LMFDB - Embedded newform 1920.2.b.c.191.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1920,2,Mod(191,1920)] 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("1920.191"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.10070523904.11
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{4} + 81 $ x^8 - 10*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.5
Root			$0.420861 - 1.68014i$ of defining polynomial
Character	$\chi$	$=$	1920.191
Dual form			1920.2.b.c.191.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.420861 - 1.68014i) q^{3} -1.00000 q^{5} -2.16991i q^{7} +(-2.64575 - 1.41421i) q^{9} -5.15587i q^{13} +(-0.420861 + 1.68014i) q^{15} -2.32744i q^{17} +3.06871 q^{19} +(-3.64575 - 0.913230i) q^{21} -5.59388 q^{23} +1.00000 q^{25} +(-3.48957 + 3.85005i) q^{27} +1.29150 q^{29} +7.91094i q^{31} +2.16991i q^{35} -5.15587i q^{37} +(-8.66259 - 2.16991i) q^{39} +7.48331i q^{41} +8.66259 q^{43} +(2.64575 + 1.41421i) q^{45} -11.7313 q^{47} +2.29150 q^{49} +(-3.91044 - 0.979531i) q^{51} -8.58301 q^{53} +(1.29150 - 5.15587i) q^{57} -4.33981i q^{59} -8.48528i q^{61} +(-3.06871 + 5.74103i) q^{63} +5.15587i q^{65} -8.66259 q^{67} +(-2.35425 + 9.39851i) q^{69} -11.1878 q^{71} -2.00000 q^{73} +(0.420861 - 1.68014i) q^{75} +3.57113i q^{79} +(5.00000 + 7.48331i) q^{81} -5.74103i q^{83} +2.32744i q^{85} +(0.543544 - 2.16991i) q^{87} +1.00197i q^{89} -11.1878 q^{91} +(13.2915 + 3.32941i) q^{93} -3.06871 q^{95} -16.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} - 8 q^{21} + 8 q^{25} - 32 q^{29} - 24 q^{49} + 16 q^{53} - 32 q^{57} - 40 q^{69} - 16 q^{73} + 40 q^{81} + 64 q^{93} - 48 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 8 * q^21 + 8 * q^25 - 32 * q^29 - 24 * q^49 + 16 * q^53 - 32 * q^57 - 40 * q^69 - 16 * q^73 + 40 * q^81 + 64 * q^93 - 48 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	0.420861	−	1.68014i	0.242984	−	0.970030i
$3$	0.420861	−	1.68014i	0.242984	−	0.970030i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	2.16991i		−	0.820148i	−0.912052	−	0.410074i	$-0.865503\pi$
$7$		−	2.16991i		−	0.820148i	0.912052	−	0.410074i	$-0.134497\pi$
$8$		0			0
$9$	−2.64575	−	1.41421i	−0.881917	−	0.471405i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		−	5.15587i		−	1.42998i	−0.699134	−	0.714991i	$-0.746431\pi$
$13$		−	5.15587i		−	1.42998i	0.699134	−	0.714991i	$-0.253569\pi$
$14$		0			0
$15$	−0.420861	+	1.68014i	−0.108666	+	0.433811i
$16$		0			0
$17$		−	2.32744i		−	0.564488i	−0.959343	−	0.282244i	$-0.908921\pi$
$17$		−	2.32744i		−	0.564488i	0.959343	−	0.282244i	$-0.0910788\pi$
$18$		0			0
$19$	3.06871			0.704011			0.352005	−	0.935998i	$-0.385500\pi$
$19$	3.06871			0.704011			0.352005	+	0.935998i	$0.385500\pi$
$20$		0			0
$21$	−3.64575	−	0.913230i	−0.795568	−	0.199283i
$22$		0			0
$23$	−5.59388			−1.16640			−0.583202	−	0.812327i	$-0.698201\pi$
$23$	−5.59388			−1.16640			−0.583202	+	0.812327i	$0.698201\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−3.48957	+	3.85005i	−0.671569	+	0.740942i
$28$		0			0
$29$	1.29150			0.239826			0.119913	−	0.992784i	$-0.461738\pi$
$29$	1.29150			0.239826			0.119913	+	0.992784i	$0.461738\pi$
$30$		0			0
$31$			7.91094i			1.42085i	0.703774	+	0.710424i	$0.251497\pi$
$31$			7.91094i			1.42085i	−0.703774	+	0.710424i	$0.748503\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.16991i			0.366781i
$36$		0			0
$37$		−	5.15587i		−	0.847620i	−0.905751	−	0.423810i	$-0.860692\pi$
$37$		−	5.15587i		−	0.847620i	0.905751	−	0.423810i	$-0.139308\pi$
$38$		0			0
$39$	−8.66259	−	2.16991i	−1.38713	−	0.347463i
$40$		0			0
$41$			7.48331i			1.16870i	0.811503	+	0.584349i	$0.198650\pi$
$41$			7.48331i			1.16870i	−0.811503	+	0.584349i	$0.801350\pi$
$42$		0			0
$43$	8.66259			1.32103			0.660517	−	0.750812i	$-0.270337\pi$
$43$	8.66259			1.32103			0.660517	+	0.750812i	$0.270337\pi$
$44$		0			0
$45$	2.64575	+	1.41421i	0.394405	+	0.210819i
$46$		0			0
$47$	−11.7313			−1.71119			−0.855593	−	0.517649i	$-0.826807\pi$
$47$	−11.7313			−1.71119			−0.855593	+	0.517649i	$0.826807\pi$
$48$		0			0
$49$	2.29150			0.327358
$50$		0			0
$51$	−3.91044	−	0.979531i	−0.547570	−	0.137162i
$52$		0			0
$53$	−8.58301			−1.17897			−0.589483	−	0.807781i	$-0.700669\pi$
$53$	−8.58301			−1.17897			−0.589483	+	0.807781i	$0.700669\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.29150	−	5.15587i	0.171064	−	0.682912i
$58$		0			0
$59$		−	4.33981i		−	0.564996i	−0.959268	−	0.282498i	$-0.908837\pi$
$59$		−	4.33981i		−	0.564996i	0.959268	−	0.282498i	$-0.0911630\pi$
$60$		0			0
$61$		−	8.48528i		−	1.08643i	−0.839594	−	0.543214i	$-0.817207\pi$
$61$		−	8.48528i		−	1.08643i	0.839594	−	0.543214i	$-0.182793\pi$
$62$		0			0
$63$	−3.06871	+	5.74103i	−0.386621	+	0.723302i
$64$		0			0
$65$			5.15587i			0.639507i
$66$		0			0
$67$	−8.66259			−1.05830			−0.529152	−	0.848527i	$-0.677490\pi$
$67$	−8.66259			−1.05830			−0.529152	+	0.848527i	$0.677490\pi$
$68$		0			0
$69$	−2.35425	+	9.39851i	−0.283418	+	1.13145i
$70$		0			0
$71$	−11.1878			−1.32774			−0.663872	−	0.747847i	$-0.731088\pi$
$71$	−11.1878			−1.32774			−0.663872	+	0.747847i	$0.731088\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$	0.420861	−	1.68014i	0.0485969	−	0.194006i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			3.57113i			0.401783i	0.979613	+	0.200892i	$0.0643839\pi$
$79$			3.57113i			0.401783i	−0.979613	+	0.200892i	$0.935616\pi$
$80$		0			0
$81$	5.00000	+	7.48331i	0.555556	+	0.831479i
$82$		0			0
$83$		−	5.74103i		−	0.630160i	−0.949065	−	0.315080i	$-0.897969\pi$
$83$		−	5.74103i		−	0.630160i	0.949065	−	0.315080i	$-0.102031\pi$
$84$		0			0
$85$			2.32744i			0.252447i
$86$		0			0
$87$	0.543544	−	2.16991i	0.0582740	−	0.232638i
$88$		0			0
$89$			1.00197i			0.106208i	0.998589	+	0.0531041i	$0.0169115\pi$
$89$			1.00197i			0.106208i	−0.998589	+	0.0531041i	$0.983088\pi$
$90$		0			0
$91$	−11.1878			−1.17280
$92$		0			0
$93$	13.2915	+	3.32941i	1.37826	+	0.345244i
$94$		0			0
$95$	−3.06871			−0.314843
$96$		0			0
$97$	−16.5830			−1.68375			−0.841875	−	0.539673i	$-0.818548\pi$
$97$	−16.5830			−1.68375			−0.841875	+	0.539673i	$0.818548\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	13.2915			1.32255			0.661277	−	0.750142i	$-0.270015\pi$
$101$	13.2915			1.32255			0.661277	+	0.750142i	$0.270015\pi$
$102$		0			0
$103$		−	6.50972i		−	0.641422i	−0.947177	−	0.320711i	$-0.896078\pi$
$103$		−	6.50972i		−	0.641422i	0.947177	−	0.320711i	$-0.103922\pi$
$104$		0			0
$105$	3.64575	+	0.913230i	0.355789	+	0.0891221i
$106$		0			0
$107$			5.74103i			0.555007i	0.960725	+	0.277503i	$0.0895070\pi$
$107$			5.74103i			0.555007i	−0.960725	+	0.277503i	$0.910493\pi$
$108$		0			0
$109$			1.82646i			0.174943i	0.996167	+	0.0874716i	$0.0278787\pi$
$109$			1.82646i			0.174943i	−0.996167	+	0.0874716i	$0.972121\pi$
$110$		0			0
$111$	−8.66259	−	2.16991i	−0.822217	−	0.205958i
$112$		0			0
$113$		−	7.98430i		−	0.751100i	−0.926802	−	0.375550i	$-0.877454\pi$
$113$		−	7.98430i		−	0.751100i	0.926802	−	0.375550i	$-0.122546\pi$
$114$		0			0
$115$	5.59388			0.521632
$116$		0			0
$117$	−7.29150	+	13.6412i	−0.674100	+	1.26112i
$118$		0			0
$119$	−5.05034			−0.462964
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	12.5730	+	3.14944i	1.13367	+	0.283975i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			17.9918i			1.59651i	0.602317	+	0.798257i	$0.294244\pi$
$127$			17.9918i			1.59651i	−0.602317	+	0.798257i	$0.705756\pi$
$128$		0			0
$129$	3.64575	−	14.5544i	0.320991	−	1.28144i
$130$		0			0
$131$			15.8219i			1.38236i	0.722681	+	0.691182i	$0.242910\pi$
$131$			15.8219i			1.38236i	−0.722681	+	0.691182i	$0.757090\pi$
$132$		0			0
$133$		−	6.65882i		−	0.577393i
$134$		0			0
$135$	3.48957	−	3.85005i	0.300335	−	0.331359i
$136$		0			0
$137$		−	18.2960i		−	1.56314i	−0.623820	−	0.781568i	$-0.714420\pi$
$137$		−	18.2960i		−	1.56314i	0.623820	−	0.781568i	$-0.285580\pi$
$138$		0			0
$139$	−9.20614			−0.780854			−0.390427	−	0.920634i	$-0.627673\pi$
$139$	−9.20614			−0.780854			−0.390427	+	0.920634i	$0.627673\pi$
$140$		0			0
$141$	−4.93725	+	19.7103i	−0.415792	+	1.65990i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−1.29150			−0.107253
$146$		0			0
$147$	0.964405	−	3.85005i	0.0795428	−	0.317547i
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$		−	16.5906i		−	1.35012i	−0.737762	−	0.675061i	$-0.764117\pi$
$151$		−	16.5906i		−	1.35012i	0.737762	−	0.675061i	$-0.235883\pi$
$152$		0			0
$153$	−3.29150	+	6.15784i	−0.266102	+	0.497832i
$154$		0			0
$155$		−	7.91094i		−	0.635422i
$156$		0			0
$157$		−	1.50295i		−	0.119948i	−0.998200	−	0.0599742i	$-0.980898\pi$
$157$		−	1.50295i		−	0.119948i	0.998200	−	0.0599742i	$-0.0191019\pi$
$158$		0			0
$159$	−3.61226	+	14.4207i	−0.286471	+	1.14363i
$160$		0			0
$161$			12.1382i			0.956624i
$162$		0			0
$163$	19.8504			1.55480			0.777400	−	0.629007i	$-0.216538\pi$
$163$	19.8504			1.55480			0.777400	+	0.629007i	$0.216538\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.7313			−0.907796			−0.453898	−	0.891054i	$-0.649967\pi$
$167$	−11.7313			−0.907796			−0.453898	+	0.891054i	$0.649967\pi$
$168$		0			0
$169$	−13.5830			−1.04485
$170$		0			0
$171$	−8.11905	−	4.33981i	−0.620879	−	0.331874i
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$		−	2.16991i		−	0.164030i
$176$		0			0
$177$	−7.29150	−	1.82646i	−0.548063	−	0.137285i
$178$		0			0
$179$			4.33981i			0.324373i	0.986760	+	0.162186i	$0.0518546\pi$
$179$			4.33981i			0.324373i	−0.986760	+	0.162186i	$0.948145\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−14.2565	−	3.57113i	−1.05387	−	0.263985i
$184$		0			0
$185$			5.15587i			0.379067i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	8.35425	+	7.57205i	0.607682	+	0.550786i
$190$		0			0
$191$	5.05034			0.365430			0.182715	−	0.983166i	$-0.441512\pi$
$191$	5.05034			0.365430			0.182715	+	0.983166i	$0.441512\pi$
$192$		0			0
$193$	12.5830			0.905745			0.452872	−	0.891575i	$-0.350399\pi$
$193$	12.5830			0.905745			0.452872	+	0.891575i	$0.350399\pi$
$194$		0			0
$195$	8.66259	+	2.16991i	0.620341	+	0.155390i
$196$		0			0
$197$	20.5830			1.46648			0.733239	−	0.679971i	$-0.238008\pi$
$197$	20.5830			1.46648			0.733239	+	0.679971i	$0.238008\pi$
$198$		0			0
$199$		−	3.57113i		−	0.253151i	−0.991957	−	0.126575i	$-0.959601\pi$
$199$		−	3.57113i		−	0.253151i	0.991957	−	0.126575i	$-0.0403985\pi$
$200$		0			0
$201$	−3.64575	+	14.5544i	−0.257151	+	1.02659i
$202$		0			0
$203$		−	2.80244i		−	0.196693i
$204$		0			0
$205$		−	7.48331i		−	0.522657i
$206$		0			0
$207$	14.8000	+	7.91094i	1.02867	+	0.549848i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−26.5313			−1.82649			−0.913246	−	0.407409i	$-0.866432\pi$
$211$	−26.5313			−1.82649			−0.913246	+	0.407409i	$0.866432\pi$
$212$		0			0
$213$	−4.70850	+	18.7970i	−0.322621	+	1.28795i
$214$		0			0
$215$	−8.66259			−0.590784
$216$		0			0
$217$	17.1660			1.16530
$218$		0			0
$219$	−0.841723	+	3.36028i	−0.0568784	+	0.227067i
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$			26.6714i			1.78605i	0.450007	+	0.893025i	$0.351422\pi$
$223$			26.6714i			1.78605i	−0.450007	+	0.893025i	$0.648578\pi$
$224$		0			0
$225$	−2.64575	−	1.41421i	−0.176383	−	0.0942809i
$226$		0			0
$227$		−	18.7605i		−	1.24518i	−0.782550	−	0.622588i	$-0.786081\pi$
$227$		−	18.7605i		−	1.24518i	0.782550	−	0.622588i	$-0.213919\pi$
$228$		0			0
$229$		−	27.2823i		−	1.80287i	−0.432919	−	0.901433i	$-0.642516\pi$
$229$		−	27.2823i		−	1.80287i	0.432919	−	0.901433i	$-0.357484\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.98626i		−	0.588710i	−0.955696	−	0.294355i	$-0.904895\pi$
$233$		−	8.98626i		−	0.588710i	0.955696	−	0.294355i	$-0.0951048\pi$
$234$		0			0
$235$	11.7313			0.765266
$236$		0			0
$237$	6.00000	+	1.50295i	0.389742	+	0.0976271i
$238$		0			0
$239$	6.13742			0.396997			0.198498	−	0.980101i	$-0.436394\pi$
$239$	6.13742			0.396997			0.198498	+	0.980101i	$0.436394\pi$
$240$		0			0
$241$	5.29150			0.340856			0.170428	−	0.985370i	$-0.445485\pi$
$241$	5.29150			0.340856			0.170428	+	0.985370i	$0.445485\pi$
$242$		0			0
$243$	14.6773	−	5.25127i	0.941551	−	0.336869i
$244$		0			0
$245$	−2.29150			−0.146399
$246$		0			0
$247$		−	15.8219i		−	1.00672i
$248$		0			0
$249$	−9.64575	−	2.41618i	−0.611275	−	0.153119i
$250$		0			0
$251$		−	17.3593i		−	1.09571i	−0.836574	−	0.547853i	$-0.815445\pi$
$251$		−	17.3593i		−	1.09571i	0.836574	−	0.547853i	$-0.184555\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	3.91044	+	0.979531i	0.244881	+	0.0613406i
$256$		0			0
$257$		−	1.32548i		−	0.0826810i	−0.999145	−	0.0413405i	$-0.986837\pi$
$257$		−	1.32548i		−	0.0826810i	0.999145	−	0.0413405i	$-0.0131628\pi$
$258$		0			0
$259$	−11.1878			−0.695174
$260$		0			0
$261$	−3.41699	−	1.82646i	−0.211507	−	0.113055i
$262$		0			0
$263$	17.8687			1.10183			0.550917	−	0.834560i	$-0.314278\pi$
$263$	17.8687			1.10183			0.550917	+	0.834560i	$0.314278\pi$
$264$		0			0
$265$	8.58301			0.527250
$266$		0			0
$267$	1.68345	+	0.421689i	0.103025	+	0.0258070i
$268$		0			0
$269$	−23.1660			−1.41246			−0.706228	−	0.707984i	$-0.749605\pi$
$269$	−23.1660			−1.41246			−0.706228	+	0.707984i	$0.749605\pi$
$270$		0			0
$271$			0.768687i			0.0466944i	0.999727	+	0.0233472i	$0.00743232\pi$
$271$			0.768687i			0.0466944i	−0.999727	+	0.0233472i	$0.992568\pi$
$272$		0			0
$273$	−4.70850	+	18.7970i	−0.284971	+	1.13765i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	22.1264i		−	1.32945i	−0.747089	−	0.664724i	$-0.768549\pi$
$277$		−	22.1264i		−	1.32945i	0.747089	−	0.664724i	$-0.231451\pi$
$278$		0			0
$279$	11.1878	−	20.9304i	0.669794	−	1.25307i
$280$		0			0
$281$		−	3.83039i		−	0.228502i	−0.993452	−	0.114251i	$-0.963553\pi$
$281$		−	3.83039i		−	0.228502i	0.993452	−	0.114251i	$-0.0364468\pi$
$282$		0			0
$283$	24.9007			1.48019			0.740096	−	0.672501i	$-0.234780\pi$
$283$	24.9007			1.48019			0.740096	+	0.672501i	$0.234780\pi$
$284$		0			0
$285$	−1.29150	+	5.15587i	−0.0765020	+	0.305407i
$286$		0			0
$287$	16.2381			0.958505
$288$		0			0
$289$	11.5830			0.681353
$290$		0			0
$291$	−6.97915	+	27.8618i	−0.409125	+	1.63329i
$292$		0			0
$293$	23.1660			1.35337			0.676686	−	0.736271i	$-0.263415\pi$
$293$	23.1660			1.35337			0.676686	+	0.736271i	$0.263415\pi$
$294$		0			0
$295$			4.33981i			0.252674i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			28.8413i			1.66794i
$300$		0			0
$301$		−	18.7970i		−	1.08344i
$302$		0			0
$303$	5.59388	−	22.3316i	0.321360	−	1.28292i
$304$		0			0
$305$			8.48528i			0.485866i
$306$		0			0
$307$	−7.57551			−0.432357			−0.216178	−	0.976354i	$-0.569359\pi$
$307$	−7.57551			−0.432357			−0.216178	+	0.976354i	$0.569359\pi$
$308$		0			0
$309$	−10.9373	−	2.73969i	−0.622199	−	0.155856i
$310$		0			0
$311$	23.4626			1.33044			0.665221	−	0.746646i	$-0.268337\pi$
$311$	23.4626			1.33044			0.665221	+	0.746646i	$0.268337\pi$
$312$		0			0
$313$	−27.1660			−1.53551			−0.767757	−	0.640741i	$-0.778627\pi$
$313$	−27.1660			−1.53551			−0.767757	+	0.640741i	$0.778627\pi$
$314$		0			0
$315$	3.06871	−	5.74103i	0.172902	−	0.323471i
$316$		0			0
$317$	−32.5830			−1.83004			−0.915022	−	0.403404i	$-0.867827\pi$
$317$	−32.5830			−1.83004			−0.915022	+	0.403404i	$0.867827\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	9.64575	+	2.41618i	0.538373	+	0.134858i
$322$		0			0
$323$		−	7.14226i		−	0.397406i
$324$		0			0
$325$		−	5.15587i		−	0.285996i
$326$		0			0
$327$	3.06871	+	0.768687i	0.169700	+	0.0425085i
$328$		0			0
$329$			25.4558i			1.40343i
$330$		0			0
$331$	−30.4946			−1.67613			−0.838067	−	0.545568i	$-0.816314\pi$
$331$	−30.4946			−1.67613			−0.838067	+	0.545568i	$0.816314\pi$
$332$		0			0
$333$	−7.29150	+	13.6412i	−0.399572	+	0.747531i
$334$		0			0
$335$	8.66259			0.473288
$336$		0			0
$337$	−12.5830			−0.685440			−0.342720	−	0.939438i	$-0.611348\pi$
$337$	−12.5830			−0.685440			−0.342720	+	0.939438i	$0.611348\pi$
$338$		0			0
$339$	−13.4148	−	3.36028i	−0.728589	−	0.182506i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	20.1617i		−	1.08863i
$344$		0			0
$345$	2.35425	−	9.39851i	0.126748	−	0.505999i
$346$		0			0
$347$			30.2425i			1.62351i	0.584002	+	0.811753i	$0.301486\pi$
$347$			30.2425i			1.62351i	−0.584002	+	0.811753i	$0.698514\pi$
$348$		0			0
$349$		−	30.9352i		−	1.65592i	−0.560784	−	0.827962i	$-0.689500\pi$
$349$		−	30.9352i		−	1.65592i	0.560784	−	0.827962i	$-0.310500\pi$
$350$		0			0
$351$	19.8504	+	17.9918i	1.05953	+	0.960331i
$352$		0			0
$353$		−	24.9549i		−	1.32821i	−0.747638	−	0.664107i	$-0.768812\pi$
$353$		−	24.9549i		−	1.32821i	0.747638	−	0.664107i	$-0.231188\pi$
$354$		0			0
$355$	11.1878			0.593785
$356$		0			0
$357$	−2.12549	+	8.48528i	−0.112493	+	0.449089i
$358$		0			0
$359$	28.5129			1.50486			0.752428	−	0.658675i	$-0.228883\pi$
$359$	28.5129			1.50486			0.752428	+	0.658675i	$0.228883\pi$
$360$		0			0
$361$	−9.58301			−0.504369
$362$		0			0
$363$	4.62948	−	18.4816i	0.242984	−	0.970030i
$364$		0			0
$365$	2.00000			0.104685
$366$		0			0
$367$		−	33.8137i		−	1.76506i	−0.470257	−	0.882530i	$-0.655839\pi$
$367$		−	33.8137i		−	1.76506i	0.470257	−	0.882530i	$-0.344161\pi$
$368$		0			0
$369$	10.5830	−	19.7990i	0.550929	−	1.03069i
$370$		0			0
$371$			18.6243i			0.966927i
$372$		0			0
$373$		−	15.4676i		−	0.800883i	−0.916322	−	0.400441i	$-0.868857\pi$
$373$		−	15.4676i		−	0.800883i	0.916322	−	0.400441i	$-0.131143\pi$
$374$		0			0
$375$	−0.420861	+	1.68014i	−0.0217332	+	0.0867621i
$376$		0			0
$377$		−	6.65882i		−	0.342947i
$378$		0			0
$379$	9.20614			0.472887			0.236444	−	0.971645i	$-0.424018\pi$
$379$	9.20614			0.472887			0.236444	+	0.971645i	$0.424018\pi$
$380$		0			0
$381$	30.2288	+	7.57205i	1.54867	+	0.387928i
$382$		0			0
$383$	10.6442			0.543894			0.271947	−	0.962312i	$-0.412332\pi$
$383$	10.6442			0.543894			0.271947	+	0.962312i	$0.412332\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−22.9191	−	12.2508i	−1.16504	−	0.622741i
$388$		0			0
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$			13.0194i			0.658422i
$392$		0			0
$393$	26.5830	+	6.65882i	1.34094	+	0.335893i
$394$		0			0
$395$		−	3.57113i		−	0.179683i
$396$		0			0
$397$			1.50295i			0.0754309i	0.999289	+	0.0377154i	$0.0120080\pi$
$397$			1.50295i			0.0754309i	−0.999289	+	0.0377154i	$0.987992\pi$
$398$		0			0
$399$	−11.1878	−	2.80244i	−0.560089	−	0.140298i
$400$		0			0
$401$		−	26.2803i		−	1.31238i	−0.754597	−	0.656189i	$-0.772168\pi$
$401$		−	26.2803i		−	1.31238i	0.754597	−	0.656189i	$-0.227832\pi$
$402$		0			0
$403$	40.7878			2.03178
$404$		0			0
$405$	−5.00000	−	7.48331i	−0.248452	−	0.371849i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−7.87451			−0.389369			−0.194685	−	0.980866i	$-0.562368\pi$
$409$	−7.87451			−0.389369			−0.194685	+	0.980866i	$0.562368\pi$
$410$		0			0
$411$	−30.7399	−	7.70010i	−1.51629	−	0.379818i
$412$		0			0
$413$	−9.41699			−0.463380
$414$		0			0
$415$			5.74103i			0.281816i
$416$		0			0
$417$	−3.87451	+	15.4676i	−0.189735	+	0.757452i
$418$		0			0
$419$		−	37.5210i		−	1.83302i	−0.400013	−	0.916509i	$-0.630994\pi$
$419$		−	37.5210i		−	1.83302i	0.400013	−	0.916509i	$-0.369006\pi$
$420$		0			0
$421$		−	4.83236i		−	0.235515i	−0.993042	−	0.117757i	$-0.962429\pi$
$421$		−	4.83236i		−	0.235515i	0.993042	−	0.117757i	$-0.0375705\pi$
$422$		0			0
$423$	31.0381	+	16.5906i	1.50912	+	0.806661i
$424$		0			0
$425$		−	2.32744i		−	0.112898i
$426$		0			0
$427$	−18.4123			−0.891032
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.08709			0.0523632			0.0261816	−	0.999657i	$-0.491665\pi$
$431$	1.08709			0.0523632			0.0261816	+	0.999657i	$0.491665\pi$
$432$		0			0
$433$	7.41699			0.356438			0.178219	−	0.983991i	$-0.442966\pi$
$433$	7.41699			0.356438			0.178219	+	0.983991i	$0.442966\pi$
$434$		0			0
$435$	−0.543544	+	2.16991i	−0.0260609	+	0.104039i
$436$		0			0
$437$	−17.1660			−0.821162
$438$		0			0
$439$			5.10850i			0.243815i	0.992541	+	0.121908i	$0.0389012\pi$
$439$			5.10850i			0.243815i	−0.992541	+	0.121908i	$0.961099\pi$
$440$		0			0
$441$	−6.06275	−	3.24067i	−0.288702	−	0.154318i
$442$		0			0
$443$		−	1.40122i		−	0.0665740i	−0.999446	−	0.0332870i	$-0.989402\pi$
$443$		−	1.40122i		−	0.0665740i	0.999446	−	0.0332870i	$-0.0105975\pi$
$444$		0			0
$445$		−	1.00197i		−	0.0474978i
$446$		0			0
$447$	2.52517	−	10.0808i	0.119436	−	0.476808i
$448$		0			0
$449$			19.7990i			0.934372i	0.884159	+	0.467186i	$0.154732\pi$
$449$			19.7990i			0.934372i	−0.884159	+	0.467186i	$0.845268\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−27.8745	−	6.98233i	−1.30966	−	0.328059i
$454$		0			0
$455$	11.1878			0.524490
$456$		0			0
$457$	16.5830			0.775720			0.387860	−	0.921718i	$-0.373214\pi$
$457$	16.5830			0.775720			0.387860	+	0.921718i	$0.373214\pi$
$458$		0			0
$459$	8.96077	+	8.12179i	0.418253	+	0.379093i
$460$		0			0
$461$	−13.2915			−0.619047			−0.309523	−	0.950892i	$-0.600170\pi$
$461$	−13.2915			−0.619047			−0.309523	+	0.950892i	$0.600170\pi$
$462$		0			0
$463$		−	9.31216i		−	0.432773i	−0.976308	−	0.216386i	$-0.930573\pi$
$463$		−	9.31216i		−	0.432773i	0.976308	−	0.216386i	$-0.0694271\pi$
$464$		0			0
$465$	−13.2915	−	3.32941i	−0.616379	−	0.154398i
$466$		0			0
$467$		−	5.74103i		−	0.265663i	−0.991139	−	0.132832i	$-0.957593\pi$
$467$		−	5.74103i		−	0.265663i	0.991139	−	0.132832i	$-0.0424070\pi$
$468$		0			0
$469$			18.7970i			0.867966i
$470$		0			0
$471$	−2.52517	−	0.632534i	−0.116354	−	0.0291456i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	3.06871			0.140802
$476$		0			0
$477$	22.7085	+	12.1382i	1.03975	+	0.555770i
$478$		0			0
$479$	16.2381			0.741938			0.370969	−	0.928645i	$-0.379026\pi$
$479$	16.2381			0.741938			0.370969	+	0.928645i	$0.379026\pi$
$480$		0			0
$481$	−26.5830			−1.21208
$482$		0			0
$483$	20.3939	+	5.10850i	0.927955	+	0.232445i
$484$		0			0
$485$	16.5830			0.752995
$486$		0			0
$487$			17.9918i			0.815286i	0.913141	+	0.407643i	$0.133649\pi$
$487$			17.9918i			0.815286i	−0.913141	+	0.407643i	$0.866351\pi$
$488$		0			0
$489$	8.35425	−	33.3514i	0.377792	−	1.50820i
$490$		0			0
$491$		−	24.5015i		−	1.10574i	−0.833268	−	0.552869i	$-0.813533\pi$
$491$		−	24.5015i		−	1.10574i	0.833268	−	0.552869i	$-0.186467\pi$
$492$		0			0
$493$		−	3.00590i		−	0.135379i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			24.2764i			1.08895i
$498$		0			0
$499$	−3.06871			−0.137374			−0.0686872	−	0.997638i	$-0.521881\pi$
$499$	−3.06871			−0.137374			−0.0686872	+	0.997638i	$0.521881\pi$
$500$		0			0
$501$	−4.93725	+	19.7103i	−0.220580	+	0.880589i
$502$		0			0
$503$	−0.543544			−0.0242354			−0.0121177	−	0.999927i	$-0.503857\pi$
$503$	−0.543544			−0.0242354			−0.0121177	+	0.999927i	$0.503857\pi$
$504$		0			0
$505$	−13.2915			−0.591464
$506$		0			0
$507$	−5.71656	+	22.8214i	−0.253881	+	1.01353i
$508$		0			0
$509$	3.87451			0.171735			0.0858673	−	0.996307i	$-0.472634\pi$
$509$	3.87451			0.171735			0.0858673	+	0.996307i	$0.472634\pi$
$510$		0			0
$511$			4.33981i			0.191982i
$512$		0			0
$513$	−10.7085	+	11.8147i	−0.472792	+	0.521631i
$514$		0			0
$515$			6.50972i			0.286853i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−2.52517	+	10.0808i	−0.110843	+	0.442500i
$520$		0			0
$521$		−	18.6196i		−	0.815737i	−0.913041	−	0.407869i	$-0.866272\pi$
$521$		−	18.6196i		−	0.815737i	0.913041	−	0.407869i	$-0.133728\pi$
$522$		0			0
$523$	38.2626			1.67311			0.836554	−	0.547885i	$-0.184567\pi$
$523$	38.2626			1.67311			0.836554	+	0.547885i	$0.184567\pi$
$524$		0			0
$525$	−3.64575	−	0.913230i	−0.159114	−	0.0398566i
$526$		0			0
$527$	18.4123			0.802051
$528$		0			0
$529$	8.29150			0.360500
$530$		0			0
$531$	−6.13742	+	11.4821i	−0.266342	+	0.498279i
$532$		0			0
$533$	38.5830			1.67122
$534$		0			0
$535$		−	5.74103i		−	0.248207i
$536$		0			0
$537$	7.29150	+	1.82646i	0.314652	+	0.0788176i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			16.9706i			0.729621i	0.931082	+	0.364811i	$0.118866\pi$
$541$			16.9706i			0.729621i	−0.931082	+	0.364811i	$0.881134\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1.82646i		−	0.0782370i
$546$		0			0
$547$	−24.9007			−1.06468			−0.532338	−	0.846532i	$-0.678687\pi$
$547$	−24.9007			−1.06468			−0.532338	+	0.846532i	$0.678687\pi$
$548$		0			0
$549$	−12.0000	+	22.4499i	−0.512148	+	0.958140i
$550$		0			0
$551$	3.96325			0.168840
$552$		0			0
$553$	7.74902			0.329522
$554$		0			0
$555$	8.66259	+	2.16991i	0.367707	+	0.0921074i
$556$		0			0
$557$	37.7490			1.59948			0.799739	−	0.600348i	$-0.204971\pi$
$557$	37.7490			1.59948			0.799739	+	0.600348i	$0.204971\pi$
$558$		0			0
$559$		−	44.6632i		−	1.88905i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	8.54348i		−	0.360065i	−0.983661	−	0.180032i	$-0.942380\pi$
$563$		−	8.54348i		−	0.360065i	0.983661	−	0.180032i	$-0.0576202\pi$
$564$		0			0
$565$			7.98430i			0.335902i
$566$		0			0
$567$	16.2381	−	10.8495i	0.681936	−	0.455638i
$568$		0			0
$569$			34.7656i			1.45745i	0.684806	+	0.728725i	$0.259887\pi$
$569$			34.7656i			1.45745i	−0.684806	+	0.728725i	$0.740113\pi$
$570$		0			0
$571$	−30.4946			−1.27616			−0.638079	−	0.769971i	$-0.720271\pi$
$571$	−30.4946			−1.27616			−0.638079	+	0.769971i	$0.720271\pi$
$572$		0			0
$573$	2.12549	−	8.48528i	0.0887937	−	0.354478i
$574$		0			0
$575$	−5.59388			−0.233281
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$	5.29570	−	21.1412i	0.220082	−	0.878600i
$580$		0			0
$581$	−12.4575			−0.516825
$582$		0			0
$583$		0			0
$584$		0			0
$585$	7.29150	−	13.6412i	0.301467	−	0.563992i
$586$		0			0
$587$			30.2425i			1.24824i	0.781327	+	0.624122i	$0.214543\pi$
$587$			30.2425i			1.24824i	−0.781327	+	0.624122i	$0.785457\pi$
$588$		0			0
$589$			24.2764i			1.00029i
$590$		0			0
$591$	8.66259	−	34.5824i	0.356331	−	1.42253i
$592$		0			0
$593$		−	4.33138i		−	0.177868i	−0.996038	−	0.0889342i	$-0.971654\pi$
$593$		−	4.33138i		−	0.177868i	0.996038	−	0.0889342i	$-0.0283461\pi$
$594$		0			0
$595$	5.05034			0.207044
$596$		0			0
$597$	−6.00000	−	1.50295i	−0.245564	−	0.0615116i
$598$		0			0
$599$	−6.13742			−0.250768			−0.125384	−	0.992108i	$-0.540016\pi$
$599$	−6.13742			−0.250768			−0.125384	+	0.992108i	$0.540016\pi$
$600$		0			0
$601$	22.4575			0.916061			0.458031	−	0.888936i	$-0.348555\pi$
$601$	22.4575			0.916061			0.458031	+	0.888936i	$0.348555\pi$
$602$		0			0
$603$	22.9191	+	12.2508i	0.933337	+	0.498889i
$604$		0			0
$605$	−11.0000			−0.447214
$606$		0			0
$607$			31.0112i			1.25871i	0.777119	+	0.629354i	$0.216680\pi$
$607$			31.0112i			1.25871i	−0.777119	+	0.629354i	$0.783320\pi$
$608$		0			0
$609$	−4.70850	−	1.17944i	−0.190798	−	0.0477933i
$610$		0			0
$611$			60.4851i			2.44697i
$612$		0			0
$613$			28.7853i			1.16263i	0.813680	+	0.581313i	$0.197461\pi$
$613$			28.7853i			1.16263i	−0.813680	+	0.581313i	$0.802539\pi$
$614$		0			0
$615$	−12.5730	−	3.14944i	−0.506993	−	0.126998i
$616$		0			0
$617$			38.9195i			1.56684i	0.621492	+	0.783421i	$0.286527\pi$
$617$			38.9195i			1.56684i	−0.621492	+	0.783421i	$0.713473\pi$
$618$		0			0
$619$	25.4442			1.02269			0.511345	−	0.859375i	$-0.329147\pi$
$619$	25.4442			1.02269			0.511345	+	0.859375i	$0.329147\pi$
$620$		0			0
$621$	19.5203	−	21.5367i	0.783321	−	0.864239i
$622$		0			0
$623$	2.17417			0.0871065
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$		−	9.44832i		−	0.376132i	−0.982156	−	0.188066i	$-0.939778\pi$
$631$		−	9.44832i		−	0.376132i	0.982156	−	0.188066i	$-0.0602218\pi$
$632$		0			0
$633$	−11.1660	+	44.5764i	−0.443809	+	1.77175i
$634$		0			0
$635$		−	17.9918i		−	0.713982i
$636$		0			0
$637$		−	11.8147i		−	0.468115i
$638$		0			0
$639$	29.6000	+	15.8219i	1.17096	+	0.625904i
$640$		0			0
$641$		−	28.1068i		−	1.11015i	−0.831800	−	0.555076i	$-0.812689\pi$
$641$		−	28.1068i		−	1.11015i	0.831800	−	0.555076i	$-0.187311\pi$
$642$		0			0
$643$	13.7129			0.540785			0.270393	−	0.962750i	$-0.412847\pi$
$643$	13.7129			0.540785			0.270393	+	0.962750i	$0.412847\pi$
$644$		0			0
$645$	−3.64575	+	14.5544i	−0.143551	+	0.573078i
$646$		0			0
$647$	29.0565			1.14233			0.571164	−	0.820836i	$-0.306492\pi$
$647$	29.0565			1.14233			0.571164	+	0.820836i	$0.306492\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	7.22451	−	28.8413i	0.283151	−	1.13038i
$652$		0			0
$653$	0.833990			0.0326365			0.0163183	−	0.999867i	$-0.494806\pi$
$653$	0.833990			0.0326365			0.0163183	+	0.999867i	$0.494806\pi$
$654$		0			0
$655$		−	15.8219i		−	0.618212i
$656$		0			0
$657$	5.29150	+	2.82843i	0.206441	+	0.110347i
$658$		0			0
$659$		−	4.33981i		−	0.169055i	−0.996421	−	0.0845276i	$-0.973062\pi$
$659$		−	4.33981i		−	0.169055i	0.996421	−	0.0845276i	$-0.0269381\pi$
$660$		0			0
$661$		−	8.48528i		−	0.330039i	−0.986290	−	0.165020i	$-0.947231\pi$
$661$		−	8.48528i		−	0.330039i	0.986290	−	0.165020i	$-0.0527687\pi$
$662$		0			0
$663$	−5.05034	+	20.1617i	−0.196139	+	0.783015i
$664$		0			0
$665$			6.65882i			0.258218i
$666$		0			0
$667$	−7.22451			−0.279734
$668$		0			0
$669$	44.8118	+	11.2250i	1.73252	+	0.433982i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	36.5830			1.41017			0.705086	−	0.709122i	$-0.250909\pi$
$673$	36.5830			1.41017			0.705086	+	0.709122i	$0.250909\pi$
$674$		0			0
$675$	−3.48957	+	3.85005i	−0.134314	+	0.148188i
$676$		0			0
$677$	0.833990			0.0320528			0.0160264	−	0.999872i	$-0.494898\pi$
$677$	0.833990			0.0320528			0.0160264	+	0.999872i	$0.494898\pi$
$678$		0			0
$679$			35.9836i			1.38092i
$680$		0			0
$681$	−31.5203	−	7.89556i	−1.20786	−	0.302559i
$682$		0			0
$683$		−	38.9222i		−	1.48932i	−0.667446	−	0.744658i	$-0.732613\pi$
$683$		−	38.9222i		−	1.48932i	0.667446	−	0.744658i	$-0.267387\pi$
$684$		0			0
$685$			18.2960i			0.699056i
$686$		0			0
$687$	−45.8381	−	11.4821i	−1.74883	−	0.438068i
$688$		0			0
$689$			44.2529i			1.68590i
$690$		0			0
$691$	−10.2932			−0.391573			−0.195786	−	0.980647i	$-0.562726\pi$
$691$	−10.2932			−0.391573			−0.195786	+	0.980647i	$0.562726\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.20614			0.349209
$696$		0			0
$697$	17.4170			0.659716
$698$		0			0
$699$	−15.0982	−	3.78197i	−0.571066	−	0.143047i
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$		−	15.8219i		−	0.596734i
$704$		0			0
$705$	4.93725	−	19.7103i	0.185948	−	0.742331i
$706$		0			0
$707$		−	28.8413i		−	1.08469i
$708$		0			0
$709$			10.3117i			0.387266i	0.981074	+	0.193633i	$0.0620270\pi$
$709$			10.3117i			0.387266i	−0.981074	+	0.193633i	$0.937973\pi$
$710$		0			0
$711$	5.05034	−	9.44832i	0.189402	−	0.354340i
$712$		0			0
$713$		−	44.2529i		−	1.65728i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	2.58301	−	10.3117i	0.0964641	−	0.385099i
$718$		0			0
$719$	−22.3755			−0.834466			−0.417233	−	0.908800i	$-0.637000\pi$
$719$	−22.3755			−0.834466			−0.417233	+	0.908800i	$0.637000\pi$
$720$		0			0
$721$	−14.1255			−0.526061
$722$		0			0
$723$	2.22699	−	8.89047i	0.0828226	−	0.330640i
$724$		0			0
$725$	1.29150			0.0479652
$726$		0			0
$727$			26.6714i			0.989188i	0.869124	+	0.494594i	$0.164683\pi$
$727$			26.6714i			0.989188i	−0.869124	+	0.494594i	$0.835317\pi$
$728$		0			0
$729$	−2.64575	−	26.8701i	−0.0979908	−	0.995187i
$730$		0			0
$731$		−	20.1617i		−	0.745707i
$732$		0			0
$733$		−	15.4676i		−	0.571309i	−0.958333	−	0.285655i	$-0.907789\pi$
$733$		−	15.4676i		−	0.571309i	0.958333	−	0.285655i	$-0.0922110\pi$
$734$		0			0
$735$	−0.964405	+	3.85005i	−0.0355726	+	0.142011i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−25.4442			−0.935981			−0.467991	−	0.883733i	$-0.655022\pi$
$739$	−25.4442			−0.935981			−0.467991	+	0.883733i	$0.655022\pi$
$740$		0			0
$741$	−26.5830	−	6.65882i	−0.976551	−	0.244618i
$742$		0			0
$743$	−36.2810			−1.33102			−0.665510	−	0.746389i	$-0.731786\pi$
$743$	−36.2810			−1.33102			−0.665510	+	0.746389i	$0.731786\pi$
$744$		0			0
$745$	−6.00000			−0.219823
$746$		0			0
$747$	−8.11905	+	15.1894i	−0.297060	+	0.555749i
$748$		0			0
$749$	12.4575			0.455188
$750$		0			0
$751$		−	41.0921i		−	1.49947i	−0.661737	−	0.749736i	$-0.730181\pi$
$751$		−	41.0921i		−	1.49947i	0.661737	−	0.749736i	$-0.269819\pi$
$752$		0			0
$753$	−29.1660	−	7.30584i	−1.06287	−	0.266240i
$754$		0			0
$755$			16.5906i			0.603793i
$756$		0			0
$757$			53.0617i			1.92856i	0.264887	+	0.964279i	$0.414665\pi$
$757$			53.0617i			1.92856i	−0.264887	+	0.964279i	$0.585335\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	48.9078i		−	1.77290i	−0.462820	−	0.886452i	$-0.653162\pi$
$761$		−	48.9078i		−	1.77290i	0.462820	−	0.886452i	$-0.346838\pi$
$762$		0			0
$763$	3.96325			0.143479
$764$		0			0
$765$	3.29150	−	6.15784i	0.119005	−	0.222637i
$766$		0			0
$767$	−22.3755			−0.807933
$768$		0			0
$769$	−4.58301			−0.165267			−0.0826337	−	0.996580i	$-0.526333\pi$
$769$	−4.58301			−0.165267			−0.0826337	+	0.996580i	$0.526333\pi$
$770$		0			0
$771$	−2.22699	−	0.557842i	−0.0802031	−	0.0200902i
$772$		0			0
$773$	−13.7490			−0.494518			−0.247259	−	0.968949i	$-0.579530\pi$
$773$	−13.7490			−0.494518			−0.247259	+	0.968949i	$0.579530\pi$
$774$		0			0
$775$			7.91094i			0.284169i
$776$		0			0
$777$	−4.70850	+	18.7970i	−0.168916	+	0.674339i
$778$		0			0
$779$			22.9641i			0.822776i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−4.50679	+	4.97235i	−0.161060	+	0.177697i
$784$		0			0
$785$			1.50295i			0.0536426i
$786$		0			0
$787$	−7.57551			−0.270038			−0.135019	−	0.990843i	$-0.543109\pi$
$787$	−7.57551			−0.270038			−0.135019	+	0.990843i	$0.543109\pi$
$788$		0			0
$789$	7.52026	−	30.0220i	0.267728	−	1.06881i
$790$		0			0
$791$	−17.3252			−0.616013
$792$		0			0
$793$	−43.7490			−1.55357
$794$		0			0
$795$	3.61226	−	14.4207i	0.128114	−	0.511448i
$796$		0			0
$797$	−32.5830			−1.15415			−0.577075	−	0.816691i	$-0.695806\pi$
$797$	−32.5830			−1.15415			−0.577075	+	0.816691i	$0.695806\pi$
$798$		0			0
$799$			27.3040i			0.965944i
$800$		0			0
$801$	1.41699	−	2.65095i	0.0500670	−	0.0936669i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		−	12.1382i		−	0.427815i
$806$		0			0
$807$	−9.74968	+	38.9222i	−0.343205	+	1.37013i
$808$		0			0
$809$			36.5921i			1.28651i	0.765652	+	0.643255i	$0.222416\pi$
$809$			36.5921i			1.28651i	−0.765652	+	0.643255i	$0.777584\pi$
$810$		0			0
$811$	−36.6320			−1.28632			−0.643162	−	0.765730i	$-0.722378\pi$
$811$	−36.6320			−1.28632			−0.643162	+	0.765730i	$0.722378\pi$
$812$		0			0
$813$	1.29150	+	0.323511i	0.0452950	+	0.0113460i
$814$		0			0
$815$	−19.8504			−0.695328
$816$		0			0
$817$	26.5830			0.930022
$818$		0			0
$819$	29.6000	+	15.8219i	1.03431	+	0.552861i
$820$		0			0
$821$	6.00000			0.209401			0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000			0.209401			0.104701	+	0.994504i	$0.466612\pi$
$822$		0			0
$823$			33.8137i			1.17867i	0.807889	+	0.589335i	$0.200610\pi$
$823$			33.8137i			1.17867i	−0.807889	+	0.589335i	$0.799390\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			8.54348i			0.297086i	0.988906	+	0.148543i	$0.0474583\pi$
$827$			8.54348i			0.297086i	−0.988906	+	0.148543i	$0.952542\pi$
$828$		0			0
$829$			32.1147i			1.11539i	0.830046	+	0.557694i	$0.188314\pi$
$829$			32.1147i			1.11539i	−0.830046	+	0.557694i	$0.811686\pi$
$830$		0			0
$831$	−37.1755	−	9.31216i	−1.28961	−	0.323035i
$832$		0			0
$833$		−	5.33334i		−	0.184789i
$834$		0			0
$835$	11.7313			0.405979
$836$		0			0
$837$	−30.4575	−	27.6058i	−1.05277	−	0.954197i
$838$		0			0
$839$	−53.0626			−1.83193			−0.915963	−	0.401263i	$-0.868571\pi$
$839$	−53.0626			−1.83193			−0.915963	+	0.401263i	$0.868571\pi$
$840$		0			0
$841$	−27.3320			−0.942483
$842$		0			0
$843$	−6.43560	−	1.61206i	−0.221654	−	0.0555225i
$844$		0			0
$845$	13.5830			0.467270
$846$		0			0
$847$		−	23.8690i		−	0.820148i
$848$		0			0
$849$	10.4797	−	41.8367i	0.359664	−	1.43583i
$850$		0			0
$851$			28.8413i			0.988668i
$852$		0			0
$853$		−	46.4028i		−	1.58880i	−0.607393	−	0.794401i	$-0.707785\pi$
$853$		−	46.4028i		−	1.58880i	0.607393	−	0.794401i	$-0.292215\pi$
$854$		0			0
$855$	8.11905	+	4.33981i	0.277666	+	0.148419i
$856$		0			0
$857$		−	18.2960i		−	0.624981i	−0.949921	−	0.312490i	$-0.898837\pi$
$857$		−	18.2960i		−	0.624981i	0.949921	−	0.312490i	$-0.101163\pi$
$858$		0			0
$859$	−25.4442			−0.868146			−0.434073	−	0.900878i	$-0.642924\pi$
$859$	−25.4442			−0.868146			−0.434073	+	0.900878i	$0.642924\pi$
$860$		0			0
$861$	6.83399	−	27.2823i	0.232902	−	0.929778i
$862$		0			0
$863$	−12.8184			−0.436343			−0.218172	−	0.975910i	$-0.570009\pi$
$863$	−12.8184			−0.436343			−0.218172	+	0.975910i	$0.570009\pi$
$864$		0			0
$865$	6.00000			0.204006
$866$		0			0
$867$	4.87484	−	19.4611i	0.165558	−	0.660933i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			44.6632i			1.51336i
$872$		0			0
$873$	43.8745	+	23.4519i	1.48493	+	0.793727i
$874$		0			0
$875$			2.16991i			0.0733563i
$876$		0			0
$877$		−	29.4323i		−	0.993857i	−0.867791	−	0.496929i	$-0.834461\pi$
$877$		−	29.4323i		−	0.993857i	0.867791	−	0.496929i	$-0.165539\pi$
$878$		0			0
$879$	9.74968	−	38.9222i	0.328849	−	1.31281i
$880$		0			0
$881$		−	13.1402i		−	0.442704i	−0.975194	−	0.221352i	$-0.928953\pi$
$881$		−	13.1402i		−	0.442704i	0.975194	−	0.221352i	$-0.0710469\pi$
$882$		0			0
$883$	−14.8000			−0.498060			−0.249030	−	0.968496i	$-0.580112\pi$
$883$	−14.8000			−0.498060			−0.249030	+	0.968496i	$0.580112\pi$
$884$		0			0
$885$	7.29150	+	1.82646i	0.245101	+	0.0613958i
$886$		0			0
$887$	6.68097			0.224325			0.112162	−	0.993690i	$-0.464222\pi$
$887$	6.68097			0.224325			0.112162	+	0.993690i	$0.464222\pi$
$888$		0			0
$889$	39.0405			1.30938
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−36.0000			−1.20469
$894$		0			0
$895$		−	4.33981i		−	0.145064i
$896$		0			0
$897$	48.4575	+	12.1382i	1.61795	+	0.405283i
$898$		0			0
$899$			10.2170i			0.340756i
$900$		0			0
$901$			19.9765i			0.665512i
$902$		0			0
$903$	−31.5817	−	7.91094i	−1.05097	−	0.263260i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	13.7129			0.455330			0.227665	−	0.973739i	$-0.426891\pi$
$907$	13.7129			0.455330			0.227665	+	0.973739i	$0.426891\pi$
$908$		0			0
$909$	−35.1660	−	18.7970i	−1.16638	−	0.623458i
$910$		0			0
$911$	−45.8381			−1.51869			−0.759343	−	0.650691i	$-0.774479\pi$
$911$	−45.8381			−1.51869			−0.759343	+	0.650691i	$0.774479\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	14.2565	+	3.57113i	0.471304	+	0.118058i
$916$		0			0
$917$	34.3320			1.13374
$918$		0			0
$919$			12.2508i			0.404115i	0.979374	+	0.202058i	$0.0647628\pi$
$919$			12.2508i			0.404115i	−0.979374	+	0.202058i	$0.935237\pi$
$920$		0			0
$921$	−3.18824	+	12.7279i	−0.105056	+	0.419399i
$922$		0			0
$923$			57.6827i			1.89865i
$924$		0			0
$925$		−	5.15587i		−	0.169524i
$926$		0			0
$927$	−9.20614	+	17.2231i	−0.302369	+	0.565681i
$928$		0			0
$929$			16.1461i			0.529735i	0.964285	+	0.264868i	$0.0853283\pi$
$929$			16.1461i			0.529735i	−0.964285	+	0.264868i	$0.914672\pi$
$930$		0			0
$931$	7.03196			0.230463
$932$		0			0
$933$	9.87451	−	39.4205i	0.323277	−	1.29057i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−36.3320			−1.18692			−0.593458	−	0.804865i	$-0.702238\pi$
$937$	−36.3320			−1.18692			−0.593458	+	0.804865i	$0.702238\pi$
$938$		0			0
$939$	−11.4331	+	45.6427i	−0.373106	+	1.48949i
$940$		0			0
$941$	42.4575			1.38408			0.692038	−	0.721861i	$-0.256713\pi$
$941$	42.4575			1.38408			0.692038	+	0.721861i	$0.256713\pi$
$942$		0			0
$943$		−	41.8608i		−	1.36317i
$944$		0			0
$945$	−8.35425	−	7.57205i	−0.271764	−	0.246319i
$946$		0			0
$947$		−	38.9222i		−	1.26480i	−0.774642	−	0.632400i	$-0.782070\pi$
$947$		−	38.9222i		−	1.26480i	0.774642	−	0.632400i	$-0.217930\pi$
$948$		0			0
$949$			10.3117i			0.334733i
$950$		0			0
$951$	−13.7129	+	54.7441i	−0.444672	+	1.77520i
$952$		0			0
$953$		−	28.6078i		−	0.926697i	−0.886176	−	0.463348i	$-0.846648\pi$
$953$		−	28.6078i		−	0.926697i	0.886176	−	0.463348i	$-0.153352\pi$
$954$		0			0
$955$	−5.05034			−0.163425
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−39.7007			−1.28200
$960$		0			0
$961$	−31.5830			−1.01881
$962$		0			0
$963$	8.11905	−	15.1894i	0.261633	−	0.489470i
$964$		0			0
$965$	−12.5830			−0.405061
$966$		0			0
$967$		−	32.5486i		−	1.04669i	−0.852120	−	0.523346i	$-0.824683\pi$
$967$		−	32.5486i		−	1.04669i	0.852120	−	0.523346i	$-0.175317\pi$
$968$		0			0
$969$	−12.0000	−	3.00590i	−0.385496	−	0.0965634i
$970$		0			0
$971$		−	49.0030i		−	1.57258i	−0.617856	−	0.786291i	$-0.711999\pi$
$971$		−	49.0030i		−	1.57258i	0.617856	−	0.786291i	$-0.288001\pi$
$972$		0			0
$973$			19.9765i			0.640416i
$974$		0			0
$975$	−8.66259	−	2.16991i	−0.277425	−	0.0694926i
$976$		0			0
$977$			1.32548i			0.0424058i	0.999775	+	0.0212029i	$0.00674959\pi$
$977$			1.32548i			0.0424058i	−0.999775	+	0.0212029i	$0.993250\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	2.58301	−	4.83236i	0.0824690	−	0.154285i
$982$		0			0
$983$	35.1939			1.12251			0.561256	−	0.827642i	$-0.310318\pi$
$983$	35.1939			1.12251			0.561256	+	0.827642i	$0.310318\pi$
$984$		0			0
$985$	−20.5830			−0.655829
$986$		0			0
$987$	42.7694	+	10.7134i	1.36137	+	0.341011i
$988$		0			0
$989$	−48.4575			−1.54086
$990$		0			0
$991$			23.7328i			0.753898i	0.926234	+	0.376949i	$0.123027\pi$
$991$			23.7328i			0.753898i	−0.926234	+	0.376949i	$0.876973\pi$
$992$		0			0
$993$	−12.8340	+	51.2352i	−0.407274	+	1.62590i
$994$		0			0
$995$			3.57113i			0.113212i
$996$		0			0
$997$			5.15587i			0.163288i	0.996662	+	0.0816440i	$0.0260171\pi$
$997$			5.15587i			0.163288i	−0.996662	+	0.0816440i	$0.973983\pi$
$998$		0			0
$999$	19.8504	+	17.9918i	0.628037	+	0.569235i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.b.c.191.5	yes	8
3.2	odd	2		1920.2.b.e.191.6	yes	8
4.3	odd	2	inner	1920.2.b.c.191.4	yes	8
8.3	odd	2		1920.2.b.e.191.5	yes	8
8.5	even	2		1920.2.b.e.191.4	yes	8
12.11	even	2		1920.2.b.e.191.3	yes	8
24.5	odd	2	inner	1920.2.b.c.191.3	✓	8
24.11	even	2	inner	1920.2.b.c.191.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.b.c.191.3	✓	8	24.5	odd	2	inner
1920.2.b.c.191.4	yes	8	4.3	odd	2	inner
1920.2.b.c.191.5	yes	8	1.1	even	1	trivial
1920.2.b.c.191.6	yes	8	24.11	even	2	inner
1920.2.b.e.191.3	yes	8	12.11	even	2
1920.2.b.e.191.4	yes	8	8.5	even	2
1920.2.b.e.191.5	yes	8	8.3	odd	2
1920.2.b.e.191.6	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.420861 - 1.68014i) q^{3} -1.00000 q^{5} -2.16991i q^{7} +(-2.64575 - 1.41421i) q^{9} -5.15587i q^{13} +(-0.420861 + 1.68014i) q^{15} -2.32744i q^{17} +3.06871 q^{19} +(-3.64575 - 0.913230i) q^{21} -5.59388 q^{23} +1.00000 q^{25} +(-3.48957 + 3.85005i) q^{27} +1.29150 q^{29} +7.91094i q^{31} +2.16991i q^{35} -5.15587i q^{37} +(-8.66259 - 2.16991i) q^{39} +7.48331i q^{41} +8.66259 q^{43} +(2.64575 + 1.41421i) q^{45} -11.7313 q^{47} +2.29150 q^{49} +(-3.91044 - 0.979531i) q^{51} -8.58301 q^{53} +(1.29150 - 5.15587i) q^{57} -4.33981i q^{59} -8.48528i q^{61} +(-3.06871 + 5.74103i) q^{63} +5.15587i q^{65} -8.66259 q^{67} +(-2.35425 + 9.39851i) q^{69} -11.1878 q^{71} -2.00000 q^{73} +(0.420861 - 1.68014i) q^{75} +3.57113i q^{79} +(5.00000 + 7.48331i) q^{81} -5.74103i q^{83} +2.32744i q^{85} +(0.543544 - 2.16991i) q^{87} +1.00197i q^{89} -11.1878 q^{91} +(13.2915 + 3.32941i) q^{93} -3.06871 q^{95} -16.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} - 8 q^{21} + 8 q^{25} - 32 q^{29} - 24 q^{49} + 16 q^{53} - 32 q^{57} - 40 q^{69} - 16 q^{73} + 40 q^{81} + 64 q^{93} - 48 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 8 * q^21 + 8 * q^25 - 32 * q^29 - 24 * q^49 + 16 * q^53 - 32 * q^57 - 40 * q^69 - 16 * q^73 + 40 * q^81 + 64 * q^93 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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