LMFDB - Embedded newform 880.2.b.h.529.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,2,Mod(529,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.529"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-3,0,0,0,-2,0,4,0,0,0,-1] 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:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.3
Root			$1.68614 - 0.396143i$ of defining polynomial
Character	$\chi$	$=$	880.529
Dual form			880.2.b.h.529.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.792287i q^{3} +(0.686141 + 2.12819i) q^{5} +3.46410i q^{7} +2.37228 q^{9} +1.00000 q^{11} +(-1.68614 + 0.543620i) q^{15} -1.58457i q^{17} +4.00000 q^{19} -2.74456 q^{21} -0.792287i q^{23} +(-4.05842 + 2.92048i) q^{25} +4.25639i q^{27} -8.74456 q^{29} -3.37228 q^{31} +0.792287i q^{33} +(-7.37228 + 2.37686i) q^{35} +1.08724i q^{37} +8.74456 q^{41} -3.46410i q^{43} +(1.62772 + 5.04868i) q^{45} -6.63325i q^{47} -5.00000 q^{49} +1.25544 q^{51} +10.0974i q^{53} +(0.686141 + 2.12819i) q^{55} +3.16915i q^{57} -7.37228 q^{59} -0.744563 q^{61} +8.21782i q^{63} +9.30506i q^{67} +0.627719 q^{69} +10.1168 q^{71} -6.92820i q^{73} +(-2.31386 - 3.21543i) q^{75} +3.46410i q^{77} +1.25544 q^{79} +3.74456 q^{81} -6.63325i q^{83} +(3.37228 - 1.08724i) q^{85} -6.92820i q^{87} -1.37228 q^{89} -2.67181i q^{93} +(2.74456 + 8.51278i) q^{95} -5.84096i q^{97} +2.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} - 2 q^{9} + 4 q^{11} - q^{15} + 16 q^{19} + 12 q^{21} + q^{25} - 12 q^{29} - 2 q^{31} - 18 q^{35} + 12 q^{41} + 18 q^{45} - 20 q^{49} + 28 q^{51} - 3 q^{55} - 18 q^{59} + 20 q^{61} + 14 q^{69}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - 3 * q^5 - 2 * q^9 + 4 * q^11 - q^15 + 16 * q^19 + 12 * q^21 + q^25 - 12 * q^29 - 2 * q^31 - 18 * q^35 + 12 * q^41 + 18 * q^45 - 20 * q^49 + 28 * q^51 - 3 * q^55 - 18 * q^59 + 20 * q^61 + 14 * q^69 + 6 * q^71 - 15 * q^75 + 28 * q^79 - 8 * q^81 + 2 * q^85 + 6 * q^89 - 12 * q^95 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$			0.792287i			0.457427i	0.973494	+	0.228714i	$0.0734519\pi$
$3$			0.792287i			0.457427i	−0.973494	+	0.228714i	$0.926548\pi$
$4$		0			0
$5$	0.686141	+	2.12819i	0.306851	+	0.951757i
$5$	0.686141	+	2.12819i	0.306851	+	0.951757i
$6$		0			0
$7$			3.46410i			1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$			3.46410i			1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$	2.37228			0.790760
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−1.68614	+	0.543620i	−0.435360	+	0.140362i
$16$		0			0
$17$		−	1.58457i		−	0.384316i	−0.981364	−	0.192158i	$-0.938451\pi$
$17$		−	1.58457i		−	0.384316i	0.981364	−	0.192158i	$-0.0615486\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	−2.74456			−0.598913
$22$		0			0
$23$		−	0.792287i		−	0.165203i	−0.996583	−	0.0826016i	$-0.973677\pi$
$23$		−	0.792287i		−	0.165203i	0.996583	−	0.0826016i	$-0.0263229\pi$
$24$		0			0
$25$	−4.05842	+	2.92048i	−0.811684	+	0.584096i
$26$		0			0
$27$			4.25639i			0.819142i
$28$		0			0
$29$	−8.74456			−1.62382			−0.811912	−	0.583779i	$-0.801573\pi$
$29$	−8.74456			−1.62382			−0.811912	+	0.583779i	$0.801573\pi$
$30$		0			0
$31$	−3.37228			−0.605680			−0.302840	−	0.953041i	$-0.597935\pi$
$31$	−3.37228			−0.605680			−0.302840	+	0.953041i	$0.597935\pi$
$32$		0			0
$33$			0.792287i			0.137919i
$34$		0			0
$35$	−7.37228	+	2.37686i	−1.24614	+	0.401763i
$36$		0			0
$37$			1.08724i			0.178741i	0.995998	+	0.0893706i	$0.0284856\pi$
$37$			1.08724i			0.178741i	−0.995998	+	0.0893706i	$0.971514\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.74456			1.36567			0.682836	−	0.730572i	$-0.260747\pi$
$41$	8.74456			1.36567			0.682836	+	0.730572i	$0.260747\pi$
$42$		0			0
$43$		−	3.46410i		−	0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$		−	3.46410i		−	0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$		0			0
$45$	1.62772	+	5.04868i	0.242646	+	0.752612i
$46$		0			0
$47$		−	6.63325i		−	0.967559i	−0.875190	−	0.483779i	$-0.839264\pi$
$47$		−	6.63325i		−	0.967559i	0.875190	−	0.483779i	$-0.160736\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$	1.25544			0.175796
$52$		0			0
$53$			10.0974i			1.38698i	0.720467	+	0.693489i	$0.243927\pi$
$53$			10.0974i			1.38698i	−0.720467	+	0.693489i	$0.756073\pi$
$54$		0			0
$55$	0.686141	+	2.12819i	0.0925192	+	0.286966i
$56$		0			0
$57$			3.16915i			0.419764i
$58$		0			0
$59$	−7.37228			−0.959789			−0.479895	−	0.877326i	$-0.659325\pi$
$59$	−7.37228			−0.959789			−0.479895	+	0.877326i	$0.659325\pi$
$60$		0			0
$61$	−0.744563			−0.0953315			−0.0476657	−	0.998863i	$-0.515178\pi$
$61$	−0.744563			−0.0953315			−0.0476657	+	0.998863i	$0.515178\pi$
$62$		0			0
$63$			8.21782i			1.03535i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			9.30506i			1.13679i	0.822754	+	0.568397i	$0.192436\pi$
$67$			9.30506i			1.13679i	−0.822754	+	0.568397i	$0.807564\pi$
$68$		0			0
$69$	0.627719			0.0755684
$70$		0			0
$71$	10.1168			1.20065			0.600324	−	0.799757i	$-0.295038\pi$
$71$	10.1168			1.20065			0.600324	+	0.799757i	$0.295038\pi$
$72$		0			0
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$		0			0
$75$	−2.31386	−	3.21543i	−0.267181	−	0.371286i
$76$		0			0
$77$			3.46410i			0.394771i
$78$		0			0
$79$	1.25544			0.141248			0.0706239	−	0.997503i	$-0.477501\pi$
$79$	1.25544			0.141248			0.0706239	+	0.997503i	$0.477501\pi$
$80$		0			0
$81$	3.74456			0.416063
$82$		0			0
$83$		−	6.63325i		−	0.728094i	−0.931381	−	0.364047i	$-0.881395\pi$
$83$		−	6.63325i		−	0.728094i	0.931381	−	0.364047i	$-0.118605\pi$
$84$		0			0
$85$	3.37228	−	1.08724i	0.365775	−	0.117928i
$86$		0			0
$87$		−	6.92820i		−	0.742781i
$88$		0			0
$89$	−1.37228			−0.145462			−0.0727308	−	0.997352i	$-0.523171\pi$
$89$	−1.37228			−0.145462			−0.0727308	+	0.997352i	$0.523171\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	2.67181i		−	0.277054i
$94$		0			0
$95$	2.74456	+	8.51278i	0.281586	+	0.873393i
$96$		0			0
$97$		−	5.84096i		−	0.593060i	−0.955024	−	0.296530i	$-0.904171\pi$
$97$		−	5.84096i		−	0.593060i	0.955024	−	0.296530i	$-0.0958295\pi$
$98$		0			0
$99$	2.37228			0.238423
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$			10.3923i			1.02398i	0.858990	+	0.511992i	$0.171092\pi$
$103$			10.3923i			1.02398i	−0.858990	+	0.511992i	$0.828908\pi$
$104$		0			0
$105$	−1.88316	−	5.84096i	−0.183777	−	0.570020i
$106$		0			0
$107$		−	6.63325i		−	0.641260i	−0.947204	−	0.320630i	$-0.896105\pi$
$107$		−	6.63325i		−	0.641260i	0.947204	−	0.320630i	$-0.103895\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$	−0.861407			−0.0817611
$112$		0			0
$113$		−	0.497333i		−	0.0467852i	−0.999726	−	0.0233926i	$-0.992553\pi$
$113$		−	0.497333i		−	0.0467852i	0.999726	−	0.0233926i	$-0.00744677\pi$
$114$		0			0
$115$	1.68614	−	0.543620i	0.157233	−	0.0506929i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.48913			0.503187
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$			6.92820i			0.624695i
$124$		0			0
$125$	−9.00000	−	6.63325i	−0.804984	−	0.593296i
$126$		0			0
$127$		−	8.21782i		−	0.729214i	−0.931162	−	0.364607i	$-0.881203\pi$
$127$		−	8.21782i		−	0.729214i	0.931162	−	0.364607i	$-0.118797\pi$
$128$		0			0
$129$	2.74456			0.241645
$130$		0			0
$131$	2.74456			0.239794			0.119897	−	0.992786i	$-0.461744\pi$
$131$	2.74456			0.239794			0.119897	+	0.992786i	$0.461744\pi$
$132$		0			0
$133$			13.8564i			1.20150i
$134$		0			0
$135$	−9.05842	+	2.92048i	−0.779625	+	0.251355i
$136$		0			0
$137$		−	14.3537i		−	1.22632i	−0.789958	−	0.613161i	$-0.789898\pi$
$137$		−	14.3537i		−	1.22632i	0.789958	−	0.613161i	$-0.210102\pi$
$138$		0			0
$139$	−16.2337			−1.37692			−0.688462	−	0.725273i	$-0.741714\pi$
$139$	−16.2337			−1.37692			−0.688462	+	0.725273i	$0.741714\pi$
$140$		0			0
$141$	5.25544			0.442588
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−6.00000	−	18.6101i	−0.498273	−	1.54549i
$146$		0			0
$147$		−	3.96143i		−	0.326734i
$148$		0			0
$149$	11.4891			0.941226			0.470613	−	0.882340i	$-0.344033\pi$
$149$	11.4891			0.941226			0.470613	+	0.882340i	$0.344033\pi$
$150$		0			0
$151$	12.2337			0.995563			0.497782	−	0.867302i	$-0.334148\pi$
$151$	12.2337			0.995563			0.497782	+	0.867302i	$0.334148\pi$
$152$		0			0
$153$		−	3.75906i		−	0.303902i
$154$		0			0
$155$	−2.31386	−	7.17687i	−0.185854	−	0.576460i
$156$		0			0
$157$			24.4511i			1.95141i	0.219090	+	0.975705i	$0.429691\pi$
$157$			24.4511i			1.95141i	−0.219090	+	0.975705i	$0.570309\pi$
$158$		0			0
$159$	−8.00000			−0.634441
$160$		0			0
$161$	2.74456			0.216302
$162$		0			0
$163$			3.46410i			0.271329i	0.990755	+	0.135665i	$0.0433170\pi$
$163$			3.46410i			0.271329i	−0.990755	+	0.135665i	$0.956683\pi$
$164$		0			0
$165$	−1.68614	+	0.543620i	−0.131266	+	0.0423208i
$166$		0			0
$167$		−	15.7359i		−	1.21768i	−0.793292	−	0.608842i	$-0.791635\pi$
$167$		−	15.7359i		−	1.21768i	0.793292	−	0.608842i	$-0.208365\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	9.48913			0.725652
$172$		0			0
$173$		−	8.51278i		−	0.647214i	−0.946192	−	0.323607i	$-0.895104\pi$
$173$		−	8.51278i		−	0.647214i	0.946192	−	0.323607i	$-0.104896\pi$
$174$		0			0
$175$	−10.1168	−	14.0588i	−0.764762	−	1.06274i
$176$		0			0
$177$		−	5.84096i		−	0.439034i
$178$		0			0
$179$	12.8614			0.961307			0.480653	−	0.876911i	$-0.340400\pi$
$179$	12.8614			0.961307			0.480653	+	0.876911i	$0.340400\pi$
$180$		0			0
$181$	24.1168			1.79259			0.896295	−	0.443457i	$-0.146248\pi$
$181$	24.1168			1.79259			0.896295	+	0.443457i	$0.146248\pi$
$182$		0			0
$183$		−	0.589907i		−	0.0436072i
$184$		0			0
$185$	−2.31386	+	0.746000i	−0.170118	+	0.0548470i
$186$		0			0
$187$		−	1.58457i		−	0.115876i
$188$		0			0
$189$	−14.7446			−1.07251
$190$		0			0
$191$	19.3723			1.40173			0.700865	−	0.713294i	$-0.252798\pi$
$191$	19.3723			1.40173			0.700865	+	0.713294i	$0.252798\pi$
$192$		0			0
$193$		−	16.4356i		−	1.18306i	−0.806282	−	0.591532i	$-0.798523\pi$
$193$		−	16.4356i		−	1.18306i	0.806282	−	0.591532i	$-0.201477\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	8.51278i		−	0.606510i	−0.952909	−	0.303255i	$-0.901927\pi$
$197$		−	8.51278i		−	0.606510i	0.952909	−	0.303255i	$-0.0980734\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$	−7.37228			−0.520001
$202$		0			0
$203$		−	30.2921i		−	2.12609i
$204$		0			0
$205$	6.00000	+	18.6101i	0.419058	+	1.29979i
$206$		0			0
$207$		−	1.87953i		−	0.130636i
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	−1.48913			−0.102516			−0.0512578	−	0.998685i	$-0.516323\pi$
$211$	−1.48913			−0.102516			−0.0512578	+	0.998685i	$0.516323\pi$
$212$		0			0
$213$			8.01544i			0.549209i
$214$		0			0
$215$	7.37228	−	2.37686i	0.502785	−	0.162101i
$216$		0			0
$217$		−	11.6819i		−	0.793021i
$218$		0			0
$219$	5.48913			0.370921
$220$		0			0
$221$		0			0
$222$		0			0
$223$			2.37686i			0.159166i	0.996828	+	0.0795832i	$0.0253589\pi$
$223$			2.37686i			0.159166i	−0.996828	+	0.0795832i	$0.974641\pi$
$224$		0			0
$225$	−9.62772	+	6.92820i	−0.641848	+	0.461880i
$226$		0			0
$227$			16.7306i			1.11045i	0.831701	+	0.555224i	$0.187368\pi$
$227$			16.7306i			1.11045i	−0.831701	+	0.555224i	$0.812632\pi$
$228$		0			0
$229$	−14.6277			−0.966627			−0.483313	−	0.875447i	$-0.660567\pi$
$229$	−14.6277			−0.966627			−0.483313	+	0.875447i	$0.660567\pi$
$230$		0			0
$231$	−2.74456			−0.180579
$232$		0			0
$233$		−	3.75906i		−	0.246264i	−0.992390	−	0.123132i	$-0.960706\pi$
$233$		−	3.75906i		−	0.246264i	0.992390	−	0.123132i	$-0.0392938\pi$
$234$		0			0
$235$	14.1168	−	4.55134i	0.920881	−	0.296897i
$236$		0			0
$237$			0.994667i			0.0646105i
$238$		0			0
$239$	−14.7446			−0.953746			−0.476873	−	0.878972i	$-0.658230\pi$
$239$	−14.7446			−0.953746			−0.476873	+	0.878972i	$0.658230\pi$
$240$		0			0
$241$	16.7446			1.07861			0.539306	−	0.842110i	$-0.318687\pi$
$241$	16.7446			1.07861			0.539306	+	0.842110i	$0.318687\pi$
$242$		0			0
$243$			15.7359i			1.00946i
$244$		0			0
$245$	−3.43070	−	10.6410i	−0.219180	−	0.679827i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	5.25544			0.333050
$250$		0			0
$251$	22.1168			1.39600			0.698001	−	0.716096i	$-0.254073\pi$
$251$	22.1168			1.39600			0.698001	+	0.716096i	$0.254073\pi$
$252$		0			0
$253$		−	0.792287i		−	0.0498107i
$254$		0			0
$255$	0.861407	+	2.67181i	0.0539434	+	0.167316i
$256$		0			0
$257$		−	10.6873i		−	0.666653i	−0.942811	−	0.333326i	$-0.891829\pi$
$257$		−	10.6873i		−	0.666653i	0.942811	−	0.333326i	$-0.108171\pi$
$258$		0			0
$259$	−3.76631			−0.234027
$260$		0			0
$261$	−20.7446			−1.28406
$262$		0			0
$263$		−	27.4179i		−	1.69066i	−0.534246	−	0.845329i	$-0.679405\pi$
$263$		−	27.4179i		−	1.69066i	0.534246	−	0.845329i	$-0.320595\pi$
$264$		0			0
$265$	−21.4891	+	6.92820i	−1.32007	+	0.425596i
$266$		0			0
$267$		−	1.08724i		−	0.0665380i
$268$		0			0
$269$	11.4891			0.700504			0.350252	−	0.936655i	$-0.386096\pi$
$269$	11.4891			0.700504			0.350252	+	0.936655i	$0.386096\pi$
$270$		0			0
$271$	−13.4891			−0.819406			−0.409703	−	0.912219i	$-0.634368\pi$
$271$	−13.4891			−0.819406			−0.409703	+	0.912219i	$0.634368\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.05842	+	2.92048i	−0.244732	+	0.176112i
$276$		0			0
$277$			11.6819i			0.701899i	0.936394	+	0.350949i	$0.114141\pi$
$277$			11.6819i			0.701899i	−0.936394	+	0.350949i	$0.885859\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	−0.510875			−0.0304762			−0.0152381	−	0.999884i	$-0.504851\pi$
$281$	−0.510875			−0.0304762			−0.0152381	+	0.999884i	$0.504851\pi$
$282$		0			0
$283$		−	15.1460i		−	0.900338i	−0.892943	−	0.450169i	$-0.851364\pi$
$283$		−	15.1460i		−	0.900338i	0.892943	−	0.450169i	$-0.148636\pi$
$284$		0			0
$285$	−6.74456	+	2.17448i	−0.399513	+	0.128805i
$286$		0			0
$287$			30.2921i			1.78808i
$288$		0			0
$289$	14.4891			0.852301
$290$		0			0
$291$	4.62772			0.271282
$292$		0			0
$293$			3.16915i			0.185144i	0.995706	+	0.0925718i	$0.0295088\pi$
$293$			3.16915i			0.185144i	−0.995706	+	0.0925718i	$0.970491\pi$
$294$		0			0
$295$	−5.05842	−	15.6896i	−0.294513	−	0.913487i
$296$		0			0
$297$			4.25639i			0.246981i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$			4.75372i			0.273094i
$304$		0			0
$305$	−0.510875	−	1.58457i	−0.0292526	−	0.0907324i
$306$		0			0
$307$			31.5817i			1.80246i	0.433340	+	0.901231i	$0.357335\pi$
$307$			31.5817i			1.80246i	−0.433340	+	0.901231i	$0.642665\pi$
$308$		0			0
$309$	−8.23369			−0.468398
$310$		0			0
$311$	−5.48913			−0.311260			−0.155630	−	0.987815i	$-0.549741\pi$
$311$	−5.48913			−0.311260			−0.155630	+	0.987815i	$0.549741\pi$
$312$		0			0
$313$		−	21.8719i		−	1.23627i	−0.786072	−	0.618135i	$-0.787888\pi$
$313$		−	21.8719i		−	1.23627i	0.786072	−	0.618135i	$-0.212112\pi$
$314$		0			0
$315$	−17.4891	+	5.63858i	−0.985401	+	0.317698i
$316$		0			0
$317$		−	32.9639i		−	1.85144i	−0.378215	−	0.925718i	$-0.623462\pi$
$317$		−	32.9639i		−	1.85144i	0.378215	−	0.925718i	$-0.376538\pi$
$318$		0			0
$319$	−8.74456			−0.489602
$320$		0			0
$321$	5.25544			0.293330
$322$		0			0
$323$		−	6.33830i		−	0.352672i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	7.92287i		−	0.438136i
$328$		0			0
$329$	22.9783			1.26683
$330$		0			0
$331$	14.1168			0.775932			0.387966	−	0.921674i	$-0.373178\pi$
$331$	14.1168			0.775932			0.387966	+	0.921674i	$0.373178\pi$
$332$		0			0
$333$			2.57924i			0.141342i
$334$		0			0
$335$	−19.8030	+	6.38458i	−1.08195	+	0.348827i
$336$		0			0
$337$			32.4665i			1.76856i	0.466952	+	0.884282i	$0.345352\pi$
$337$			32.4665i			1.76856i	−0.466952	+	0.884282i	$0.654648\pi$
$338$		0			0
$339$	0.394031			0.0214008
$340$		0			0
$341$	−3.37228			−0.182619
$342$		0			0
$343$			6.92820i			0.374088i
$344$		0			0
$345$	0.430703	+	1.33591i	0.0231883	+	0.0719228i
$346$		0			0
$347$		−	22.6641i		−	1.21667i	−0.793679	−	0.608337i	$-0.791837\pi$
$347$		−	22.6641i		−	1.21667i	0.793679	−	0.608337i	$-0.208163\pi$
$348$		0			0
$349$	−15.4891			−0.829114			−0.414557	−	0.910023i	$-0.636063\pi$
$349$	−15.4891			−0.829114			−0.414557	+	0.910023i	$0.636063\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			25.0410i			1.33280i	0.745595	+	0.666399i	$0.232165\pi$
$353$			25.0410i			1.33280i	−0.745595	+	0.666399i	$0.767835\pi$
$354$		0			0
$355$	6.94158	+	21.5306i	0.368421	+	1.14273i
$356$		0			0
$357$			4.34896i			0.230172i
$358$		0			0
$359$	29.4891			1.55638			0.778188	−	0.628031i	$-0.216139\pi$
$359$	29.4891			1.55638			0.778188	+	0.628031i	$0.216139\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$			0.792287i			0.0415843i
$364$		0			0
$365$	14.7446	−	4.75372i	0.771766	−	0.248821i
$366$		0			0
$367$			25.7407i			1.34365i	0.740708	+	0.671827i	$0.234490\pi$
$367$			25.7407i			1.34365i	−0.740708	+	0.671827i	$0.765510\pi$
$368$		0			0
$369$	20.7446			1.07992
$370$		0			0
$371$	−34.9783			−1.81598
$372$		0			0
$373$		−	11.6819i		−	0.604867i	−0.953170	−	0.302434i	$-0.902201\pi$
$373$		−	11.6819i		−	0.604867i	0.953170	−	0.302434i	$-0.0977990\pi$
$374$		0			0
$375$	5.25544	−	7.13058i	0.271390	−	0.368222i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.627719			−0.0322437			−0.0161219	−	0.999870i	$-0.505132\pi$
$379$	−0.627719			−0.0322437			−0.0161219	+	0.999870i	$0.505132\pi$
$380$		0			0
$381$	6.51087			0.333562
$382$		0			0
$383$			10.8896i			0.556435i	0.960518	+	0.278217i	$0.0897435\pi$
$383$			10.8896i			0.556435i	−0.960518	+	0.278217i	$0.910256\pi$
$384$		0			0
$385$	−7.37228	+	2.37686i	−0.375726	+	0.121136i
$386$		0			0
$387$		−	8.21782i		−	0.417735i
$388$		0			0
$389$	−18.8614			−0.956311			−0.478156	−	0.878275i	$-0.658695\pi$
$389$	−18.8614			−0.956311			−0.478156	+	0.878275i	$0.658695\pi$
$390$		0			0
$391$	−1.25544			−0.0634902
$392$		0			0
$393$			2.17448i			0.109688i
$394$		0			0
$395$	0.861407	+	2.67181i	0.0433421	+	0.134434i
$396$		0			0
$397$			16.4356i			0.824881i	0.910984	+	0.412441i	$0.135324\pi$
$397$			16.4356i			0.824881i	−0.910984	+	0.412441i	$0.864676\pi$
$398$		0			0
$399$	−10.9783			−0.549600
$400$		0			0
$401$	−11.4891			−0.573740			−0.286870	−	0.957970i	$-0.592615\pi$
$401$	−11.4891			−0.573740			−0.286870	+	0.957970i	$0.592615\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	2.56930	+	7.96916i	0.127669	+	0.395991i
$406$		0			0
$407$			1.08724i			0.0538925i
$408$		0			0
$409$	−27.4891			−1.35925			−0.679625	−	0.733560i	$-0.737857\pi$
$409$	−27.4891			−1.35925			−0.679625	+	0.733560i	$0.737857\pi$
$410$		0			0
$411$	11.3723			0.560953
$412$		0			0
$413$		−	25.5383i		−	1.25666i
$414$		0			0
$415$	14.1168	−	4.55134i	0.692969	−	0.223417i
$416$		0			0
$417$		−	12.8617i		−	0.629842i
$418$		0			0
$419$	−22.9783			−1.12256			−0.561280	−	0.827626i	$-0.689691\pi$
$419$	−22.9783			−1.12256			−0.561280	+	0.827626i	$0.689691\pi$
$420$		0			0
$421$	8.51087			0.414795			0.207397	−	0.978257i	$-0.433501\pi$
$421$	8.51087			0.414795			0.207397	+	0.978257i	$0.433501\pi$
$422$		0			0
$423$		−	15.7359i		−	0.765107i
$424$		0			0
$425$	4.62772	+	6.43087i	0.224477	+	0.311943i
$426$		0			0
$427$		−	2.57924i		−	0.124818i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	25.7228			1.23902			0.619512	−	0.784987i	$-0.287330\pi$
$431$	25.7228			1.23902			0.619512	+	0.784987i	$0.287330\pi$
$432$		0			0
$433$			29.2048i			1.40349i	0.712426	+	0.701747i	$0.247596\pi$
$433$			29.2048i			1.40349i	−0.712426	+	0.701747i	$0.752404\pi$
$434$		0			0
$435$	14.7446	−	4.75372i	0.706948	−	0.227924i
$436$		0			0
$437$		−	3.16915i		−	0.151601i
$438$		0			0
$439$	21.4891			1.02562			0.512810	−	0.858502i	$-0.328605\pi$
$439$	21.4891			1.02562			0.512810	+	0.858502i	$0.328605\pi$
$440$		0			0
$441$	−11.8614			−0.564829
$442$		0			0
$443$			31.6742i			1.50489i	0.658656	+	0.752444i	$0.271125\pi$
$443$			31.6742i			1.50489i	−0.658656	+	0.752444i	$0.728875\pi$
$444$		0			0
$445$	−0.941578	−	2.92048i	−0.0446351	−	0.138444i
$446$		0			0
$447$			9.10268i			0.430542i
$448$		0			0
$449$	−6.86141			−0.323810			−0.161905	−	0.986806i	$-0.551764\pi$
$449$	−6.86141			−0.323810			−0.161905	+	0.986806i	$0.551764\pi$
$450$		0			0
$451$	8.74456			0.411765
$452$		0			0
$453$			9.69259i			0.455398i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			20.7846i			0.972263i	0.873886	+	0.486132i	$0.161592\pi$
$457$			20.7846i			0.972263i	−0.873886	+	0.486132i	$0.838408\pi$
$458$		0			0
$459$	6.74456			0.314809
$460$		0			0
$461$	2.23369			0.104033			0.0520166	−	0.998646i	$-0.483435\pi$
$461$	2.23369			0.104033			0.0520166	+	0.998646i	$0.483435\pi$
$462$		0			0
$463$			30.0897i			1.39839i	0.714933	+	0.699193i	$0.246457\pi$
$463$			30.0897i			1.39839i	−0.714933	+	0.699193i	$0.753543\pi$
$464$		0			0
$465$	5.68614	−	1.83324i	0.263688	−	0.0850145i
$466$		0			0
$467$		−	7.72049i		−	0.357262i	−0.983916	−	0.178631i	$-0.942833\pi$
$467$		−	7.72049i		−	0.357262i	0.983916	−	0.178631i	$-0.0571668\pi$
$468$		0			0
$469$	−32.2337			−1.48841
$470$		0			0
$471$	−19.3723			−0.892628
$472$		0			0
$473$		−	3.46410i		−	0.159280i
$474$		0			0
$475$	−16.2337	+	11.6819i	−0.744853	+	0.536003i
$476$		0			0
$477$			23.9538i			1.09677i
$478$		0			0
$479$	−5.48913			−0.250805			−0.125402	−	0.992106i	$-0.540022\pi$
$479$	−5.48913			−0.250805			−0.125402	+	0.992106i	$0.540022\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$			2.17448i			0.0989423i
$484$		0			0
$485$	12.4307	−	4.00772i	0.564449	−	0.181981i
$486$		0			0
$487$		−	7.13058i		−	0.323118i	−0.986863	−	0.161559i	$-0.948348\pi$
$487$		−	7.13058i		−	0.323118i	0.986863	−	0.161559i	$-0.0516521\pi$
$488$		0			0
$489$	−2.74456			−0.124113
$490$		0			0
$491$	−6.51087			−0.293832			−0.146916	−	0.989149i	$-0.546935\pi$
$491$	−6.51087			−0.293832			−0.146916	+	0.989149i	$0.546935\pi$
$492$		0			0
$493$			13.8564i			0.624061i
$494$		0			0
$495$	1.62772	+	5.04868i	0.0731605	+	0.226921i
$496$		0			0
$497$			35.0458i			1.57202i
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$		0			0
$501$	12.4674			0.557001
$502$		0			0
$503$		−	13.5615i		−	0.604675i	−0.953201	−	0.302338i	$-0.902233\pi$
$503$		−	13.5615i		−	0.604675i	0.953201	−	0.302338i	$-0.0977670\pi$
$504$		0			0
$505$	4.11684	+	12.7692i	0.183197	+	0.568220i
$506$		0			0
$507$			10.2997i			0.457427i
$508$		0			0
$509$	−22.6277			−1.00296			−0.501478	−	0.865170i	$-0.667210\pi$
$509$	−22.6277			−1.00296			−0.501478	+	0.865170i	$0.667210\pi$
$510$		0			0
$511$	24.0000			1.06170
$512$		0			0
$513$			17.0256i			0.751697i
$514$		0			0
$515$	−22.1168	+	7.13058i	−0.974585	+	0.314211i
$516$		0			0
$517$		−	6.63325i		−	0.291730i
$518$		0			0
$519$	6.74456			0.296053
$520$		0			0
$521$	−21.6060			−0.946575			−0.473287	−	0.880908i	$-0.656933\pi$
$521$	−21.6060			−0.946575			−0.473287	+	0.880908i	$0.656933\pi$
$522$		0			0
$523$		−	29.0024i		−	1.26819i	−0.773256	−	0.634094i	$-0.781373\pi$
$523$		−	29.0024i		−	1.26819i	0.773256	−	0.634094i	$-0.218627\pi$
$524$		0			0
$525$	11.1386	−	8.01544i	0.486128	−	0.349823i
$526$		0			0
$527$			5.34363i			0.232772i
$528$		0			0
$529$	22.3723			0.972708
$530$		0			0
$531$	−17.4891			−0.758963
$532$		0			0
$533$		0			0
$534$		0			0
$535$	14.1168	−	4.55134i	0.610324	−	0.196772i
$536$		0			0
$537$			10.1899i			0.439728i
$538$		0			0
$539$	−5.00000			−0.215365
$540$		0			0
$541$	34.2337			1.47182			0.735911	−	0.677079i	$-0.236754\pi$
$541$	34.2337			1.47182			0.735911	+	0.677079i	$0.236754\pi$
$542$		0			0
$543$			19.1075i			0.819980i
$544$		0			0
$545$	−6.86141	−	21.2819i	−0.293910	−	0.911618i
$546$		0			0
$547$		−	29.0024i		−	1.24005i	−0.784580	−	0.620027i	$-0.787122\pi$
$547$		−	29.0024i		−	1.24005i	0.784580	−	0.620027i	$-0.212878\pi$
$548$		0			0
$549$	−1.76631			−0.0753844
$550$		0			0
$551$	−34.9783			−1.49012
$552$		0			0
$553$			4.34896i			0.184937i
$554$		0			0
$555$	−0.591046	−	1.83324i	−0.0250885	−	0.0778167i
$556$		0			0
$557$			0.994667i			0.0421454i	0.999778	+	0.0210727i	$0.00670814\pi$
$557$			0.994667i			0.0421454i	−0.999778	+	0.0210727i	$0.993292\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	1.25544			0.0530046
$562$		0			0
$563$			18.9051i			0.796754i	0.917222	+	0.398377i	$0.130426\pi$
$563$			18.9051i			0.796754i	−0.917222	+	0.398377i	$0.869574\pi$
$564$		0			0
$565$	1.05842	−	0.341241i	0.0445281	−	0.0143561i
$566$		0			0
$567$			12.9715i			0.544754i
$568$		0			0
$569$	27.2554			1.14261			0.571304	−	0.820739i	$-0.306438\pi$
$569$	27.2554			1.14261			0.571304	+	0.820739i	$0.306438\pi$
$570$		0			0
$571$	−1.48913			−0.0623180			−0.0311590	−	0.999514i	$-0.509920\pi$
$571$	−1.48913			−0.0623180			−0.0311590	+	0.999514i	$0.509920\pi$
$572$		0			0
$573$			15.3484i			0.641189i
$574$		0			0
$575$	2.31386	+	3.21543i	0.0964946	+	0.134093i
$576$		0			0
$577$		−	21.8719i		−	0.910537i	−0.890354	−	0.455269i	$-0.849543\pi$
$577$		−	21.8719i		−	0.910537i	0.890354	−	0.455269i	$-0.150457\pi$
$578$		0			0
$579$	13.0217			0.541165
$580$		0			0
$581$	22.9783			0.953298
$582$		0			0
$583$			10.0974i			0.418190i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	41.2743i		−	1.70357i	−0.523890	−	0.851786i	$-0.675520\pi$
$587$		−	41.2743i		−	1.70357i	0.523890	−	0.851786i	$-0.324480\pi$
$588$		0			0
$589$	−13.4891			−0.555810
$590$		0			0
$591$	6.74456			0.277434
$592$		0			0
$593$		−	22.7739i		−	0.935214i	−0.883937	−	0.467607i	$-0.845116\pi$
$593$		−	22.7739i		−	0.935214i	0.883937	−	0.467607i	$-0.154884\pi$
$594$		0			0
$595$	3.76631	+	11.6819i	0.154404	+	0.478912i
$596$		0			0
$597$		−	6.33830i		−	0.259409i
$598$		0			0
$599$	10.9783			0.448559			0.224280	−	0.974525i	$-0.427997\pi$
$599$	10.9783			0.448559			0.224280	+	0.974525i	$0.427997\pi$
$600$		0			0
$601$	−38.4674			−1.56912			−0.784558	−	0.620055i	$-0.787110\pi$
$601$	−38.4674			−1.56912			−0.784558	+	0.620055i	$0.787110\pi$
$602$		0			0
$603$			22.0742i			0.898932i
$604$		0			0
$605$	0.686141	+	2.12819i	0.0278956	+	0.0865234i
$606$		0			0
$607$			3.46410i			0.140604i	0.997526	+	0.0703018i	$0.0223962\pi$
$607$			3.46410i			0.140604i	−0.997526	+	0.0703018i	$0.977604\pi$
$608$		0			0
$609$	24.0000			0.972529
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	4.34896i		−	0.175653i	−0.996136	−	0.0878265i	$-0.972008\pi$
$613$		−	4.34896i		−	0.175653i	0.996136	−	0.0878265i	$-0.0279921\pi$
$614$		0			0
$615$	−14.7446	+	4.75372i	−0.594558	+	0.191689i
$616$		0			0
$617$			17.0256i			0.685423i	0.939441	+	0.342712i	$0.111345\pi$
$617$			17.0256i			0.685423i	−0.939441	+	0.342712i	$0.888655\pi$
$618$		0			0
$619$	14.1168			0.567404			0.283702	−	0.958913i	$-0.408437\pi$
$619$	14.1168			0.567404			0.283702	+	0.958913i	$0.408437\pi$
$620$		0			0
$621$	3.37228			0.135325
$622$		0			0
$623$		−	4.75372i		−	0.190454i
$624$		0			0
$625$	7.94158	−	23.7051i	0.317663	−	0.948204i
$626$		0			0
$627$			3.16915i			0.126564i
$628$		0			0
$629$	1.72281			0.0686931
$630$		0			0
$631$	−23.6060			−0.939739			−0.469869	−	0.882736i	$-0.655699\pi$
$631$	−23.6060			−0.939739			−0.469869	+	0.882736i	$0.655699\pi$
$632$		0			0
$633$		−	1.17981i		−	0.0468934i
$634$		0			0
$635$	17.4891	−	5.63858i	0.694035	−	0.223760i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	24.0000			0.949425
$640$		0			0
$641$	−25.3723			−1.00214			−0.501072	−	0.865405i	$-0.667061\pi$
$641$	−25.3723			−1.00214			−0.501072	+	0.865405i	$0.667061\pi$
$642$		0			0
$643$			30.4944i			1.20258i	0.799030	+	0.601292i	$0.205347\pi$
$643$			30.4944i			1.20258i	−0.799030	+	0.601292i	$0.794653\pi$
$644$		0			0
$645$	1.88316	+	5.84096i	0.0741492	+	0.229988i
$646$		0			0
$647$		−	21.9817i		−	0.864188i	−0.901829	−	0.432094i	$-0.857775\pi$
$647$		−	21.9817i		−	0.864188i	0.901829	−	0.432094i	$-0.142225\pi$
$648$		0			0
$649$	−7.37228			−0.289387
$650$		0			0
$651$	9.25544			0.362749
$652$		0			0
$653$		−	30.7894i		−	1.20488i	−0.798163	−	0.602441i	$-0.794195\pi$
$653$		−	30.7894i		−	1.20488i	0.798163	−	0.602441i	$-0.205805\pi$
$654$		0			0
$655$	1.88316	+	5.84096i	0.0735810	+	0.228225i
$656$		0			0
$657$		−	16.4356i		−	0.641216i
$658$		0			0
$659$	21.2554			0.827994			0.413997	−	0.910278i	$-0.364132\pi$
$659$	21.2554			0.827994			0.413997	+	0.910278i	$0.364132\pi$
$660$		0			0
$661$	−16.3505			−0.635962			−0.317981	−	0.948097i	$-0.603005\pi$
$661$	−16.3505			−0.635962			−0.317981	+	0.948097i	$0.603005\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−29.4891	+	9.50744i	−1.14354	+	0.368683i
$666$		0			0
$667$			6.92820i			0.268261i
$668$		0			0
$669$	−1.88316			−0.0728070
$670$		0			0
$671$	−0.744563			−0.0287435
$672$		0			0
$673$			18.6101i			0.717368i	0.933459	+	0.358684i	$0.116774\pi$
$673$			18.6101i			0.717368i	−0.933459	+	0.358684i	$0.883226\pi$
$674$		0			0
$675$	−12.4307	−	17.2742i	−0.478458	−	0.664885i
$676$		0			0
$677$		−	50.0820i		−	1.92481i	−0.271623	−	0.962404i	$-0.587560\pi$
$677$		−	50.0820i		−	1.92481i	0.271623	−	0.962404i	$-0.412440\pi$
$678$		0			0
$679$	20.2337			0.776498
$680$		0			0
$681$	−13.2554			−0.507949
$682$		0			0
$683$		−	17.9104i		−	0.685323i	−0.939459	−	0.342661i	$-0.888672\pi$
$683$		−	17.9104i		−	0.685323i	0.939459	−	0.342661i	$-0.111328\pi$
$684$		0			0
$685$	30.5475	−	9.84868i	1.16716	−	0.376299i
$686$		0			0
$687$		−	11.5894i		−	0.442161i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−44.8614			−1.70661			−0.853304	−	0.521413i	$-0.825405\pi$
$691$	−44.8614			−1.70661			−0.853304	+	0.521413i	$0.825405\pi$
$692$		0			0
$693$			8.21782i			0.312169i
$694$		0			0
$695$	−11.1386	−	34.5484i	−0.422511	−	1.31050i
$696$		0			0
$697$		−	13.8564i		−	0.524849i
$698$		0			0
$699$	2.97825			0.112648
$700$		0			0
$701$	−12.5109			−0.472529			−0.236265	−	0.971689i	$-0.575923\pi$
$701$	−12.5109			−0.472529			−0.236265	+	0.971689i	$0.575923\pi$
$702$		0			0
$703$			4.34896i			0.164024i
$704$		0			0
$705$	3.60597	+	11.1846i	0.135809	+	0.421236i
$706$		0			0
$707$			20.7846i			0.781686i
$708$		0			0
$709$	−23.8832			−0.896951			−0.448475	−	0.893795i	$-0.648033\pi$
$709$	−23.8832			−0.896951			−0.448475	+	0.893795i	$0.648033\pi$
$710$		0			0
$711$	2.97825			0.111693
$712$		0			0
$713$			2.67181i			0.100060i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	11.6819i		−	0.436269i
$718$		0			0
$719$	−30.3505			−1.13188			−0.565942	−	0.824445i	$-0.691487\pi$
$719$	−30.3505			−1.13188			−0.565942	+	0.824445i	$0.691487\pi$
$720$		0			0
$721$	−36.0000			−1.34071
$722$		0			0
$723$			13.2665i			0.493386i
$724$		0			0
$725$	35.4891	−	25.5383i	1.31803	−	0.948470i
$726$		0			0
$727$			14.0588i			0.521412i	0.965418	+	0.260706i	$0.0839552\pi$
$727$			14.0588i			0.521412i	−0.965418	+	0.260706i	$0.916045\pi$
$728$		0			0
$729$	−1.23369			−0.0456921
$730$		0			0
$731$	−5.48913			−0.203023
$732$		0			0
$733$		−	9.50744i		−	0.351165i	−0.984465	−	0.175583i	$-0.943819\pi$
$733$		−	9.50744i		−	0.351165i	0.984465	−	0.175583i	$-0.0561809\pi$
$734$		0			0
$735$	8.43070	−	2.71810i	0.310971	−	0.100259i
$736$		0			0
$737$			9.30506i			0.342756i
$738$		0			0
$739$	−10.7446			−0.395245			−0.197623	−	0.980278i	$-0.563322\pi$
$739$	−10.7446			−0.395245			−0.197623	+	0.980278i	$0.563322\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	11.3870i		−	0.417747i	−0.977943	−	0.208874i	$-0.933020\pi$
$743$		−	11.3870i		−	0.417747i	0.977943	−	0.208874i	$-0.0669798\pi$
$744$		0			0
$745$	7.88316	+	24.4511i	0.288816	+	0.895819i
$746$		0			0
$747$		−	15.7359i		−	0.575748i
$748$		0			0
$749$	22.9783			0.839607
$750$		0			0
$751$	−27.3723			−0.998829			−0.499414	−	0.866363i	$-0.666451\pi$
$751$	−27.3723			−0.998829			−0.499414	+	0.866363i	$0.666451\pi$
$752$		0			0
$753$			17.5229i			0.638570i
$754$		0			0
$755$	8.39403	+	26.0357i	0.305490	+	0.947535i
$756$		0			0
$757$		−	39.7995i		−	1.44654i	−0.690567	−	0.723269i	$-0.742639\pi$
$757$		−	39.7995i		−	1.44654i	0.690567	−	0.723269i	$-0.257361\pi$
$758$		0			0
$759$	0.627719			0.0227847
$760$		0			0
$761$	32.7446			1.18699			0.593495	−	0.804838i	$-0.297748\pi$
$761$	32.7446			1.18699			0.593495	+	0.804838i	$0.297748\pi$
$762$		0			0
$763$		−	34.6410i		−	1.25409i
$764$		0			0
$765$	8.00000	−	2.57924i	0.289241	−	0.0932526i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−29.2119			−1.05341			−0.526705	−	0.850048i	$-0.676573\pi$
$769$	−29.2119			−1.05341			−0.526705	+	0.850048i	$0.676573\pi$
$770$		0			0
$771$	8.46738			0.304945
$772$		0			0
$773$		−	17.6155i		−	0.633584i	−0.948495	−	0.316792i	$-0.897394\pi$
$773$		−	17.6155i		−	0.633584i	0.948495	−	0.316792i	$-0.102606\pi$
$774$		0			0
$775$	13.6861	−	9.84868i	0.491621	−	0.353775i
$776$		0			0
$777$		−	2.98400i		−	0.107050i
$778$		0			0
$779$	34.9783			1.25323
$780$		0			0
$781$	10.1168			0.362009
$782$		0			0
$783$		−	37.2203i		−	1.33014i
$784$		0			0
$785$	−52.0367	+	16.7769i	−1.85727	+	0.598793i
$786$		0			0
$787$		−	15.1460i		−	0.539898i	−0.962875	−	0.269949i	$-0.912993\pi$
$787$		−	15.1460i		−	0.539898i	0.962875	−	0.269949i	$-0.0870068\pi$
$788$		0			0
$789$	21.7228			0.773353
$790$		0			0
$791$	1.72281			0.0612562
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−5.48913	−	17.0256i	−0.194679	−	0.603834i
$796$		0			0
$797$		−	5.25106i		−	0.186002i	−0.995666	−	0.0930010i	$-0.970354\pi$
$797$		−	5.25106i		−	0.186002i	0.995666	−	0.0930010i	$-0.0296460\pi$
$798$		0			0
$799$	−10.5109			−0.371848
$800$		0			0
$801$	−3.25544			−0.115025
$802$		0			0
$803$		−	6.92820i		−	0.244491i
$804$		0			0
$805$	1.88316	+	5.84096i	0.0663725	+	0.205867i
$806$		0			0
$807$			9.10268i			0.320430i
$808$		0			0
$809$	32.7446			1.15124			0.575619	−	0.817718i	$-0.304761\pi$
$809$	32.7446			1.15124			0.575619	+	0.817718i	$0.304761\pi$
$810$		0			0
$811$	0.233688			0.00820589			0.00410295	−	0.999992i	$-0.498694\pi$
$811$	0.233688			0.00820589			0.00410295	+	0.999992i	$0.498694\pi$
$812$		0			0
$813$		−	10.6873i		−	0.374819i
$814$		0			0
$815$	−7.37228	+	2.37686i	−0.258240	+	0.0832578i
$816$		0			0
$817$		−	13.8564i		−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.0000			−0.628204			−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000			−0.628204			−0.314102	+	0.949389i	$0.601703\pi$
$822$		0			0
$823$		−	56.0328i		−	1.95318i	−0.215111	−	0.976590i	$-0.569011\pi$
$823$		−	56.0328i		−	1.95318i	0.215111	−	0.976590i	$-0.430989\pi$
$824$		0			0
$825$	−2.31386	−	3.21543i	−0.0805582	−	0.111947i
$826$		0			0
$827$			28.4125i			0.988000i	0.869462	+	0.494000i	$0.164466\pi$
$827$			28.4125i			0.988000i	−0.869462	+	0.494000i	$0.835534\pi$
$828$		0			0
$829$	20.3505			0.706803			0.353402	−	0.935472i	$-0.385025\pi$
$829$	20.3505			0.706803			0.353402	+	0.935472i	$0.385025\pi$
$830$		0			0
$831$	−9.25544			−0.321068
$832$		0			0
$833$			7.92287i			0.274511i
$834$		0			0
$835$	33.4891	−	10.7971i	1.15894	−	0.373648i
$836$		0			0
$837$		−	14.3537i		−	0.496138i
$838$		0			0
$839$	−10.1168			−0.349272			−0.174636	−	0.984633i	$-0.555875\pi$
$839$	−10.1168			−0.349272			−0.174636	+	0.984633i	$0.555875\pi$
$840$		0			0
$841$	47.4674			1.63681
$842$		0			0
$843$		−	0.404759i		−	0.0139407i
$844$		0			0
$845$	8.91983	+	27.6665i	0.306851	+	0.951757i
$846$		0			0
$847$			3.46410i			0.119028i
$848$		0			0
$849$	12.0000			0.411839
$850$		0			0
$851$	0.861407			0.0295286
$852$		0			0
$853$			35.0458i			1.19994i	0.800021	+	0.599972i	$0.204822\pi$
$853$			35.0458i			1.19994i	−0.800021	+	0.599972i	$0.795178\pi$
$854$		0			0
$855$	6.51087	+	20.1947i	0.222667	+	0.690644i
$856$		0			0
$857$			23.9538i			0.818245i	0.912480	+	0.409122i	$0.134165\pi$
$857$			23.9538i			0.818245i	−0.912480	+	0.409122i	$0.865835\pi$
$858$		0			0
$859$	−6.11684			−0.208704			−0.104352	−	0.994540i	$-0.533277\pi$
$859$	−6.11684			−0.208704			−0.104352	+	0.994540i	$0.533277\pi$
$860$		0			0
$861$	−24.0000			−0.817918
$862$		0			0
$863$			2.87419i			0.0978387i	0.998803	+	0.0489194i	$0.0155777\pi$
$863$			2.87419i			0.0978387i	−0.998803	+	0.0489194i	$0.984422\pi$
$864$		0			0
$865$	18.1168	−	5.84096i	0.615991	−	0.198599i
$866$		0			0
$867$			11.4795i			0.389866i
$868$		0			0
$869$	1.25544			0.0425878
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	13.8564i		−	0.468968i
$874$		0			0
$875$	22.9783	−	31.1769i	0.776807	−	1.05397i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	−2.51087			−0.0846897
$880$		0			0
$881$	−6.86141			−0.231167			−0.115583	−	0.993298i	$-0.536874\pi$
$881$	−6.86141			−0.231167			−0.115583	+	0.993298i	$0.536874\pi$
$882$		0			0
$883$		−	24.2487i		−	0.816034i	−0.912974	−	0.408017i	$-0.866220\pi$
$883$		−	24.2487i		−	0.816034i	0.912974	−	0.408017i	$-0.133780\pi$
$884$		0			0
$885$	12.4307	−	4.00772i	0.417854	−	0.134718i
$886$		0			0
$887$		−	27.4179i		−	0.920602i	−0.887763	−	0.460301i	$-0.847742\pi$
$887$		−	27.4179i		−	0.920602i	0.887763	−	0.460301i	$-0.152258\pi$
$888$		0			0
$889$	28.4674			0.954765
$890$		0			0
$891$	3.74456			0.125448
$892$		0			0
$893$		−	26.5330i		−	0.887893i
$894$		0			0
$895$	8.82473	+	27.3716i	0.294978	+	0.914931i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	29.4891			0.983517
$900$		0			0
$901$	16.0000			0.533037
$902$		0			0
$903$			9.50744i			0.316388i
$904$		0			0
$905$	16.5475	+	51.3253i	0.550059	+	1.70611i
$906$		0			0
$907$		−	19.8997i		−	0.660760i	−0.943848	−	0.330380i	$-0.892823\pi$
$907$		−	19.8997i		−	0.660760i	0.943848	−	0.330380i	$-0.107177\pi$
$908$		0			0
$909$	14.2337			0.472102
$910$		0			0
$911$	53.4891			1.77217			0.886087	−	0.463519i	$-0.153413\pi$
$911$	53.4891			1.77217			0.886087	+	0.463519i	$0.153413\pi$
$912$		0			0
$913$		−	6.63325i		−	0.219529i
$914$		0			0
$915$	1.25544	−	0.404759i	0.0415035	−	0.0133809i
$916$		0			0
$917$			9.50744i			0.313963i
$918$		0			0
$919$	−28.2337			−0.931343			−0.465672	−	0.884958i	$-0.654187\pi$
$919$	−28.2337			−0.931343			−0.465672	+	0.884958i	$0.654187\pi$
$920$		0			0
$921$	−25.0217			−0.824495
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−3.17527	−	4.41248i	−0.104402	−	0.145081i
$926$		0			0
$927$			24.6535i			0.809726i
$928$		0			0
$929$	7.02175			0.230376			0.115188	−	0.993344i	$-0.463253\pi$
$929$	7.02175			0.230376			0.115188	+	0.993344i	$0.463253\pi$
$930$		0			0
$931$	−20.0000			−0.655474
$932$		0			0
$933$		−	4.34896i		−	0.142379i
$934$		0			0
$935$	3.37228	−	1.08724i	0.110285	−	0.0355566i
$936$		0			0
$937$		−	53.6559i		−	1.75286i	−0.481527	−	0.876431i	$-0.659918\pi$
$937$		−	53.6559i		−	1.75286i	0.481527	−	0.876431i	$-0.340082\pi$
$938$		0			0
$939$	17.3288			0.565503
$940$		0			0
$941$	−58.4674			−1.90598			−0.952991	−	0.302999i	$-0.902012\pi$
$941$	−58.4674			−1.90598			−0.952991	+	0.302999i	$0.902012\pi$
$942$		0			0
$943$		−	6.92820i		−	0.225613i
$944$		0			0
$945$	−10.1168	−	31.3793i	−0.329101	−	1.02077i
$946$		0			0
$947$		−	26.7354i		−	0.868783i	−0.900724	−	0.434392i	$-0.856963\pi$
$947$		−	26.7354i		−	0.868783i	0.900724	−	0.434392i	$-0.143037\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	26.1168			0.846897
$952$		0			0
$953$			31.2867i			1.01348i	0.862100	+	0.506738i	$0.169149\pi$
$953$			31.2867i			1.01348i	−0.862100	+	0.506738i	$0.830851\pi$
$954$		0			0
$955$	13.2921	+	41.2280i	0.430123	+	1.33411i
$956$		0			0
$957$		−	6.92820i		−	0.223957i
$958$		0			0
$959$	49.7228			1.60563
$960$		0			0
$961$	−19.6277			−0.633152
$962$		0			0
$963$		−	15.7359i		−	0.507083i
$964$		0			0
$965$	34.9783	−	11.2772i	1.12599	−	0.363025i
$966$		0			0
$967$		−	26.4232i		−	0.849713i	−0.905261	−	0.424856i	$-0.860325\pi$
$967$		−	26.4232i		−	0.849713i	0.905261	−	0.424856i	$-0.139675\pi$
$968$		0			0
$969$	5.02175			0.161322
$970$		0			0
$971$	9.09509			0.291875			0.145938	−	0.989294i	$-0.453380\pi$
$971$	9.09509			0.291875			0.145938	+	0.989294i	$0.453380\pi$
$972$		0			0
$973$		−	56.2351i		−	1.80282i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			50.5793i			1.61818i	0.587687	+	0.809088i	$0.300038\pi$
$977$			50.5793i			1.61818i	−0.587687	+	0.809088i	$0.699962\pi$
$978$		0			0
$979$	−1.37228			−0.0438583
$980$		0			0
$981$	−23.7228			−0.757411
$982$		0			0
$983$			24.7460i			0.789276i	0.918837	+	0.394638i	$0.129130\pi$
$983$			24.7460i			0.789276i	−0.918837	+	0.394638i	$0.870870\pi$
$984$		0			0
$985$	18.1168	−	5.84096i	0.577251	−	0.186109i
$986$		0			0
$987$			18.2054i			0.579483i
$988$		0			0
$989$	−2.74456			−0.0872720
$990$		0			0
$991$	−18.9783			−0.602864			−0.301432	−	0.953488i	$-0.597465\pi$
$991$	−18.9783			−0.602864			−0.301432	+	0.953488i	$0.597465\pi$
$992$		0			0
$993$			11.1846i			0.354932i
$994$		0			0
$995$	−5.48913	−	17.0256i	−0.174017	−	0.539746i
$996$		0			0
$997$		−	2.17448i		−	0.0688665i	−0.999407	−	0.0344333i	$-0.989037\pi$
$997$		−	2.17448i		−	0.0688665i	0.999407	−	0.0344333i	$-0.0109626\pi$
$998$		0			0
$999$	−4.62772			−0.146415

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.b.h.529.3		4
4.3	odd	2		55.2.b.a.34.4	yes	4
5.2	odd	4		4400.2.a.cc.1.3		4
5.3	odd	4		4400.2.a.cc.1.2		4
5.4	even	2	inner	880.2.b.h.529.2		4
12.11	even	2		495.2.c.a.199.1		4
20.3	even	4		275.2.a.h.1.4		4
20.7	even	4		275.2.a.h.1.1		4
20.19	odd	2		55.2.b.a.34.1	✓	4
44.3	odd	10		605.2.j.i.9.4		16
44.7	even	10		605.2.j.j.269.4		16
44.15	odd	10		605.2.j.i.269.1		16
44.19	even	10		605.2.j.j.9.1		16
44.27	odd	10		605.2.j.i.124.1		16
44.31	odd	10		605.2.j.i.444.4		16
44.35	even	10		605.2.j.j.444.1		16
44.39	even	10		605.2.j.j.124.4		16
44.43	even	2		605.2.b.c.364.1		4
60.23	odd	4		2475.2.a.bi.1.1		4
60.47	odd	4		2475.2.a.bi.1.4		4
60.59	even	2		495.2.c.a.199.4		4
220.19	even	10		605.2.j.j.9.4		16
220.39	even	10		605.2.j.j.124.1		16
220.43	odd	4		3025.2.a.ba.1.1		4
220.59	odd	10		605.2.j.i.269.4		16
220.79	even	10		605.2.j.j.444.4		16
220.87	odd	4		3025.2.a.ba.1.4		4
220.119	odd	10		605.2.j.i.444.1		16
220.139	even	10		605.2.j.j.269.1		16
220.159	odd	10		605.2.j.i.124.4		16
220.179	odd	10		605.2.j.i.9.1		16
220.219	even	2		605.2.b.c.364.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.b.a.34.1	✓	4	20.19	odd	2
55.2.b.a.34.4	yes	4	4.3	odd	2
275.2.a.h.1.1		4	20.7	even	4
275.2.a.h.1.4		4	20.3	even	4
495.2.c.a.199.1		4	12.11	even	2
495.2.c.a.199.4		4	60.59	even	2
605.2.b.c.364.1		4	44.43	even	2
605.2.b.c.364.4		4	220.219	even	2
605.2.j.i.9.1		16	220.179	odd	10
605.2.j.i.9.4		16	44.3	odd	10
605.2.j.i.124.1		16	44.27	odd	10
605.2.j.i.124.4		16	220.159	odd	10
605.2.j.i.269.1		16	44.15	odd	10
605.2.j.i.269.4		16	220.59	odd	10
605.2.j.i.444.1		16	220.119	odd	10
605.2.j.i.444.4		16	44.31	odd	10
605.2.j.j.9.1		16	44.19	even	10
605.2.j.j.9.4		16	220.19	even	10
605.2.j.j.124.1		16	220.39	even	10
605.2.j.j.124.4		16	44.39	even	10
605.2.j.j.269.1		16	220.139	even	10
605.2.j.j.269.4		16	44.7	even	10
605.2.j.j.444.1		16	44.35	even	10
605.2.j.j.444.4		16	220.79	even	10
880.2.b.h.529.2		4	5.4	even	2	inner
880.2.b.h.529.3		4	1.1	even	1	trivial
2475.2.a.bi.1.1		4	60.23	odd	4
2475.2.a.bi.1.4		4	60.47	odd	4
3025.2.a.ba.1.1		4	220.43	odd	4
3025.2.a.ba.1.4		4	220.87	odd	4
4400.2.a.cc.1.2		4	5.3	odd	4
4400.2.a.cc.1.3		4	5.2	odd	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:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.792287i q^{3} +(0.686141 + 2.12819i) q^{5} +3.46410i q^{7} +2.37228 q^{9} +1.00000 q^{11} +(-1.68614 + 0.543620i) q^{15} -1.58457i q^{17} +4.00000 q^{19} -2.74456 q^{21} -0.792287i q^{23} +(-4.05842 + 2.92048i) q^{25} +4.25639i q^{27} -8.74456 q^{29} -3.37228 q^{31} +0.792287i q^{33} +(-7.37228 + 2.37686i) q^{35} +1.08724i q^{37} +8.74456 q^{41} -3.46410i q^{43} +(1.62772 + 5.04868i) q^{45} -6.63325i q^{47} -5.00000 q^{49} +1.25544 q^{51} +10.0974i q^{53} +(0.686141 + 2.12819i) q^{55} +3.16915i q^{57} -7.37228 q^{59} -0.744563 q^{61} +8.21782i q^{63} +9.30506i q^{67} +0.627719 q^{69} +10.1168 q^{71} -6.92820i q^{73} +(-2.31386 - 3.21543i) q^{75} +3.46410i q^{77} +1.25544 q^{79} +3.74456 q^{81} -6.63325i q^{83} +(3.37228 - 1.08724i) q^{85} -6.92820i q^{87} -1.37228 q^{89} -2.67181i q^{93} +(2.74456 + 8.51278i) q^{95} -5.84096i q^{97} +2.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} - 2 q^{9} + 4 q^{11} - q^{15} + 16 q^{19} + 12 q^{21} + q^{25} - 12 q^{29} - 2 q^{31} - 18 q^{35} + 12 q^{41} + 18 q^{45} - 20 q^{49} + 28 q^{51} - 3 q^{55} - 18 q^{59} + 20 q^{61} + 14 q^{69}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - 3 * q^5 - 2 * q^9 + 4 * q^11 - q^15 + 16 * q^19 + 12 * q^21 + q^25 - 12 * q^29 - 2 * q^31 - 18 * q^35 + 12 * q^41 + 18 * q^45 - 20 * q^49 + 28 * q^51 - 3 * q^55 - 18 * q^59 + 20 * q^61 + 14 * q^69 + 6 * q^71 - 15 * q^75 + 28 * q^79 - 8 * q^81 + 2 * q^85 + 6 * q^89 - 12 * q^95 - 2 * q^99

Introduction

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Modular forms

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Representations

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Database

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