	− 9.56155i	− 0.453261i
$446$	0	0
$447$	−6.63068	−0.313621
$448$	0	0
$449$	−36.5464	−1.72473	−0.862366	−	0.506286i	$-0.831018\pi$
$449$	−36.5464	−1.72473	−0.862366	+	0.506286i	$0.831018\pi$
$450$	0	0
$451$	− 1.12311i	− 0.0528850i
$452$	0	0
$453$	14.6307i	0.687409i
$454$	0	0
$455$	56.9848	2.67149
$456$	0	0
$457$	−23.8617	−1.11621	−0.558103	−	0.829772i	$-0.688470\pi$
$457$	−23.8617	−1.11621	−0.558103	+	0.829772i	$0.688470\pi$
$458$	0	0
$459$	11.1231i	0.519182i
$460$	0	0
$461$	− 1.12311i	− 0.0523082i	−0.999658	−	0.0261541i	$-0.991674\pi$
$461$	− 1.12311i	− 0.0523082i	0.999658	−	0.0261541i	$-0.00832606\pi$
$462$	0	0
$463$	15.3153	0.711764	0.355882	−	0.934531i	$-0.384180\pi$
$463$	15.3153	0.711764	0.355882	+	0.934531i	$0.384180\pi$
$464$	0	0
$465$	30.9309	1.43438
$466$	0	0
$467$	− 28.3002i	− 1.30958i	−0.755812	−	0.654788i	$-0.772758\pi$
$467$	− 28.3002i	− 1.30958i	0.755812	−	0.654788i	$-0.227242\pi$
$468$	0	0
$469$	29.8617i	1.37889i
$470$	0	0
$471$	−6.93087	−0.319358
$472$	0	0
$473$	−7.12311	−0.327521
$474$	0	0
$475$	− 30.7386i	− 1.41039i
$476$	0	0
$477$	6.87689i	0.314871i
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	38.7386	1.76633
$482$	0	0
$483$	11.8920i	0.541107i
$484$	0	0
$485$	55.4233i	2.51664i
$486$	0	0
$487$	−14.9309	−0.676582	−0.338291	−	0.941041i	$-0.609849\pi$
$487$	−14.9309	−0.676582	−0.338291	+	0.941041i	$0.609849\pi$
$488$	0	0
$489$	−6.24621	−0.282463
$490$	0	0
$491$	13.7538i	0.620700i	0.950622	+	0.310350i	$0.100446\pi$
$491$	13.7538i	0.620700i	−0.950622	+	0.310350i	$0.899554\pi$
$492$	0	0
$493$	10.2462i	0.461466i
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	−27.1231	−1.21664
$498$	0	0
$499$	28.9848i	1.29754i	0.760985	+	0.648770i	$0.224716\pi$
$499$	28.9848i	1.29754i	−0.760985	+	0.648770i	$0.775284\pi$
$500$	0	0
$501$	− 12.4924i	− 0.558120i
$502$	0	0
$503$	31.6155	1.40967	0.704833	−	0.709373i	$-0.251022\pi$
$503$	31.6155	1.40967	0.704833	+	0.709373i	$0.251022\pi$
$504$	0	0
$505$	7.12311	0.316974
$506$	0	0
$507$	− 20.6847i	− 0.918638i
$508$	0	0
$509$	18.3002i	0.811142i	0.914064	+	0.405571i	$0.132927\pi$
$509$	18.3002i	0.811142i	−0.914064	+	0.405571i	$0.867073\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	−22.2462	−0.982194
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	19.1231	0.839411
$520$	0	0
$521$	−1.31534	−0.0576262	−0.0288131	−	0.999585i	$-0.509173\pi$
$521$	−1.31534	−0.0576262	−0.0288131	+	0.999585i	$0.509173\pi$
$522$	0	0
$523$	− 12.0000i	− 0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$	− 12.0000i	− 0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$	0	0
$525$	37.4773i	1.63564i
$526$	0	0
$527$	−11.1231	−0.484530
$528$	0	0
$529$	−17.0540	−0.741477
$530$	0	0
$531$	4.38447i	0.190270i
$532$	0	0
$533$	5.75379i	0.249224i
$534$	0	0
$535$	47.6155	2.05860
$536$	0	0
$537$	−10.0540	−0.433861
$538$	0	0
$539$	− 2.75379i	− 0.118614i
$540$	0	0
$541$	23.8617i	1.02590i	0.858420	+	0.512948i	$0.171447\pi$
$541$	23.8617i	1.02590i	−0.858420	+	0.512948i	$0.828553\pi$
$542$	0	0
$543$	2.05398	0.0881445
$544$	0	0
$545$	43.6155	1.86828
$546$	0	0
$547$	42.2462i	1.80632i	0.429307	+	0.903159i	$0.358758\pi$
$547$	42.2462i	1.80632i	−0.429307	+	0.903159i	$0.641242\pi$
$548$	0	0
$549$	− 0.630683i	− 0.0269169i
$550$	0	0
$551$	−20.4924	−0.873007
$552$	0	0
$553$	34.7386	1.47724
$554$	0	0
$555$	42.0540i	1.78509i
$556$	0	0
$557$	3.75379i	0.159053i	0.996833	+	0.0795266i	$0.0253409\pi$
$557$	3.75379i	0.159053i	−0.996833	+	0.0795266i	$0.974659\pi$
$558$	0	0
$559$	36.4924	1.54347
$560$	0	0
$561$	3.12311	0.131858
$562$	0	0
$563$	− 24.4924i	− 1.03223i	−0.856519	−	0.516116i	$-0.827377\pi$
$563$	− 24.4924i	− 1.03223i	0.856519	−	0.516116i	$-0.172623\pi$
$564$	0	0
$565$	− 1.56155i	− 0.0656950i
$566$	0	0
$567$	21.8617	0.918107
$568$	0	0
$569$	26.8769	1.12674	0.563369	−	0.826205i	$-0.309505\pi$
$569$	26.8769	1.12674	0.563369	+	0.826205i	$0.309505\pi$
$570$	0	0
$571$	16.4924i	0.690186i	0.938568	+	0.345093i	$0.112153\pi$
$571$	16.4924i	0.690186i	−0.938568	+	0.345093i	$0.887847\pi$
$572$	0	0
$573$	− 16.3002i	− 0.680950i
$574$	0	0
$575$	18.7386	0.781455
$576$	0	0
$577$	15.5616	0.647836	0.323918	−	0.946085i	$-0.395000\pi$
$577$	15.5616	0.647836	0.323918	+	0.946085i	$0.395000\pi$
$578$	0	0
$579$	− 14.2462i	− 0.592052i
$580$	0	0
$581$	2.73863i	0.113618i
$582$	0	0
$583$	12.2462	0.507186
$584$	0	0
$585$	−10.2462	−0.423629
$586$	0	0
$587$	− 24.4924i	− 1.01091i	−0.862853	−	0.505455i	$-0.831325\pi$
$587$	− 24.4924i	− 1.01091i	0.862853	−	0.505455i	$-0.168675\pi$
$588$	0	0
$589$	− 22.2462i	− 0.916639i
$590$	0	0
$591$	22.6307	0.930902
$592$	0	0
$593$	3.36932	0.138361	0.0691806	−	0.997604i	$-0.477962\pi$
$593$	3.36932	0.138361	0.0691806	+	0.997604i	$0.477962\pi$
$594$	0	0
$595$	− 22.2462i	− 0.912006i
$596$	0	0
$597$	19.5076i	0.798392i
$598$	0	0
$599$	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
$601$	−3.75379	−0.153120	−0.0765601	−	0.997065i	$-0.524394\pi$
$601$	−3.75379	−0.153120	−0.0765601	+	0.997065i	$0.524394\pi$
$602$	0	0
$603$	− 5.36932i	− 0.218655i
$604$	0	0
$605$	− 3.56155i	− 0.144798i
$606$	0	0
$607$	−45.8617	−1.86147	−0.930735	−	0.365694i	$-0.880832\pi$
$607$	−45.8617	−1.86147	−0.930735	+	0.365694i	$0.880832\pi$
$608$	0	0
$609$	24.9848	1.01244
$610$	0	0
$611$	40.9848i	1.65807i
$612$	0	0
$613$	11.8617i	0.479091i	0.970885	+	0.239546i	$0.0769985\pi$
$613$	11.8617i	0.479091i	−0.970885	+	0.239546i	$0.923002\pi$
$614$	0	0
$615$	−6.24621	−0.251872
$616$	0	0
$617$	2.49242	0.100341	0.0501706	−	0.998741i	$-0.484024\pi$
$617$	2.49242	0.100341	0.0501706	+	0.998741i	$0.484024\pi$
$618$	0	0
$619$	18.9309i	0.760896i	0.924802	+	0.380448i	$0.124230\pi$
$619$	18.9309i	0.760896i	−0.924802	+	0.380448i	$0.875770\pi$
$620$	0	0
$621$	− 13.5616i	− 0.544206i
$622$	0	0
$623$	8.38447	0.335917
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	6.24621i	0.249450i
$628$	0	0
$629$	− 15.1231i	− 0.602998i
$630$	0	0
$631$	42.0540	1.67414	0.837071	−	0.547094i	$-0.184266\pi$
$631$	42.0540	1.67414	0.837071	+	0.547094i	$0.184266\pi$
$632$	0	0
$633$	13.2614	0.527092
$634$	0	0
$635$	22.2462i	0.882814i
$636$	0	0
$637$	14.1080i	0.558977i
$638$	0	0
$639$	4.87689	0.192927
$640$	0	0
$641$	−46.3002	−1.82875	−0.914374	−	0.404871i	$-0.867316\pi$
$641$	−46.3002	−1.82875	−0.914374	+	0.404871i	$0.867316\pi$
$642$	0	0
$643$	9.17708i	0.361909i	0.983491	+	0.180954i	$0.0579186\pi$
$643$	9.17708i	0.361909i	−0.983491	+	0.180954i	$0.942081\pi$
$644$	0	0
$645$	39.6155i	1.55986i
$646$	0	0
$647$	−13.5616	−0.533160	−0.266580	−	0.963813i	$-0.585894\pi$
$647$	−13.5616	−0.533160	−0.266580	+	0.963813i	$0.585894\pi$
$648$	0	0
$649$	7.80776	0.306482
$650$	0	0
$651$	27.1231i	1.06304i
$652$	0	0
$653$	− 35.1771i	− 1.37659i	−0.725433	−	0.688293i	$-0.758360\pi$
$653$	− 35.1771i	− 1.37659i	0.725433	−	0.688293i	$-0.241640\pi$
$654$	0	0
$655$	47.6155	1.86049
$656$	0	0
$657$	−2.87689	−0.112238
$658$	0	0
$659$	11.6155i	0.452477i	0.974072	+	0.226238i	$0.0726428\pi$
$659$	11.6155i	0.452477i	−0.974072	+	0.226238i	$0.927357\pi$
$660$	0	0
$661$	41.8078i	1.62613i	0.582170	+	0.813067i	$0.302204\pi$
$661$	41.8078i	1.62613i	−0.582170	+	0.813067i	$0.697796\pi$
$662$	0	0
$663$	−16.0000	−0.621389
$664$	0	0
$665$	44.4924	1.72534
$666$	0	0
$667$	− 12.4924i	− 0.483709i
$668$	0	0
$669$	− 18.4384i	− 0.712872i
$670$	0	0
$671$	−1.12311	−0.0433570
$672$	0	0
$673$	33.2311	1.28096	0.640482	−	0.767974i	$-0.278735\pi$
$673$	33.2311	1.28096	0.640482	+	0.767974i	$0.278735\pi$
$674$	0	0
$675$	− 42.7386i	− 1.64501i
$676$	0	0
$677$	− 20.7386i	− 0.797050i	−0.917157	−	0.398525i	$-0.869522\pi$
$677$	− 20.7386i	− 0.797050i	0.917157	−	0.398525i	$-0.130478\pi$
$678$	0	0
$679$	−48.6004	−1.86511
$680$	0	0
$681$	36.1080	1.38366
$682$	0	0
$683$	6.73863i	0.257847i	0.991655	+	0.128923i	$0.0411521\pi$
$683$	6.73863i	0.257847i	−0.991655	+	0.128923i	$0.958848\pi$
$684$	0	0
$685$	30.0540i	1.14830i
$686$	0	0
$687$	−22.9309	−0.874867
$688$	0	0
$689$	−62.7386	−2.39015
$690$	0	0
$691$	9.94602i	0.378365i	0.981942	+	0.189182i	$0.0605837\pi$
$691$	9.94602i	0.378365i	−0.981942	+	0.189182i	$0.939416\pi$
$692$	0	0
$693$	1.75379i	0.0666209i
$694$	0	0
$695$	−53.8617	−2.04309
$696$	0	0
$697$	2.24621	0.0850813
$698$	0	0
$699$	11.5076i	0.435257i
$700$	0	0
$701$	− 50.4924i	− 1.90707i	−0.301278	−	0.953536i	$-0.597413\pi$
$701$	− 50.4924i	− 1.90707i	0.301278	−	0.953536i	$-0.402587\pi$
$702$	0	0
$703$	30.2462	1.14076
$704$	0	0
$705$	−44.4924	−1.67568
$706$	0	0
$707$	6.24621i	0.234913i
$708$	0	0
$709$	2.19224i	0.0823311i	0.999152	+	0.0411656i	$0.0131071\pi$
$709$	2.19224i	0.0823311i	−0.999152	+	0.0411656i	$0.986893\pi$
$710$	0	0
$711$	−6.24621	−0.234251
$712$	0	0
$713$	13.5616	0.507884
$714$	0	0
$715$	18.2462i	0.682370i
$716$	0	0
$717$	− 7.61553i	− 0.284407i
$718$	0	0
$719$	35.4233	1.32107	0.660533	−	0.750797i	$-0.270330\pi$
$719$	35.4233	1.32107	0.660533	+	0.750797i	$0.270330\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	45.4773i	1.69132i
$724$	0	0
$725$	− 39.3693i	− 1.46214i
$726$	0	0
$727$	23.3153	0.864718	0.432359	−	0.901702i	$-0.357681\pi$
$727$	23.3153	0.864718	0.432359	+	0.901702i	$0.357681\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	− 14.2462i	− 0.526915i
$732$	0	0
$733$	− 1.12311i	− 0.0414829i	−0.999785	−	0.0207414i	$-0.993397\pi$
$733$	− 1.12311i	− 0.0414829i	0.999785	−	0.0207414i	$-0.00660267\pi$
$734$	0	0
$735$	−15.3153	−0.564915
$736$	0	0
$737$	−9.56155	−0.352204
$738$	0	0
$739$	2.63068i	0.0967712i	0.998829	+	0.0483856i	$0.0154076\pi$
$739$	2.63068i	0.0967712i	−0.998829	+	0.0483856i	$0.984592\pi$
$740$	0	0
$741$	− 32.0000i	− 1.17555i
$742$	0	0
$743$	−10.7386	−0.393962	−0.196981	−	0.980407i	$-0.563114\pi$
$743$	−10.7386	−0.393962	−0.196981	+	0.980407i	$0.563114\pi$
$744$	0	0
$745$	−15.1231	−0.554068
$746$	0	0
$747$	− 0.492423i	− 0.0180168i
$748$	0	0
$749$	41.7538i	1.52565i
$750$	0	0
$751$	5.56155	0.202944	0.101472	−	0.994838i	$-0.467645\pi$
$751$	5.56155	0.202944	0.101472	+	0.994838i	$0.467645\pi$
$752$	0	0
$753$	−2.43845	−0.0888620
$754$	0	0
$755$	33.3693i	1.21443i
$756$	0	0
$757$	15.7538i	0.572581i	0.958143	+	0.286291i	$0.0924223\pi$
$757$	15.7538i	0.572581i	−0.958143	+	0.286291i	$0.907578\pi$
$758$	0	0
$759$	−3.80776	−0.138213
$760$	0	0
$761$	−5.12311	−0.185712	−0.0928562	−	0.995680i	$-0.529600\pi$
$761$	−5.12311	−0.185712	−0.0928562	+	0.995680i	$0.529600\pi$
$762$	0	0
$763$	38.2462i	1.38461i
$764$	0	0
$765$	4.00000i	0.144620i
$766$	0	0
$767$	−40.0000	−1.44432
$768$	0	0
$769$	25.6155	0.923720	0.461860	−	0.886953i	$-0.347182\pi$
$769$	25.6155	0.923720	0.461860	+	0.886953i	$0.347182\pi$
$770$	0	0
$771$	18.3542i	0.661009i
$772$	0	0
$773$	40.7386i	1.46527i	0.680623	+	0.732633i	$0.261709\pi$
$773$	40.7386i	1.46527i	−0.680623	+	0.732633i	$0.738291\pi$
$774$	0	0
$775$	42.7386	1.53522
$776$	0	0
$777$	−36.8769	−1.32295
$778$	0	0
$779$	4.49242i	0.160958i
$780$	0	0
$781$	− 8.68466i	− 0.310762i
$782$	0	0
$783$	−28.4924	−1.01824
$784$	0	0
$785$	−15.8078	−0.564203
$786$	0	0
$787$	29.7538i	1.06061i	0.847808	+	0.530304i	$0.177922\pi$
$787$	29.7538i	1.06061i	−0.847808	+	0.530304i	$0.822078\pi$
$788$	0	0
$789$	− 29.8617i	− 1.06311i
$790$	0	0
$791$	1.36932	0.0486873
$792$	0	0
$793$	5.75379	0.204323
$794$	0	0
$795$	− 68.1080i	− 2.41554i
$796$	0	0
$797$	14.1922i	0.502715i	0.967894	+	0.251357i	$0.0808769\pi$
$797$	14.1922i	0.502715i	−0.967894	+	0.251357i	$0.919123\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−1.50758	−0.0532676
$802$	0	0
$803$	5.12311i	0.180790i
$804$	0	0
$805$	27.1231i	0.955964i
$806$	0	0
$807$	−32.3845	−1.13999
$808$	0	0
$809$	45.6155	1.60376	0.801878	−	0.597487i	$-0.203834\pi$
$809$	45.6155	1.60376	0.801878	+	0.597487i	$0.203834\pi$
$810$	0	0
$811$	− 7.12311i	− 0.250126i	−0.992149	−	0.125063i	$-0.960087\pi$
$811$	− 7.12311i	− 0.250126i	0.992149	−	0.125063i	$-0.0399133\pi$
$812$	0	0
$813$	− 44.4924i	− 1.56042i
$814$	0	0
$815$	−14.2462	−0.499023
$816$	0	0
$817$	28.4924	0.996824
$818$	0	0
$819$	− 8.98485i	− 0.313956i
$820$	0	0
$821$	− 42.9848i	− 1.50018i	−0.661335	−	0.750091i	$-0.730010\pi$
$821$	− 42.9848i	− 1.50018i	0.661335	−	0.750091i	$-0.269990\pi$
$822$	0	0
$823$	54.5464	1.90137	0.950684	−	0.310161i	$-0.100383\pi$
$823$	54.5464	1.90137	0.950684	+	0.310161i	$0.100383\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	38.7386i	1.34707i	0.739153	+	0.673537i	$0.235226\pi$
$827$	38.7386i	1.34707i	−0.739153	+	0.673537i	$0.764774\pi$
$828$	0	0
$829$	− 15.0691i	− 0.523373i	−0.965153	−	0.261686i	$-0.915721\pi$
$829$	− 15.0691i	− 0.523373i	0.965153	−	0.261686i	$-0.0842786\pi$
$830$	0	0
$831$	28.1080	0.975054
$832$	0	0
$833$	5.50758	0.190826
$834$	0	0
$835$	− 28.4924i	− 0.986021i
$836$	0	0
$837$	− 30.9309i	− 1.06913i
$838$	0	0
$839$	19.8078	0.683840	0.341920	−	0.939729i	$-0.388923\pi$
$839$	19.8078	0.683840	0.341920	+	0.939729i	$0.388923\pi$
$840$	0	0
$841$	2.75379	0.0949582
$842$	0	0
$843$	− 25.3693i	− 0.873766i
$844$	0	0
$845$	− 47.1771i	− 1.62294i
$846$	0	0
$847$	3.12311	0.107311
$848$	0	0
$849$	−31.2311	−1.07185
$850$	0	0
$851$	18.4384i	0.632062i
$852$	0	0
$853$	− 46.4924i	− 1.59187i	−0.605382	−	0.795935i	$-0.706980\pi$
$853$	− 46.4924i	− 1.59187i	0.605382	−	0.795935i	$-0.293020\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	−30.1080	−1.02847	−0.514234	−	0.857650i	$-0.671924\pi$
$857$	−30.1080	−1.02847	−0.514234	+	0.857650i	$0.671924\pi$
$858$	0	0
$859$	30.0540i	1.02543i	0.858559	+	0.512714i	$0.171360\pi$
$859$	30.0540i	1.02543i	−0.858559	+	0.512714i	$0.828640\pi$
$860$	0	0
$861$	− 5.47727i	− 0.186665i
$862$	0	0
$863$	−36.4924	−1.24222	−0.621108	−	0.783725i	$-0.713317\pi$
$863$	−36.4924	−1.24222	−0.621108	+	0.783725i	$0.713317\pi$
$864$	0	0
$865$	43.6155	1.48297
$866$	0	0
$867$	− 20.3002i	− 0.689430i
$868$	0	0
$869$	11.1231i	0.377326i
$870$	0	0
$871$	48.9848	1.65979
$872$	0	0
$873$	8.73863	0.295758
$874$	0	0
$875$	29.8617i	1.00951i
$876$	0	0
$877$	− 55.3693i	− 1.86969i	−0.355057	−	0.934844i	$-0.615539\pi$
$877$	− 55.3693i	− 1.86969i	0.355057	−	0.934844i	$-0.384461\pi$
$878$	0	0
$879$	5.26137	0.177461
$880$	0	0
$881$	34.3002	1.15560	0.577801	−	0.816177i	$-0.303911\pi$
$881$	34.3002	1.15560	0.577801	+	0.816177i	$0.303911\pi$
$882$	0	0
$883$	− 8.49242i	− 0.285793i	−0.989738	−	0.142896i	$-0.954358\pi$
$883$	− 8.49242i	− 0.285793i	0.989738	−	0.142896i	$-0.0456416\pi$
$884$	0	0
$885$	− 43.4233i	− 1.45966i
$886$	0	0
$887$	−31.6155	−1.06155	−0.530773	−	0.847514i	$-0.678098\pi$
$887$	−31.6155	−1.06155	−0.530773	+	0.847514i	$0.678098\pi$
$888$	0	0
$889$	−19.5076	−0.654263
$890$	0	0
$891$	7.00000i	0.234509i
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	−22.9309	−0.766494
$896$	0	0
$897$	19.5076	0.651339
$898$	0	0
$899$	− 28.4924i	− 0.950275i
$900$	0	0
$901$	24.4924i	0.815961i
$902$	0	0
$903$	−34.7386	−1.15603
$904$	0	0
$905$	4.68466	0.155723
$906$	0	0
$907$	16.4924i	0.547622i	0.961784	+	0.273811i	$0.0882843\pi$
$907$	16.4924i	0.547622i	−0.961784	+	0.273811i	$0.911716\pi$
$908$	0	0
$909$	− 1.12311i	− 0.0372511i
$910$	0	0
$911$	−26.7386	−0.885890	−0.442945	−	0.896549i	$-0.646066\pi$
$911$	−26.7386	−0.885890	−0.442945	+	0.896549i	$0.646066\pi$
$912$	0	0
$913$	−0.876894	−0.0290210
$914$	0	0
$915$	6.24621i	0.206493i
$916$	0	0
$917$	41.7538i	1.37883i
$918$	0	0
$919$	−6.63068	−0.218726	−0.109363	−	0.994002i	$-0.534881\pi$
$919$	−6.63068	−0.218726	−0.109363	+	0.994002i	$0.534881\pi$
$920$	0	0
$921$	−50.7386	−1.67189
$922$	0	0
$923$	44.4924i	1.46449i
$924$	0	0
$925$	58.1080i	1.91058i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.4924	1.52537	0.762683	−	0.646772i	$-0.223881\pi$
$929$	46.4924	1.52537	0.762683	+	0.646772i	$0.223881\pi$
$930$	0	0
$931$	11.0152i	0.361007i
$932$	0	0
$933$	− 15.2311i	− 0.498642i
$934$	0	0
$935$	7.12311	0.232950
$936$	0	0
$937$	42.1080	1.37561	0.687803	−	0.725897i	$-0.258575\pi$
$937$	42.1080	1.37561	0.687803	+	0.725897i	$0.258575\pi$
$938$	0	0
$939$	15.3153i	0.499797i
$940$	0	0
$941$	32.2462i	1.05120i	0.850733	+	0.525598i	$0.176158\pi$
$941$	32.2462i	1.05120i	−0.850733	+	0.525598i	$0.823842\pi$
$942$	0	0
$943$	−2.73863	−0.0891822
$944$	0	0
$945$	61.8617	2.01236
$946$	0	0
$947$	12.6847i	0.412196i	0.978531	+	0.206098i	$0.0660766\pi$
$947$	12.6847i	0.412196i	−0.978531	+	0.206098i	$0.933923\pi$
$948$	0	0
$949$	− 26.2462i	− 0.851988i
$950$	0	0
$951$	−22.1619	−0.718650
$952$	0	0
$953$	−0.246211	−0.00797556	−0.00398778	−	0.999992i	$-0.501269\pi$
$953$	−0.246211	−0.00797556	−0.00398778	+	0.999992i	$0.501269\pi$
$954$	0	0
$955$	− 37.1771i	− 1.20302i
$956$	0	0
$957$	8.00000i	0.258603i
$958$	0	0
$959$	−26.3542	−0.851020
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	− 7.50758i	− 0.241928i
$964$	0	0
$965$	− 32.4924i	− 1.04597i
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	−12.4924	−0.401314
$970$	0	0
$971$	− 34.5464i	− 1.10865i	−0.832301	−	0.554323i	$-0.812977\pi$
$971$	− 34.5464i	− 1.10865i	0.832301	−	0.554323i	$-0.187023\pi$
$972$	0	0
$973$	− 47.2311i	− 1.51416i
$974$	0	0
$975$	61.4773	1.96885
$976$	0	0
$977$	53.8078	1.72146	0.860731	−	0.509059i	$-0.170007\pi$
$977$	53.8078	1.72146	0.860731	+	0.509059i	$0.170007\pi$
$978$	0	0
$979$	2.68466i	0.0858021i
$980$	0	0
$981$	− 6.87689i	− 0.219562i
$982$	0	0
$983$	−30.9309	−0.986542	−0.493271	−	0.869876i	$-0.664199\pi$
$983$	−30.9309	−0.986542	−0.493271	+	0.869876i	$0.664199\pi$
$984$	0	0
$985$	51.6155	1.64461
$986$	0	0
$987$	− 39.0152i	− 1.24187i
$988$	0	0
$989$	17.3693i	0.552312i
$990$	0	0
$991$	−4.49242	−0.142707	−0.0713533	−	0.997451i	$-0.522732\pi$
$991$	−4.49242	−0.142707	−0.0713533	+	0.997451i	$0.522732\pi$
$992$	0	0
$993$	−54.5464	−1.73098
$994$	0	0
$995$	44.4924i	1.41050i
$996$	0	0
$997$	52.2462i	1.65465i	0.561721	+	0.827327i	$0.310140\pi$
$997$	52.2462i	1.65465i	−0.561721	+	0.827327i	$0.689860\pi$
$998$	0	0
$999$	42.0540	1.33053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.c.w.1409.3		4
4.3	odd	2		2816.2.c.p.1409.2		4
8.3	odd	2		2816.2.c.p.1409.3		4
8.5	even	2	inner	2816.2.c.w.1409.2		4
16.3	odd	4		176.2.a.d.1.2		2
16.5	even	4		704.2.a.m.1.2		2
16.11	odd	4		704.2.a.p.1.1		2
16.13	even	4		88.2.a.b.1.1	✓	2
48.5	odd	4		6336.2.a.cu.1.2		2
48.11	even	4		6336.2.a.cx.1.2		2
48.29	odd	4		792.2.a.h.1.1		2
48.35	even	4		1584.2.a.t.1.1		2
80.3	even	4		4400.2.b.v.4049.3		4
80.13	odd	4		2200.2.b.g.1849.2		4
80.19	odd	4		4400.2.a.bp.1.1		2
80.29	even	4		2200.2.a.o.1.2		2
80.67	even	4		4400.2.b.v.4049.2		4
80.77	odd	4		2200.2.b.g.1849.3		4
112.13	odd	4		4312.2.a.n.1.2		2
112.83	even	4		8624.2.a.cb.1.1		2
176.13	odd	20		968.2.i.q.81.1		8
176.21	odd	4		7744.2.a.by.1.2		2
176.29	odd	20		968.2.i.q.753.2		8
176.43	even	4		7744.2.a.cl.1.1		2
176.61	odd	20		968.2.i.q.729.1		8
176.93	even	20		968.2.i.r.729.1		8
176.109	odd	4		968.2.a.j.1.1		2
176.125	even	20		968.2.i.r.753.2		8
176.131	even	4		1936.2.a.r.1.2		2
176.141	even	20		968.2.i.r.81.1		8
176.157	even	20		968.2.i.r.9.2		8
176.173	odd	20		968.2.i.q.9.2		8
528.461	even	4		8712.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.a.b.1.1	✓	2	16.13	even	4
176.2.a.d.1.2		2	16.3	odd	4
704.2.a.m.1.2		2	16.5	even	4
704.2.a.p.1.1		2	16.11	odd	4
792.2.a.h.1.1		2	48.29	odd	4
968.2.a.j.1.1		2	176.109	odd	4
968.2.i.q.9.2		8	176.173	odd	20
968.2.i.q.81.1		8	176.13	odd	20
968.2.i.q.729.1		8	176.61	odd	20
968.2.i.q.753.2		8	176.29	odd	20
968.2.i.r.9.2		8	176.157	even	20
968.2.i.r.81.1		8	176.141	even	20
968.2.i.r.729.1		8	176.93	even	20
968.2.i.r.753.2		8	176.125	even	20
1584.2.a.t.1.1		2	48.35	even	4
1936.2.a.r.1.2		2	176.131	even	4
2200.2.a.o.1.2		2	80.29	even	4
2200.2.b.g.1849.2		4	80.13	odd	4
2200.2.b.g.1849.3		4	80.77	odd	4
2816.2.c.p.1409.2		4	4.3	odd	2
2816.2.c.p.1409.3		4	8.3	odd	2
2816.2.c.w.1409.2		4	8.5	even	2	inner
2816.2.c.w.1409.3		4	1.1	even	1	trivial
4312.2.a.n.1.2		2	112.13	odd	4
4400.2.a.bp.1.1		2	80.19	odd	4
4400.2.b.v.4049.2		4	80.67	even	4
4400.2.b.v.4049.3		4	80.3	even	4
6336.2.a.cu.1.2		2	48.5	odd	4
6336.2.a.cx.1.2		2	48.11	even	4
7744.2.a.by.1.2		2	176.21	odd	4
7744.2.a.cl.1.1		2	176.43	even	4
8624.2.a.cb.1.1		2	112.83	even	4
8712.2.a.bb.1.1		2	528.461	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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{3} +3.56155i q^{5} -3.12311 q^{7} +0.561553 q^{9} -1.00000i q^{11} +5.12311i q^{13} -5.56155 q^{15} +2.00000 q^{17} +4.00000i q^{19} -4.87689i q^{21} -2.43845 q^{23} -7.68466 q^{25} +5.56155i q^{27} +5.12311i q^{29} -5.56155 q^{31} +1.56155 q^{33} -11.1231i q^{35} -7.56155i q^{37} -8.00000 q^{39} +1.12311 q^{41} -7.12311i q^{43} +2.00000i q^{45} +8.00000 q^{47} +2.75379 q^{49} +3.12311i q^{51} +12.2462i q^{53} +3.56155 q^{55} -6.24621 q^{57} +7.80776i q^{59} -1.12311i q^{61} -1.75379 q^{63} -18.2462 q^{65} -9.56155i q^{67} -3.80776i q^{69} +8.68466 q^{71} -5.12311 q^{73} -12.0000i q^{75} +3.12311i q^{77} -11.1231 q^{79} -7.00000 q^{81} -0.876894i q^{83} +7.12311i q^{85} -8.00000 q^{87} -2.68466 q^{89} -16.0000i q^{91} -8.68466i q^{93} -14.2462 q^{95} +15.5616 q^{97} -0.561553i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 6 q^{9} - 14 q^{15} + 8 q^{17} - 18 q^{23} - 6 q^{25} - 14 q^{31} - 2 q^{33} - 32 q^{39} - 12 q^{41} + 32 q^{47} + 44 q^{49} + 6 q^{55} + 8 q^{57} - 40 q^{63} - 40 q^{65} + 10 q^{71} - 4 q^{73}+ \cdots + 54 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 6 * q^9 - 14 * q^15 + 8 * q^17 - 18 * q^23 - 6 * q^25 - 14 * q^31 - 2 * q^33 - 32 * q^39 - 12 * q^41 + 32 * q^47 + 44 * q^49 + 6 * q^55 + 8 * q^57 - 40 * q^63 - 40 * q^65 + 10 * q^71 - 4 * q^73 - 28 * q^79 - 28 * q^81 - 32 * q^87 + 14 * q^89 - 24 * q^95 + 54 * q^97

Introduction

L-functions

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