LMFDB - Embedded newform 900.2.e.f.251.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,2,Mod(251,900)] 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("900.251"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	900.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,4,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$7.18653618192$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.207360000.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} + 32x^{4} - 24x^{2} + 16 $ x^8 - 6x^6 + 32x^4 - 24*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.5
Root			$1.98168 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	900.251
Dual form			900.2.e.f.251.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.11803 - 0.866025i) q^{2} +(0.500000 - 1.93649i) q^{4} -3.16228i q^{7} +(-1.11803 - 2.59808i) q^{8} -5.47723 q^{11} -2.44949 q^{13} +(-2.73861 - 3.53553i) q^{14} +(-3.50000 - 1.93649i) q^{16} +(-6.12372 + 4.74342i) q^{22} +4.47214 q^{23} +(-2.73861 + 2.12132i) q^{26} +(-6.12372 - 1.58114i) q^{28} +2.82843i q^{29} -7.74597i q^{31} +(-5.59017 + 0.866025i) q^{32} +7.34847 q^{37} +1.41421i q^{41} -6.32456i q^{43} +(-2.73861 + 10.6066i) q^{44} +(5.00000 - 3.87298i) q^{46} +8.94427 q^{47} -3.00000 q^{49} +(-1.22474 + 4.74342i) q^{52} +(-8.21584 + 3.53553i) q^{56} +(2.44949 + 3.16228i) q^{58} -5.47723 q^{59} -10.0000 q^{61} +(-6.70820 - 8.66025i) q^{62} +(-5.50000 + 5.80948i) q^{64} -12.6491i q^{67} +14.6969 q^{73} +(8.21584 - 6.36396i) q^{74} +17.3205i q^{77} -7.74597i q^{79} +(1.22474 + 1.58114i) q^{82} -8.94427 q^{83} +(-5.47723 - 7.07107i) q^{86} +(6.12372 + 14.2302i) q^{88} -9.89949i q^{89} +7.74597i q^{91} +(2.23607 - 8.66025i) q^{92} +(10.0000 - 7.74597i) q^{94} +4.89898 q^{97} +(-3.35410 + 2.59808i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 28 q^{16} + 40 q^{46} - 24 q^{49} - 80 q^{61} - 44 q^{64} + 80 q^{94}+O(q^{100}) $ 8 * q + 4 * q^4 - 28 * q^16 + 40 * q^46 - 24 * q^49 - 80 * q^61 - 44 * q^64 + 80 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	1.11803	−	0.866025i	0.790569	−	0.612372i
$2$	1.11803	−	0.866025i	0.790569	−	0.612372i
$3$		0			0
$3$		0			0
$4$	0.500000	−	1.93649i	0.250000	−	0.968246i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	3.16228i		−	1.19523i	−0.801784	−	0.597614i	$-0.796115\pi$
$7$		−	3.16228i		−	1.19523i	0.801784	−	0.597614i	$-0.203885\pi$
$8$	−1.11803	−	2.59808i	−0.395285	−	0.918559i
$9$		0			0
$10$		0			0
$11$	−5.47723			−1.65145			−0.825723	−	0.564076i	$-0.809232\pi$
$11$	−5.47723			−1.65145			−0.825723	+	0.564076i	$0.809232\pi$
$12$		0			0
$13$	−2.44949			−0.679366			−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949			−0.679366			−0.339683	+	0.940540i	$0.610320\pi$
$14$	−2.73861	−	3.53553i	−0.731925	−	0.944911i
$15$		0			0
$16$	−3.50000	−	1.93649i	−0.875000	−	0.484123i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$	−6.12372	+	4.74342i	−1.30558	+	1.01130i
$23$	4.47214			0.932505			0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214			0.932505			0.466252	+	0.884652i	$0.345604\pi$
$24$		0			0
$25$		0			0
$26$	−2.73861	+	2.12132i	−0.537086	+	0.416025i
$27$		0			0
$28$	−6.12372	−	1.58114i	−1.15728	−	0.298807i
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$		−	7.74597i		−	1.39122i	−0.718421	−	0.695608i	$-0.755135\pi$
$31$		−	7.74597i		−	1.39122i	0.718421	−	0.695608i	$-0.244865\pi$
$32$	−5.59017	+	0.866025i	−0.988212	+	0.153093i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.34847			1.20808			0.604040	−	0.796954i	$-0.293557\pi$
$37$	7.34847			1.20808			0.604040	+	0.796954i	$0.293557\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.41421i			0.220863i	0.993884	+	0.110432i	$0.0352233\pi$
$41$			1.41421i			0.220863i	−0.993884	+	0.110432i	$0.964777\pi$
$42$		0			0
$43$		−	6.32456i		−	0.964486i	−0.876038	−	0.482243i	$-0.839822\pi$
$43$		−	6.32456i		−	0.964486i	0.876038	−	0.482243i	$-0.160178\pi$
$44$	−2.73861	+	10.6066i	−0.412861	+	1.59901i
$45$		0			0
$46$	5.00000	−	3.87298i	0.737210	−	0.571040i
$47$	8.94427			1.30466			0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427			1.30466			0.652328	+	0.757937i	$0.273792\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$		0			0
$52$	−1.22474	+	4.74342i	−0.169842	+	0.657794i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$	−8.21584	+	3.53553i	−1.09789	+	0.472456i
$57$		0			0
$58$	2.44949	+	3.16228i	0.321634	+	0.415227i
$59$	−5.47723			−0.713074			−0.356537	−	0.934281i	$-0.616043\pi$
$59$	−5.47723			−0.713074			−0.356537	+	0.934281i	$0.616043\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$	−6.70820	−	8.66025i	−0.851943	−	1.09985i
$63$		0			0
$64$	−5.50000	+	5.80948i	−0.687500	+	0.726184i
$65$		0			0
$66$		0			0
$67$		−	12.6491i		−	1.54533i	−0.634811	−	0.772667i	$-0.718922\pi$
$67$		−	12.6491i		−	1.54533i	0.634811	−	0.772667i	$-0.281078\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	14.6969			1.72015			0.860073	−	0.510171i	$-0.170418\pi$
$73$	14.6969			1.72015			0.860073	+	0.510171i	$0.170418\pi$
$74$	8.21584	−	6.36396i	0.955072	−	0.739795i
$75$		0			0
$76$		0			0
$77$			17.3205i			1.97386i
$78$		0			0
$79$		−	7.74597i		−	0.871489i	−0.900070	−	0.435745i	$-0.856485\pi$
$79$		−	7.74597i		−	0.871489i	0.900070	−	0.435745i	$-0.143515\pi$
$80$		0			0
$81$		0			0
$82$	1.22474	+	1.58114i	0.135250	+	0.174608i
$83$	−8.94427			−0.981761			−0.490881	−	0.871227i	$-0.663325\pi$
$83$	−8.94427			−0.981761			−0.490881	+	0.871227i	$0.663325\pi$
$84$		0			0
$85$		0			0
$86$	−5.47723	−	7.07107i	−0.590624	−	0.762493i
$87$		0			0
$88$	6.12372	+	14.2302i	0.652791	+	1.51695i
$89$		−	9.89949i		−	1.04934i	−0.851304	−	0.524672i	$-0.824188\pi$
$89$		−	9.89949i		−	1.04934i	0.851304	−	0.524672i	$-0.175812\pi$
$90$		0			0
$91$			7.74597i			0.811998i
$92$	2.23607	−	8.66025i	0.233126	−	0.902894i
$93$		0			0
$94$	10.0000	−	7.74597i	1.03142	−	0.798935i
$95$		0			0
$96$		0			0
$97$	4.89898			0.497416			0.248708	−	0.968579i	$-0.419994\pi$
$97$	4.89898			0.497416			0.248708	+	0.968579i	$0.419994\pi$
$98$	−3.35410	+	2.59808i	−0.338815	+	0.262445i
$99$		0			0
$100$		0			0
$101$			5.65685i			0.562878i	0.959579	+	0.281439i	$0.0908117\pi$
$101$			5.65685i			0.562878i	−0.959579	+	0.281439i	$0.909188\pi$
$102$		0			0
$103$			3.16228i			0.311588i	0.987790	+	0.155794i	$0.0497937\pi$
$103$			3.16228i			0.311588i	−0.987790	+	0.155794i	$0.950206\pi$
$104$	2.73861	+	6.36396i	0.268543	+	0.624038i
$105$		0			0
$106$		0			0
$107$	8.94427			0.864675			0.432338	−	0.901712i	$-0.357689\pi$
$107$	8.94427			0.864675			0.432338	+	0.901712i	$0.357689\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$	−6.12372	+	11.0680i	−0.578638	+	1.04583i
$113$			13.8564i			1.30350i	0.758433	+	0.651751i	$0.225965\pi$
$113$			13.8564i			1.30350i	−0.758433	+	0.651751i	$0.774035\pi$
$114$		0			0
$115$		0			0
$116$	5.47723	+	1.41421i	0.508548	+	0.131306i
$117$		0			0
$118$	−6.12372	+	4.74342i	−0.563735	+	0.436667i
$119$		0			0
$120$		0			0
$121$	19.0000			1.72727
$122$	−11.1803	+	8.66025i	−1.01222	+	0.784063i
$123$		0			0
$124$	−15.0000	−	3.87298i	−1.34704	−	0.347804i
$125$		0			0
$126$		0			0
$127$		−	3.16228i		−	0.280607i	−0.990109	−	0.140303i	$-0.955192\pi$
$127$		−	3.16228i		−	0.280607i	0.990109	−	0.140303i	$-0.0448078\pi$
$128$	−1.11803	+	11.2583i	−0.0988212	+	0.995105i
$129$		0			0
$130$		0			0
$131$	5.47723			0.478547			0.239274	−	0.970952i	$-0.423091\pi$
$131$	5.47723			0.478547			0.239274	+	0.970952i	$0.423091\pi$
$132$		0			0
$133$		0			0
$134$	−10.9545	−	14.1421i	−0.946320	−	1.22169i
$135$		0			0
$136$		0			0
$137$		−	6.92820i		−	0.591916i	−0.955201	−	0.295958i	$-0.904361\pi$
$137$		−	6.92820i		−	0.591916i	0.955201	−	0.295958i	$-0.0956389\pi$
$138$		0			0
$139$			7.74597i			0.657004i	0.944503	+	0.328502i	$0.106544\pi$
$139$			7.74597i			0.657004i	−0.944503	+	0.328502i	$0.893456\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	13.4164			1.12194
$144$		0			0
$145$		0			0
$146$	16.4317	−	12.7279i	1.35990	−	1.05337i
$147$		0			0
$148$	3.67423	−	14.2302i	0.302020	−	1.16972i
$149$			11.3137i			0.926855i	0.886135	+	0.463428i	$0.153381\pi$
$149$			11.3137i			0.926855i	−0.886135	+	0.463428i	$0.846619\pi$
$150$		0			0
$151$			7.74597i			0.630358i	0.949032	+	0.315179i	$0.102065\pi$
$151$			7.74597i			0.630358i	−0.949032	+	0.315179i	$0.897935\pi$
$152$		0			0
$153$		0			0
$154$	15.0000	+	19.3649i	1.20873	+	1.56047i
$155$		0			0
$156$		0			0
$157$	17.1464			1.36843			0.684217	−	0.729279i	$-0.260144\pi$
$157$	17.1464			1.36843			0.684217	+	0.729279i	$0.260144\pi$
$158$	−6.70820	−	8.66025i	−0.533676	−	0.688973i
$159$		0			0
$160$		0			0
$161$		−	14.1421i		−	1.11456i
$162$		0			0
$163$		−	6.32456i		−	0.495377i	−0.968840	−	0.247689i	$-0.920329\pi$
$163$		−	6.32456i		−	0.495377i	0.968840	−	0.247689i	$-0.0796710\pi$
$164$	2.73861	+	0.707107i	0.213850	+	0.0552158i
$165$		0			0
$166$	−10.0000	+	7.74597i	−0.776151	+	0.601204i
$167$	−4.47214			−0.346064			−0.173032	−	0.984916i	$-0.555356\pi$
$167$	−4.47214			−0.346064			−0.173032	+	0.984916i	$0.555356\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$		0			0
$172$	−12.2474	−	3.16228i	−0.933859	−	0.241121i
$173$		−	13.8564i		−	1.05348i	−0.850026	−	0.526742i	$-0.823414\pi$
$173$		−	13.8564i		−	1.05348i	0.850026	−	0.526742i	$-0.176586\pi$
$174$		0			0
$175$		0			0
$176$	19.1703	+	10.6066i	1.44501	+	0.799503i
$177$		0			0
$178$	−8.57321	−	11.0680i	−0.642590	−	0.829580i
$179$	16.4317			1.22816			0.614081	−	0.789243i	$-0.289527\pi$
$179$	16.4317			1.22816			0.614081	+	0.789243i	$0.289527\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	6.70820	+	8.66025i	0.497245	+	0.641941i
$183$		0			0
$184$	−5.00000	−	11.6190i	−0.368605	−	0.856560i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	4.47214	−	17.3205i	0.326164	−	1.26323i
$189$		0			0
$190$		0			0
$191$	10.9545			0.792636			0.396318	−	0.918113i	$-0.370288\pi$
$191$	10.9545			0.792636			0.396318	+	0.918113i	$0.370288\pi$
$192$		0			0
$193$	−9.79796			−0.705273			−0.352636	−	0.935760i	$-0.614715\pi$
$193$	−9.79796			−0.705273			−0.352636	+	0.935760i	$0.614715\pi$
$194$	5.47723	−	4.24264i	0.393242	−	0.304604i
$195$		0			0
$196$	−1.50000	+	5.80948i	−0.107143	+	0.414963i
$197$		−	10.3923i		−	0.740421i	−0.928948	−	0.370211i	$-0.879286\pi$
$197$		−	10.3923i		−	0.740421i	0.928948	−	0.370211i	$-0.120714\pi$
$198$		0			0
$199$			23.2379i			1.64729i	0.567105	+	0.823646i	$0.308063\pi$
$199$			23.2379i			1.64729i	−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$	4.89898	+	6.32456i	0.344691	+	0.444994i
$203$	8.94427			0.627765
$204$		0			0
$205$		0			0
$206$	2.73861	+	3.53553i	0.190808	+	0.246332i
$207$		0			0
$208$	8.57321	+	4.74342i	0.594445	+	0.328897i
$209$		0			0
$210$		0			0
$211$		−	7.74597i		−	0.533254i	−0.963800	−	0.266627i	$-0.914091\pi$
$211$		−	7.74597i		−	0.533254i	0.963800	−	0.266627i	$-0.0859092\pi$
$212$		0			0
$213$		0			0
$214$	10.0000	−	7.74597i	0.683586	−	0.529503i
$215$		0			0
$216$		0			0
$217$	−24.4949			−1.66282
$218$	11.1803	−	8.66025i	0.757228	−	0.586546i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			3.16228i			0.211762i	0.994379	+	0.105881i	$0.0337662\pi$
$223$			3.16228i			0.211762i	−0.994379	+	0.105881i	$0.966234\pi$
$224$	2.73861	+	17.6777i	0.182981	+	1.18114i
$225$		0			0
$226$	12.0000	+	15.4919i	0.798228	+	1.03051i
$227$	−17.8885			−1.18730			−0.593652	−	0.804722i	$-0.702314\pi$
$227$	−17.8885			−1.18730			−0.593652	+	0.804722i	$0.702314\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$		0			0
$232$	7.34847	−	3.16228i	0.482451	−	0.207614i
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$	−2.73861	+	10.6066i	−0.178269	+	0.690431i
$237$		0			0
$238$		0			0
$239$	10.9545			0.708585			0.354292	−	0.935135i	$-0.384722\pi$
$239$	10.9545			0.708585			0.354292	+	0.935135i	$0.384722\pi$
$240$		0			0
$241$	8.00000			0.515325			0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000			0.515325			0.257663	+	0.966235i	$0.417048\pi$
$242$	21.2426	−	16.4545i	1.36553	−	1.05773i
$243$		0			0
$244$	−5.00000	+	19.3649i	−0.320092	+	1.23971i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−20.1246	+	8.66025i	−1.27791	+	0.549927i
$249$		0			0
$250$		0			0
$251$	16.4317			1.03716			0.518579	−	0.855030i	$-0.326461\pi$
$251$	16.4317			1.03716			0.518579	+	0.855030i	$0.326461\pi$
$252$		0			0
$253$	−24.4949			−1.53998
$254$	−2.73861	−	3.53553i	−0.171836	−	0.221839i
$255$		0			0
$256$	8.50000	+	13.5554i	0.531250	+	0.847215i
$257$			6.92820i			0.432169i	0.976375	+	0.216085i	$0.0693287\pi$
$257$			6.92820i			0.432169i	−0.976375	+	0.216085i	$0.930671\pi$
$258$		0			0
$259$		−	23.2379i		−	1.44393i
$260$		0			0
$261$		0			0
$262$	6.12372	−	4.74342i	0.378325	−	0.293049i
$263$	−22.3607			−1.37882			−0.689409	−	0.724372i	$-0.742130\pi$
$263$	−22.3607			−1.37882			−0.689409	+	0.724372i	$0.742130\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−24.4949	−	6.32456i	−1.49626	−	0.386334i
$269$			2.82843i			0.172452i	0.996276	+	0.0862261i	$0.0274808\pi$
$269$			2.82843i			0.172452i	−0.996276	+	0.0862261i	$0.972519\pi$
$270$		0			0
$271$			23.2379i			1.41160i	0.708410	+	0.705801i	$0.249413\pi$
$271$			23.2379i			1.41160i	−0.708410	+	0.705801i	$0.750587\pi$
$272$		0			0
$273$		0			0
$274$	−6.00000	−	7.74597i	−0.362473	−	0.467951i
$275$		0			0
$276$		0			0
$277$	−17.1464			−1.03023			−0.515115	−	0.857121i	$-0.672251\pi$
$277$	−17.1464			−1.03023			−0.515115	+	0.857121i	$0.672251\pi$
$278$	6.70820	+	8.66025i	0.402331	+	0.519408i
$279$		0			0
$280$		0			0
$281$		−	15.5563i		−	0.928014i	−0.885832	−	0.464007i	$-0.846411\pi$
$281$		−	15.5563i		−	0.928014i	0.885832	−	0.464007i	$-0.153589\pi$
$282$		0			0
$283$			12.6491i			0.751912i	0.926638	+	0.375956i	$0.122686\pi$
$283$			12.6491i			0.751912i	−0.926638	+	0.375956i	$0.877314\pi$
$284$		0			0
$285$		0			0
$286$	15.0000	−	11.6190i	0.886969	−	0.687043i
$287$	4.47214			0.263982
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$	7.34847	−	28.4605i	0.430037	−	1.66552i
$293$			3.46410i			0.202375i	0.994867	+	0.101187i	$0.0322642\pi$
$293$			3.46410i			0.202375i	−0.994867	+	0.101187i	$0.967736\pi$
$294$		0			0
$295$		0			0
$296$	−8.21584	−	19.0919i	−0.477536	−	1.10969i
$297$		0			0
$298$	9.79796	+	12.6491i	0.567581	+	0.732743i
$299$	−10.9545			−0.633512
$300$		0			0
$301$	−20.0000			−1.15278
$302$	6.70820	+	8.66025i	0.386014	+	0.498342i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			25.2982i			1.44385i	0.691974	+	0.721923i	$0.256741\pi$
$307$			25.2982i			1.44385i	−0.691974	+	0.721923i	$0.743259\pi$
$308$	33.5410	+	8.66025i	1.91118	+	0.493464i
$309$		0			0
$310$		0			0
$311$	−10.9545			−0.621170			−0.310585	−	0.950546i	$-0.600525\pi$
$311$	−10.9545			−0.621170			−0.310585	+	0.950546i	$0.600525\pi$
$312$		0			0
$313$	9.79796			0.553813			0.276907	−	0.960897i	$-0.410691\pi$
$313$	9.79796			0.553813			0.276907	+	0.960897i	$0.410691\pi$
$314$	19.1703	−	14.8492i	1.08184	−	0.837991i
$315$		0			0
$316$	−15.0000	−	3.87298i	−0.843816	−	0.217872i
$317$		−	17.3205i		−	0.972817i	−0.873732	−	0.486408i	$-0.838307\pi$
$317$		−	17.3205i		−	0.972817i	0.873732	−	0.486408i	$-0.161693\pi$
$318$		0			0
$319$		−	15.4919i		−	0.867382i
$320$		0			0
$321$		0			0
$322$	−12.2474	−	15.8114i	−0.682524	−	0.881134i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−5.47723	−	7.07107i	−0.303355	−	0.391630i
$327$		0			0
$328$	3.67423	−	1.58114i	0.202876	−	0.0873038i
$329$		−	28.2843i		−	1.55936i
$330$		0			0
$331$		−	15.4919i		−	0.851514i	−0.904838	−	0.425757i	$-0.860008\pi$
$331$		−	15.4919i		−	0.851514i	0.904838	−	0.425757i	$-0.139992\pi$
$332$	−4.47214	+	17.3205i	−0.245440	+	0.950586i
$333$		0			0
$334$	−5.00000	+	3.87298i	−0.273588	+	0.211920i
$335$		0			0
$336$		0			0
$337$	−4.89898			−0.266864			−0.133432	−	0.991058i	$-0.542600\pi$
$337$	−4.89898			−0.266864			−0.133432	+	0.991058i	$0.542600\pi$
$338$	−7.82624	+	6.06218i	−0.425691	+	0.329739i
$339$		0			0
$340$		0			0
$341$			42.4264i			2.29752i
$342$		0			0
$343$		−	12.6491i		−	0.682988i
$344$	−16.4317	+	7.07107i	−0.885937	+	0.381246i
$345$		0			0
$346$	−12.0000	−	15.4919i	−0.645124	−	0.832851i
$347$	8.94427			0.480154			0.240077	−	0.970754i	$-0.422827\pi$
$347$	8.94427			0.480154			0.240077	+	0.970754i	$0.422827\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$		0			0
$352$	30.6186	−	4.74342i	1.63198	−	0.252825i
$353$		−	34.6410i		−	1.84376i	−0.387481	−	0.921878i	$-0.626655\pi$
$353$		−	34.6410i		−	1.84376i	0.387481	−	0.921878i	$-0.373345\pi$
$354$		0			0
$355$		0			0
$356$	−19.1703	−	4.94975i	−1.01602	−	0.262336i
$357$		0			0
$358$	18.3712	−	14.2302i	0.970947	−	0.752092i
$359$	−32.8634			−1.73446			−0.867231	−	0.497906i	$-0.834102\pi$
$359$	−32.8634			−1.73446			−0.867231	+	0.497906i	$0.834102\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$	−11.1803	+	8.66025i	−0.587626	+	0.455173i
$363$		0			0
$364$	15.0000	+	3.87298i	0.786214	+	0.202999i
$365$		0			0
$366$		0			0
$367$			34.7851i			1.81577i	0.419225	+	0.907883i	$0.362302\pi$
$367$			34.7851i			1.81577i	−0.419225	+	0.907883i	$0.637698\pi$
$368$	−15.6525	−	8.66025i	−0.815942	−	0.451447i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	26.9444			1.39513			0.697564	−	0.716523i	$-0.254267\pi$
$373$	26.9444			1.39513			0.697564	+	0.716523i	$0.254267\pi$
$374$		0			0
$375$		0			0
$376$	−10.0000	−	23.2379i	−0.515711	−	1.19840i
$377$		−	6.92820i		−	0.356821i
$378$		0			0
$379$			23.2379i			1.19365i	0.802371	+	0.596825i	$0.203571\pi$
$379$			23.2379i			1.19365i	−0.802371	+	0.596825i	$0.796429\pi$
$380$		0			0
$381$		0			0
$382$	12.2474	−	9.48683i	0.626634	−	0.485389i
$383$	−8.94427			−0.457031			−0.228515	−	0.973540i	$-0.573387\pi$
$383$	−8.94427			−0.457031			−0.228515	+	0.973540i	$0.573387\pi$
$384$		0			0
$385$		0			0
$386$	−10.9545	+	8.48528i	−0.557567	+	0.431889i
$387$		0			0
$388$	2.44949	−	9.48683i	0.124354	−	0.481621i
$389$			2.82843i			0.143407i	0.997426	+	0.0717035i	$0.0228435\pi$
$389$			2.82843i			0.143407i	−0.997426	+	0.0717035i	$0.977156\pi$
$390$		0			0
$391$		0			0
$392$	3.35410	+	7.79423i	0.169408	+	0.393668i
$393$		0			0
$394$	−9.00000	−	11.6190i	−0.453413	−	0.585354i
$395$		0			0
$396$		0			0
$397$	−7.34847			−0.368809			−0.184405	−	0.982850i	$-0.559036\pi$
$397$	−7.34847			−0.368809			−0.184405	+	0.982850i	$0.559036\pi$
$398$	20.1246	+	25.9808i	1.00876	+	1.30230i
$399$		0			0
$400$		0			0
$401$		−	15.5563i		−	0.776847i	−0.921481	−	0.388424i	$-0.873020\pi$
$401$		−	15.5563i		−	0.776847i	0.921481	−	0.388424i	$-0.126980\pi$
$402$		0			0
$403$			18.9737i			0.945146i
$404$	10.9545	+	2.82843i	0.545004	+	0.140720i
$405$		0			0
$406$	10.0000	−	7.74597i	0.496292	−	0.384426i
$407$	−40.2492			−1.99508
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		0			0
$412$	6.12372	+	1.58114i	0.301694	+	0.0778971i
$413$			17.3205i			0.852286i
$414$		0			0
$415$		0			0
$416$	13.6931	−	2.12132i	0.671358	−	0.104006i
$417$		0			0
$418$		0			0
$419$	−5.47723			−0.267580			−0.133790	−	0.991010i	$-0.542715\pi$
$419$	−5.47723			−0.267580			−0.133790	+	0.991010i	$0.542715\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$	−6.70820	−	8.66025i	−0.326550	−	0.421575i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$			31.6228i			1.53033i
$428$	4.47214	−	17.3205i	0.216169	−	0.837218i
$429$		0			0
$430$		0			0
$431$	−32.8634			−1.58297			−0.791486	−	0.611187i	$-0.790692\pi$
$431$	−32.8634			−1.58297			−0.791486	+	0.611187i	$0.790692\pi$
$432$		0			0
$433$	−14.6969			−0.706290			−0.353145	−	0.935569i	$-0.614888\pi$
$433$	−14.6969			−0.706290			−0.353145	+	0.935569i	$0.614888\pi$
$434$	−27.3861	+	21.2132i	−1.31458	+	1.01827i
$435$		0			0
$436$	5.00000	−	19.3649i	0.239457	−	0.927411i
$437$		0			0
$438$		0			0
$439$		−	7.74597i		−	0.369695i	−0.982767	−	0.184847i	$-0.940821\pi$
$439$		−	7.74597i		−	0.369695i	0.982767	−	0.184847i	$-0.0591791\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.94427			−0.424955			−0.212478	−	0.977166i	$-0.568153\pi$
$443$	−8.94427			−0.424955			−0.212478	+	0.977166i	$0.568153\pi$
$444$		0			0
$445$		0			0
$446$	2.73861	+	3.53553i	0.129677	+	0.167412i
$447$		0			0
$448$	18.3712	+	17.3925i	0.867956	+	0.821720i
$449$			32.5269i			1.53504i	0.641025	+	0.767520i	$0.278509\pi$
$449$			32.5269i			1.53504i	−0.641025	+	0.767520i	$0.721491\pi$
$450$		0			0
$451$		−	7.74597i		−	0.364743i
$452$	26.8328	+	6.92820i	1.26211	+	0.325875i
$453$		0			0
$454$	−20.0000	+	15.4919i	−0.938647	+	0.727072i
$455$		0			0
$456$		0			0
$457$	4.89898			0.229165			0.114582	−	0.993414i	$-0.463447\pi$
$457$	4.89898			0.229165			0.114582	+	0.993414i	$0.463447\pi$
$458$	−15.6525	+	12.1244i	−0.731392	+	0.566534i
$459$		0			0
$460$		0			0
$461$			22.6274i			1.05386i	0.849907	+	0.526932i	$0.176658\pi$
$461$			22.6274i			1.05386i	−0.849907	+	0.526932i	$0.823342\pi$
$462$		0			0
$463$			3.16228i			0.146964i	0.997297	+	0.0734818i	$0.0234111\pi$
$463$			3.16228i			0.146964i	−0.997297	+	0.0734818i	$0.976589\pi$
$464$	5.47723	−	9.89949i	0.254274	−	0.459573i
$465$		0			0
$466$		0			0
$467$	−17.8885			−0.827783			−0.413892	−	0.910326i	$-0.635831\pi$
$467$	−17.8885			−0.827783			−0.413892	+	0.910326i	$0.635831\pi$
$468$		0			0
$469$	−40.0000			−1.84703
$470$		0			0
$471$		0			0
$472$	6.12372	+	14.2302i	0.281867	+	0.655000i
$473$			34.6410i			1.59280i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	12.2474	−	9.48683i	0.560185	−	0.433918i
$479$	21.9089			1.00104			0.500522	−	0.865724i	$-0.333142\pi$
$479$	21.9089			1.00104			0.500522	+	0.865724i	$0.333142\pi$
$480$		0			0
$481$	−18.0000			−0.820729
$482$	8.94427	−	6.92820i	0.407400	−	0.315571i
$483$		0			0
$484$	9.50000	−	36.7933i	0.431818	−	1.67242i
$485$		0			0
$486$		0			0
$487$		−	22.1359i		−	1.00308i	−0.865136	−	0.501538i	$-0.832768\pi$
$487$		−	22.1359i		−	1.00308i	0.865136	−	0.501538i	$-0.167232\pi$
$488$	11.1803	+	25.9808i	0.506110	+	1.17609i
$489$		0			0
$490$		0			0
$491$	−27.3861			−1.23592			−0.617959	−	0.786210i	$-0.712040\pi$
$491$	−27.3861			−1.23592			−0.617959	+	0.786210i	$0.712040\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−15.0000	+	27.1109i	−0.673520	+	1.21731i
$497$		0			0
$498$		0			0
$499$		−	15.4919i		−	0.693514i	−0.937955	−	0.346757i	$-0.887283\pi$
$499$		−	15.4919i		−	0.693514i	0.937955	−	0.346757i	$-0.112717\pi$
$500$		0			0
$501$		0			0
$502$	18.3712	−	14.2302i	0.819946	−	0.635127i
$503$	−8.94427			−0.398805			−0.199403	−	0.979918i	$-0.563900\pi$
$503$	−8.94427			−0.398805			−0.199403	+	0.979918i	$0.563900\pi$
$504$		0			0
$505$		0			0
$506$	−27.3861	+	21.2132i	−1.21746	+	0.943042i
$507$		0			0
$508$	−6.12372	−	1.58114i	−0.271696	−	0.0701517i
$509$			11.3137i			0.501471i	0.968056	+	0.250736i	$0.0806725\pi$
$509$			11.3137i			0.501471i	−0.968056	+	0.250736i	$0.919328\pi$
$510$		0			0
$511$		−	46.4758i		−	2.05597i
$512$	21.2426	+	7.79423i	0.938801	+	0.344459i
$513$		0			0
$514$	6.00000	+	7.74597i	0.264649	+	0.341660i
$515$		0			0
$516$		0			0
$517$	−48.9898			−2.15457
$518$	−20.1246	−	25.9808i	−0.884225	−	1.14153i
$519$		0			0
$520$		0			0
$521$		−	41.0122i		−	1.79678i	−0.439202	−	0.898388i	$-0.644739\pi$
$521$		−	41.0122i		−	1.79678i	0.439202	−	0.898388i	$-0.355261\pi$
$522$		0			0
$523$			31.6228i			1.38277i	0.722488	+	0.691384i	$0.242999\pi$
$523$			31.6228i			1.38277i	−0.722488	+	0.691384i	$0.757001\pi$
$524$	2.73861	−	10.6066i	0.119637	−	0.463352i
$525$		0			0
$526$	−25.0000	+	19.3649i	−1.09005	+	0.844350i
$527$		0			0
$528$		0			0
$529$	−3.00000			−0.130435
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	3.46410i		−	0.150047i
$534$		0			0
$535$		0			0
$536$	−32.8634	+	14.1421i	−1.41948	+	0.610847i
$537$		0			0
$538$	2.44949	+	3.16228i	0.105605	+	0.136335i
$539$	16.4317			0.707762
$540$		0			0
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$	20.1246	+	25.9808i	0.864426	+	1.11597i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	31.6228i		−	1.35209i	−0.736859	−	0.676046i	$-0.763692\pi$
$547$		−	31.6228i		−	1.35209i	0.736859	−	0.676046i	$-0.236308\pi$
$548$	−13.4164	−	3.46410i	−0.573121	−	0.147979i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−24.4949			−1.04163
$554$	−19.1703	+	14.8492i	−0.814468	+	0.630884i
$555$		0			0
$556$	15.0000	+	3.87298i	0.636142	+	0.164251i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$			15.4919i			0.655239i
$560$		0			0
$561$		0			0
$562$	−13.4722	−	17.3925i	−0.568290	−	0.733659i
$563$	44.7214			1.88478			0.942390	−	0.334515i	$-0.108573\pi$
$563$	44.7214			1.88478			0.942390	+	0.334515i	$0.108573\pi$
$564$		0			0
$565$		0			0
$566$	10.9545	+	14.1421i	0.460450	+	0.594438i
$567$		0			0
$568$		0			0
$569$		−	18.3848i		−	0.770730i	−0.922764	−	0.385365i	$-0.874076\pi$
$569$		−	18.3848i		−	0.770730i	0.922764	−	0.385365i	$-0.125924\pi$
$570$		0			0
$571$			15.4919i			0.648317i	0.946003	+	0.324159i	$0.105081\pi$
$571$			15.4919i			0.648317i	−0.946003	+	0.324159i	$0.894919\pi$
$572$	6.70820	−	25.9808i	0.280484	−	1.08631i
$573$		0			0
$574$	5.00000	−	3.87298i	0.208696	−	0.161655i
$575$		0			0
$576$		0			0
$577$	44.0908			1.83552			0.917762	−	0.397130i	$-0.129994\pi$
$577$	44.0908			1.83552			0.917762	+	0.397130i	$0.129994\pi$
$578$	19.0066	−	14.7224i	0.790569	−	0.612372i
$579$		0			0
$580$		0			0
$581$			28.2843i			1.17343i
$582$		0			0
$583$		0			0
$584$	−16.4317	−	38.1838i	−0.679948	−	1.58006i
$585$		0			0
$586$	3.00000	+	3.87298i	0.123929	+	0.159991i
$587$	8.94427			0.369170			0.184585	−	0.982817i	$-0.440906\pi$
$587$	8.94427			0.369170			0.184585	+	0.982817i	$0.440906\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−25.7196	−	14.2302i	−1.05707	−	0.584860i
$593$		−	20.7846i		−	0.853522i	−0.904365	−	0.426761i	$-0.859655\pi$
$593$		−	20.7846i		−	0.853522i	0.904365	−	0.426761i	$-0.140345\pi$
$594$		0			0
$595$		0			0
$596$	21.9089	+	5.65685i	0.897424	+	0.231714i
$597$		0			0
$598$	−12.2474	+	9.48683i	−0.500835	+	0.387945i
$599$	10.9545			0.447587			0.223793	−	0.974637i	$-0.428156\pi$
$599$	10.9545			0.447587			0.223793	+	0.974637i	$0.428156\pi$
$600$		0			0
$601$	20.0000			0.815817			0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000			0.815817			0.407909	+	0.913023i	$0.366258\pi$
$602$	−22.3607	+	17.3205i	−0.911353	+	0.705931i
$603$		0			0
$604$	15.0000	+	3.87298i	0.610341	+	0.157589i
$605$		0			0
$606$		0			0
$607$			15.8114i			0.641764i	0.947119	+	0.320882i	$0.103979\pi$
$607$			15.8114i			0.641764i	−0.947119	+	0.320882i	$0.896021\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21.9089			−0.886339
$612$		0			0
$613$	22.0454			0.890406			0.445203	−	0.895430i	$-0.353132\pi$
$613$	22.0454			0.890406			0.445203	+	0.895430i	$0.353132\pi$
$614$	21.9089	+	28.2843i	0.884171	+	1.14146i
$615$		0			0
$616$	45.0000	−	19.3649i	1.81310	−	0.780235i
$617$		−	34.6410i		−	1.39459i	−0.716782	−	0.697297i	$-0.754386\pi$
$617$		−	34.6410i		−	1.39459i	0.716782	−	0.697297i	$-0.245614\pi$
$618$		0			0
$619$		−	7.74597i		−	0.311337i	−0.987809	−	0.155668i	$-0.950247\pi$
$619$		−	7.74597i		−	0.311337i	0.987809	−	0.155668i	$-0.0497531\pi$
$620$		0			0
$621$		0			0
$622$	−12.2474	+	9.48683i	−0.491078	+	0.380387i
$623$	−31.3050			−1.25421
$624$		0			0
$625$		0			0
$626$	10.9545	−	8.48528i	0.437828	−	0.339140i
$627$		0			0
$628$	8.57321	−	33.2039i	0.342108	−	1.32498i
$629$		0			0
$630$		0			0
$631$		−	23.2379i		−	0.925086i	−0.886597	−	0.462543i	$-0.846937\pi$
$631$		−	23.2379i		−	0.925086i	0.886597	−	0.462543i	$-0.153063\pi$
$632$	−20.1246	+	8.66025i	−0.800514	+	0.344486i
$633$		0			0
$634$	−15.0000	−	19.3649i	−0.595726	−	0.769079i
$635$		0			0
$636$		0			0
$637$	7.34847			0.291157
$638$	−13.4164	−	17.3205i	−0.531161	−	0.685725i
$639$		0			0
$640$		0			0
$641$		−	41.0122i		−	1.61988i	−0.586510	−	0.809942i	$-0.699498\pi$
$641$		−	41.0122i		−	1.61988i	0.586510	−	0.809942i	$-0.300502\pi$
$642$		0			0
$643$		−	44.2719i		−	1.74591i	−0.487798	−	0.872956i	$-0.662200\pi$
$643$		−	44.2719i		−	1.74591i	0.487798	−	0.872956i	$-0.337800\pi$
$644$	−27.3861	−	7.07107i	−1.07916	−	0.278639i
$645$		0			0
$646$		0			0
$647$	8.94427			0.351636			0.175818	−	0.984423i	$-0.443743\pi$
$647$	8.94427			0.351636			0.175818	+	0.984423i	$0.443743\pi$
$648$		0			0
$649$	30.0000			1.17760
$650$		0			0
$651$		0			0
$652$	−12.2474	−	3.16228i	−0.479647	−	0.123844i
$653$		−	17.3205i		−	0.677804i	−0.940822	−	0.338902i	$-0.889945\pi$
$653$		−	17.3205i		−	0.677804i	0.940822	−	0.338902i	$-0.110055\pi$
$654$		0			0
$655$		0			0
$656$	2.73861	−	4.94975i	0.106925	−	0.193255i
$657$		0			0
$658$	−24.4949	−	31.6228i	−0.954911	−	1.23278i
$659$	−27.3861			−1.06681			−0.533406	−	0.845859i	$-0.679088\pi$
$659$	−27.3861			−1.06681			−0.533406	+	0.845859i	$0.679088\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$	−13.4164	−	17.3205i	−0.521443	−	0.673181i
$663$		0			0
$664$	10.0000	+	23.2379i	0.388075	+	0.901805i
$665$		0			0
$666$		0			0
$667$			12.6491i			0.489776i
$668$	−2.23607	+	8.66025i	−0.0865161	+	0.335075i
$669$		0			0
$670$		0			0
$671$	54.7723			2.11446
$672$		0			0
$673$	−34.2929			−1.32189			−0.660946	−	0.750433i	$-0.729845\pi$
$673$	−34.2929			−1.32189			−0.660946	+	0.750433i	$0.729845\pi$
$674$	−5.47723	+	4.24264i	−0.210975	+	0.163420i
$675$		0			0
$676$	−3.50000	+	13.5554i	−0.134615	+	0.521363i
$677$		−	17.3205i		−	0.665681i	−0.942983	−	0.332841i	$-0.891993\pi$
$677$		−	17.3205i		−	0.665681i	0.942983	−	0.332841i	$-0.108007\pi$
$678$		0			0
$679$		−	15.4919i		−	0.594526i
$680$		0			0
$681$		0			0
$682$	36.7423	+	47.4342i	1.40694	+	1.81635i
$683$	17.8885			0.684486			0.342243	−	0.939611i	$-0.388813\pi$
$683$	17.8885			0.684486			0.342243	+	0.939611i	$0.388813\pi$
$684$		0			0
$685$		0			0
$686$	−10.9545	−	14.1421i	−0.418243	−	0.539949i
$687$		0			0
$688$	−12.2474	+	22.1359i	−0.466930	+	0.843925i
$689$		0			0
$690$		0			0
$691$			30.9839i			1.17868i	0.807884	+	0.589341i	$0.200613\pi$
$691$			30.9839i			1.17868i	−0.807884	+	0.589341i	$0.799387\pi$
$692$	−26.8328	−	6.92820i	−1.02003	−	0.263371i
$693$		0			0
$694$	10.0000	−	7.74597i	0.379595	−	0.294033i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	11.1803	−	8.66025i	0.423182	−	0.327795i
$699$		0			0
$700$		0			0
$701$			22.6274i			0.854626i	0.904104	+	0.427313i	$0.140540\pi$
$701$			22.6274i			0.854626i	−0.904104	+	0.427313i	$0.859460\pi$
$702$		0			0
$703$		0			0
$704$	30.1247	−	31.8198i	1.13537	−	1.19925i
$705$		0			0
$706$	−30.0000	−	38.7298i	−1.12906	−	1.45762i
$707$	17.8885			0.672768
$708$		0			0
$709$	−26.0000			−0.976450			−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000			−0.976450			−0.488225	+	0.872718i	$0.662356\pi$
$710$		0			0
$711$		0			0
$712$	−25.7196	+	11.0680i	−0.963884	+	0.414790i
$713$		−	34.6410i		−	1.29732i
$714$		0			0
$715$		0			0
$716$	8.21584	−	31.8198i	0.307040	−	1.18916i
$717$		0			0
$718$	−36.7423	+	28.4605i	−1.37121	+	1.06214i
$719$	−32.8634			−1.22560			−0.612798	−	0.790239i	$-0.709956\pi$
$719$	−32.8634			−1.22560			−0.612798	+	0.790239i	$0.709956\pi$
$720$		0			0
$721$	10.0000			0.372419
$722$	21.2426	−	16.4545i	0.790569	−	0.612372i
$723$		0			0
$724$	−5.00000	+	19.3649i	−0.185824	+	0.719691i
$725$		0			0
$726$		0			0
$727$		−	3.16228i		−	0.117282i	−0.998279	−	0.0586412i	$-0.981323\pi$
$727$		−	3.16228i		−	0.117282i	0.998279	−	0.0586412i	$-0.0186768\pi$
$728$	20.1246	−	8.66025i	0.745868	−	0.320970i
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−2.44949			−0.0904740			−0.0452370	−	0.998976i	$-0.514404\pi$
$733$	−2.44949			−0.0904740			−0.0452370	+	0.998976i	$0.514404\pi$
$734$	30.1247	+	38.8909i	1.11192	+	1.43549i
$735$		0			0
$736$	−25.0000	+	3.87298i	−0.921512	+	0.142760i
$737$			69.2820i			2.55204i
$738$		0			0
$739$			46.4758i			1.70964i	0.518925	+	0.854820i	$0.326333\pi$
$739$			46.4758i			1.70964i	−0.518925	+	0.854820i	$0.673667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	44.7214			1.64067			0.820334	−	0.571885i	$-0.193788\pi$
$743$	44.7214			1.64067			0.820334	+	0.571885i	$0.193788\pi$
$744$		0			0
$745$		0			0
$746$	30.1247	−	23.3345i	1.10295	−	0.854338i
$747$		0			0
$748$		0			0
$749$		−	28.2843i		−	1.03348i
$750$		0			0
$751$			38.7298i			1.41327i	0.707577	+	0.706636i	$0.249788\pi$
$751$			38.7298i			1.41327i	−0.707577	+	0.706636i	$0.750212\pi$
$752$	−31.3050	−	17.3205i	−1.14157	−	0.631614i
$753$		0			0
$754$	−6.00000	−	7.74597i	−0.218507	−	0.282091i
$755$		0			0
$756$		0			0
$757$	−7.34847			−0.267085			−0.133542	−	0.991043i	$-0.542635\pi$
$757$	−7.34847			−0.267085			−0.133542	+	0.991043i	$0.542635\pi$
$758$	20.1246	+	25.9808i	0.730959	+	0.943664i
$759$		0			0
$760$		0			0
$761$			1.41421i			0.0512652i	0.999671	+	0.0256326i	$0.00816000\pi$
$761$			1.41421i			0.0512652i	−0.999671	+	0.0256326i	$0.991840\pi$
$762$		0			0
$763$		−	31.6228i		−	1.14482i
$764$	5.47723	−	21.2132i	0.198159	−	0.767467i
$765$		0			0
$766$	−10.0000	+	7.74597i	−0.361315	+	0.279873i
$767$	13.4164			0.484438
$768$		0			0
$769$	40.0000			1.44244			0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000			1.44244			0.721218	+	0.692708i	$0.243582\pi$
$770$		0			0
$771$		0			0
$772$	−4.89898	+	18.9737i	−0.176318	+	0.682877i
$773$			51.9615i			1.86893i	0.356060	+	0.934463i	$0.384120\pi$
$773$			51.9615i			1.86893i	−0.356060	+	0.934463i	$0.615880\pi$
$774$		0			0
$775$		0			0
$776$	−5.47723	−	12.7279i	−0.196621	−	0.456906i
$777$		0			0
$778$	2.44949	+	3.16228i	0.0878185	+	0.113373i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	10.5000	+	5.80948i	0.375000	+	0.207481i
$785$		0			0
$786$		0			0
$787$		−	31.6228i		−	1.12723i	−0.826038	−	0.563615i	$-0.809410\pi$
$787$		−	31.6228i		−	1.12723i	0.826038	−	0.563615i	$-0.190590\pi$
$788$	−20.1246	−	5.19615i	−0.716910	−	0.185105i
$789$		0			0
$790$		0			0
$791$	43.8178			1.55798
$792$		0			0
$793$	24.4949			0.869839
$794$	−8.21584	+	6.36396i	−0.291569	+	0.225849i
$795$		0			0
$796$	45.0000	+	11.6190i	1.59498	+	0.411823i
$797$			27.7128i			0.981638i	0.871262	+	0.490819i	$0.163302\pi$
$797$			27.7128i			0.981638i	−0.871262	+	0.490819i	$0.836698\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	−13.4722	−	17.3925i	−0.475720	−	0.614151i
$803$	−80.4984			−2.84073
$804$		0			0
$805$		0			0
$806$	16.4317	+	21.2132i	0.578781	+	0.747203i
$807$		0			0
$808$	14.6969	−	6.32456i	0.517036	−	0.222497i
$809$		−	18.3848i		−	0.646374i	−0.946335	−	0.323187i	$-0.895246\pi$
$809$		−	18.3848i		−	0.646374i	0.946335	−	0.323187i	$-0.104754\pi$
$810$		0			0
$811$			23.2379i			0.815993i	0.912983	+	0.407997i	$0.133772\pi$
$811$			23.2379i			0.815993i	−0.912983	+	0.407997i	$0.866228\pi$
$812$	4.47214	−	17.3205i	0.156941	−	0.607831i
$813$		0			0
$814$	−45.0000	+	34.8569i	−1.57725	+	1.22173i
$815$		0			0
$816$		0			0
$817$		0			0
$818$	11.1803	−	8.66025i	0.390911	−	0.302799i
$819$		0			0
$820$		0			0
$821$			48.0833i			1.67812i	0.544041	+	0.839059i	$0.316894\pi$
$821$			48.0833i			1.67812i	−0.544041	+	0.839059i	$0.683106\pi$
$822$		0			0
$823$		−	15.8114i		−	0.551150i	−0.961280	−	0.275575i	$-0.911132\pi$
$823$		−	15.8114i		−	0.551150i	0.961280	−	0.275575i	$-0.0888683\pi$
$824$	8.21584	−	3.53553i	0.286212	−	0.123166i
$825$		0			0
$826$	15.0000	+	19.3649i	0.521917	+	0.673792i
$827$	8.94427			0.311023			0.155511	−	0.987834i	$-0.450297\pi$
$827$	8.94427			0.311023			0.155511	+	0.987834i	$0.450297\pi$
$828$		0			0
$829$	10.0000			0.347314			0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000			0.347314			0.173657	+	0.984806i	$0.444442\pi$
$830$		0			0
$831$		0			0
$832$	13.4722	−	14.2302i	0.467064	−	0.493345i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−6.12372	+	4.74342i	−0.211541	+	0.163859i
$839$	54.7723			1.89095			0.945474	−	0.325697i	$-0.105599\pi$
$839$	54.7723			1.89095			0.945474	+	0.325697i	$0.105599\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$	2.23607	−	1.73205i	0.0770600	−	0.0596904i
$843$		0			0
$844$	−15.0000	−	3.87298i	−0.516321	−	0.133314i
$845$		0			0
$846$		0			0
$847$		−	60.0833i		−	2.06449i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	32.8634			1.12654
$852$		0			0
$853$	2.44949			0.0838689			0.0419345	−	0.999120i	$-0.486648\pi$
$853$	2.44949			0.0838689			0.0419345	+	0.999120i	$0.486648\pi$
$854$	27.3861	+	35.3553i	0.937134	+	1.20983i
$855$		0			0
$856$	−10.0000	−	23.2379i	−0.341793	−	0.794255i
$857$			34.6410i			1.18331i	0.806190	+	0.591657i	$0.201526\pi$
$857$			34.6410i			1.18331i	−0.806190	+	0.591657i	$0.798474\pi$
$858$		0			0
$859$		−	38.7298i		−	1.32144i	−0.750630	−	0.660722i	$-0.770250\pi$
$859$		−	38.7298i		−	1.32144i	0.750630	−	0.660722i	$-0.229750\pi$
$860$		0			0
$861$		0			0
$862$	−36.7423	+	28.4605i	−1.25145	+	0.969368i
$863$	31.3050			1.06563			0.532816	−	0.846231i	$-0.321134\pi$
$863$	31.3050			1.06563			0.532816	+	0.846231i	$0.321134\pi$
$864$		0			0
$865$		0			0
$866$	−16.4317	+	12.7279i	−0.558371	+	0.432512i
$867$		0			0
$868$	−12.2474	+	47.4342i	−0.415705	+	1.61002i
$869$			42.4264i			1.43922i
$870$		0			0
$871$			30.9839i			1.04985i
$872$	−11.1803	−	25.9808i	−0.378614	−	0.879820i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	31.8434			1.07527			0.537637	−	0.843176i	$-0.319317\pi$
$877$	31.8434			1.07527			0.537637	+	0.843176i	$0.319317\pi$
$878$	−6.70820	−	8.66025i	−0.226391	−	0.292269i
$879$		0			0
$880$		0			0
$881$		−	15.5563i		−	0.524107i	−0.965053	−	0.262053i	$-0.915600\pi$
$881$		−	15.5563i		−	0.524107i	0.965053	−	0.262053i	$-0.0843996\pi$
$882$		0			0
$883$			31.6228i			1.06419i	0.846684	+	0.532096i	$0.178595\pi$
$883$			31.6228i			1.06419i	−0.846684	+	0.532096i	$0.821405\pi$
$884$		0			0
$885$		0			0
$886$	−10.0000	+	7.74597i	−0.335957	+	0.260231i
$887$	49.1935			1.65176			0.825878	−	0.563849i	$-0.190680\pi$
$887$	49.1935			1.65176			0.825878	+	0.563849i	$0.190680\pi$
$888$		0			0
$889$	−10.0000			−0.335389
$890$		0			0
$891$		0			0
$892$	6.12372	+	1.58114i	0.205037	+	0.0529404i
$893$		0			0
$894$		0			0
$895$		0			0
$896$	35.6020	+	3.53553i	1.18938	+	0.118114i
$897$		0			0
$898$	28.1691	+	36.3662i	0.940016	+	1.21356i
$899$	21.9089			0.730703
$900$		0			0
$901$		0			0
$902$	−6.70820	−	8.66025i	−0.223359	−	0.288355i
$903$		0			0
$904$	36.0000	−	15.4919i	1.19734	−	0.515254i
$905$		0			0
$906$		0			0
$907$		−	31.6228i		−	1.05002i	−0.851097	−	0.525009i	$-0.824062\pi$
$907$		−	31.6228i		−	1.05002i	0.851097	−	0.525009i	$-0.175938\pi$
$908$	−8.94427	+	34.6410i	−0.296826	+	1.14960i
$909$		0			0
$910$		0			0
$911$	−21.9089			−0.725874			−0.362937	−	0.931814i	$-0.618226\pi$
$911$	−21.9089			−0.725874			−0.362937	+	0.931814i	$0.618226\pi$
$912$		0			0
$913$	48.9898			1.62133
$914$	5.47723	−	4.24264i	0.181171	−	0.140334i
$915$		0			0
$916$	−7.00000	+	27.1109i	−0.231287	+	0.895769i
$917$		−	17.3205i		−	0.571974i
$918$		0			0
$919$		−	23.2379i		−	0.766548i	−0.923635	−	0.383274i	$-0.874797\pi$
$919$		−	23.2379i		−	0.766548i	0.923635	−	0.383274i	$-0.125203\pi$
$920$		0			0
$921$		0			0
$922$	19.5959	+	25.2982i	0.645357	+	0.833153i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	2.73861	+	3.53553i	0.0899964	+	0.116185i
$927$		0			0
$928$	−2.44949	−	15.8114i	−0.0804084	−	0.519034i
$929$			32.5269i			1.06717i	0.845745	+	0.533587i	$0.179156\pi$
$929$			32.5269i			1.06717i	−0.845745	+	0.533587i	$0.820844\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−20.0000	+	15.4919i	−0.654420	+	0.506912i
$935$		0			0
$936$		0			0
$937$	29.3939			0.960256			0.480128	−	0.877198i	$-0.340590\pi$
$937$	29.3939			0.960256			0.480128	+	0.877198i	$0.340590\pi$
$938$	−44.7214	+	34.6410i	−1.46020	+	1.13107i
$939$		0			0
$940$		0			0
$941$		−	36.7696i		−	1.19865i	−0.800505	−	0.599327i	$-0.795435\pi$
$941$		−	36.7696i		−	1.19865i	0.800505	−	0.599327i	$-0.204565\pi$
$942$		0			0
$943$			6.32456i			0.205956i
$944$	19.1703	+	10.6066i	0.623940	+	0.345215i
$945$		0			0
$946$	30.0000	+	38.7298i	0.975384	+	1.25922i
$947$	8.94427			0.290650			0.145325	−	0.989384i	$-0.453577\pi$
$947$	8.94427			0.290650			0.145325	+	0.989384i	$0.453577\pi$
$948$		0			0
$949$	−36.0000			−1.16861
$950$		0			0
$951$		0			0
$952$		0			0
$953$			20.7846i			0.673280i	0.941634	+	0.336640i	$0.109290\pi$
$953$			20.7846i			0.673280i	−0.941634	+	0.336640i	$0.890710\pi$
$954$		0			0
$955$		0			0
$956$	5.47723	−	21.2132i	0.177146	−	0.686084i
$957$		0			0
$958$	24.4949	−	18.9737i	0.791394	−	0.613011i
$959$	−21.9089			−0.707475
$960$		0			0
$961$	−29.0000			−0.935484
$962$	−20.1246	+	15.5885i	−0.648844	+	0.502592i
$963$		0			0
$964$	4.00000	−	15.4919i	0.128831	−	0.498962i
$965$		0			0
$966$		0			0
$967$			15.8114i			0.508460i	0.967144	+	0.254230i	$0.0818220\pi$
$967$			15.8114i			0.508460i	−0.967144	+	0.254230i	$0.918178\pi$
$968$	−21.2426	−	49.3634i	−0.682764	−	1.58660i
$969$		0			0
$970$		0			0
$971$	49.2950			1.58195			0.790976	−	0.611847i	$-0.209573\pi$
$971$	49.2950			1.58195			0.790976	+	0.611847i	$0.209573\pi$
$972$		0			0
$973$	24.4949			0.785270
$974$	−19.1703	−	24.7487i	−0.614256	−	0.793001i
$975$		0			0
$976$	35.0000	+	19.3649i	1.12032	+	0.619856i
$977$		−	34.6410i		−	1.10826i	−0.832429	−	0.554132i	$-0.813050\pi$
$977$		−	34.6410i		−	1.10826i	0.832429	−	0.554132i	$-0.186950\pi$
$978$		0			0
$979$			54.2218i			1.73294i
$980$		0			0
$981$		0			0
$982$	−30.6186	+	23.7171i	−0.977079	+	0.756843i
$983$	−8.94427			−0.285278			−0.142639	−	0.989775i	$-0.545559\pi$
$983$	−8.94427			−0.285278			−0.142639	+	0.989775i	$0.545559\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	28.2843i		−	0.899388i
$990$		0			0
$991$		−	23.2379i		−	0.738176i	−0.929394	−	0.369088i	$-0.879670\pi$
$991$		−	23.2379i		−	0.738176i	0.929394	−	0.369088i	$-0.120330\pi$
$992$	6.70820	+	43.3013i	0.212986	+	1.37482i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	56.3383			1.78425			0.892125	−	0.451788i	$-0.149214\pi$
$997$	56.3383			1.78425			0.892125	+	0.451788i	$0.149214\pi$
$998$	−13.4164	−	17.3205i	−0.424689	−	0.548271i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.2.e.f.251.5		8
3.2	odd	2	inner	900.2.e.f.251.3		8
4.3	odd	2	inner	900.2.e.f.251.2		8
5.2	odd	4		180.2.h.b.179.7	yes	8
5.3	odd	4		180.2.h.b.179.1	✓	8
5.4	even	2	inner	900.2.e.f.251.4		8
12.11	even	2	inner	900.2.e.f.251.8		8
15.2	even	4		180.2.h.b.179.2	yes	8
15.8	even	4		180.2.h.b.179.8	yes	8
15.14	odd	2	inner	900.2.e.f.251.6		8
20.3	even	4		180.2.h.b.179.3	yes	8
20.7	even	4		180.2.h.b.179.5	yes	8
20.19	odd	2	inner	900.2.e.f.251.7		8
40.3	even	4		2880.2.o.c.2879.8		8
40.13	odd	4		2880.2.o.c.2879.7		8
40.27	even	4		2880.2.o.c.2879.3		8
40.37	odd	4		2880.2.o.c.2879.4		8
60.23	odd	4		180.2.h.b.179.6	yes	8
60.47	odd	4		180.2.h.b.179.4	yes	8
60.59	even	2	inner	900.2.e.f.251.1		8
120.53	even	4		2880.2.o.c.2879.1		8
120.77	even	4		2880.2.o.c.2879.6		8
120.83	odd	4		2880.2.o.c.2879.2		8
120.107	odd	4		2880.2.o.c.2879.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.h.b.179.1	✓	8	5.3	odd	4
180.2.h.b.179.2	yes	8	15.2	even	4
180.2.h.b.179.3	yes	8	20.3	even	4
180.2.h.b.179.4	yes	8	60.47	odd	4
180.2.h.b.179.5	yes	8	20.7	even	4
180.2.h.b.179.6	yes	8	60.23	odd	4
180.2.h.b.179.7	yes	8	5.2	odd	4
180.2.h.b.179.8	yes	8	15.8	even	4
900.2.e.f.251.1		8	60.59	even	2	inner
900.2.e.f.251.2		8	4.3	odd	2	inner
900.2.e.f.251.3		8	3.2	odd	2	inner
900.2.e.f.251.4		8	5.4	even	2	inner
900.2.e.f.251.5		8	1.1	even	1	trivial
900.2.e.f.251.6		8	15.14	odd	2	inner
900.2.e.f.251.7		8	20.19	odd	2	inner
900.2.e.f.251.8		8	12.11	even	2	inner
2880.2.o.c.2879.1		8	120.53	even	4
2880.2.o.c.2879.2		8	120.83	odd	4
2880.2.o.c.2879.3		8	40.27	even	4
2880.2.o.c.2879.4		8	40.37	odd	4
2880.2.o.c.2879.5		8	120.107	odd	4
2880.2.o.c.2879.6		8	120.77	even	4
2880.2.o.c.2879.7		8	40.13	odd	4
2880.2.o.c.2879.8		8	40.3	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.11803 - 0.866025i) q^{2} +(0.500000 - 1.93649i) q^{4} -3.16228i q^{7} +(-1.11803 - 2.59808i) q^{8} -5.47723 q^{11} -2.44949 q^{13} +(-2.73861 - 3.53553i) q^{14} +(-3.50000 - 1.93649i) q^{16} +(-6.12372 + 4.74342i) q^{22} +4.47214 q^{23} +(-2.73861 + 2.12132i) q^{26} +(-6.12372 - 1.58114i) q^{28} +2.82843i q^{29} -7.74597i q^{31} +(-5.59017 + 0.866025i) q^{32} +7.34847 q^{37} +1.41421i q^{41} -6.32456i q^{43} +(-2.73861 + 10.6066i) q^{44} +(5.00000 - 3.87298i) q^{46} +8.94427 q^{47} -3.00000 q^{49} +(-1.22474 + 4.74342i) q^{52} +(-8.21584 + 3.53553i) q^{56} +(2.44949 + 3.16228i) q^{58} -5.47723 q^{59} -10.0000 q^{61} +(-6.70820 - 8.66025i) q^{62} +(-5.50000 + 5.80948i) q^{64} -12.6491i q^{67} +14.6969 q^{73} +(8.21584 - 6.36396i) q^{74} +17.3205i q^{77} -7.74597i q^{79} +(1.22474 + 1.58114i) q^{82} -8.94427 q^{83} +(-5.47723 - 7.07107i) q^{86} +(6.12372 + 14.2302i) q^{88} -9.89949i q^{89} +7.74597i q^{91} +(2.23607 - 8.66025i) q^{92} +(10.0000 - 7.74597i) q^{94} +4.89898 q^{97} +(-3.35410 + 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 28 q^{16} + 40 q^{46} - 24 q^{49} - 80 q^{61} - 44 q^{64} + 80 q^{94}+O(q^{100}) \) 8 * q + 4 * q^4 - 28 * q^16 + 40 * q^46 - 24 * q^49 - 80 * q^61 - 44 * q^64 + 80 * q^94

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