LMFDB - Embedded newform 900.2.k.a.307.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,2,Mod(307,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.307"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	900.k (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,0,0,0,-6,4,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,-2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(26)] == traces)

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

Self dual:	no
Analytic conductor:	$7.18653618192$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			307.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	900.307
Dual form			900.2.k.a.343.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(-3.00000 + 3.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} -6.00000i q^{14} -4.00000 q^{16} +(-1.00000 - 1.00000i) q^{23} +(6.00000 + 6.00000i) q^{28} -6.00000i q^{29} +(4.00000 - 4.00000i) q^{32} -12.0000 q^{41} +(-9.00000 - 9.00000i) q^{43} +2.00000 q^{46} +(-7.00000 + 7.00000i) q^{47} -11.0000i q^{49} -12.0000 q^{56} +(6.00000 + 6.00000i) q^{58} -8.00000 q^{61} +8.00000i q^{64} +(-3.00000 + 3.00000i) q^{67} +(12.0000 - 12.0000i) q^{82} +(-11.0000 - 11.0000i) q^{83} +18.0000 q^{86} -6.00000i q^{89} +(-2.00000 + 2.00000i) q^{92} -14.0000i q^{94} +(11.0000 + 11.0000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 6 q^{7} + 4 q^{8} - 8 q^{16} - 2 q^{23} + 12 q^{28} + 8 q^{32} - 24 q^{41} - 18 q^{43} + 4 q^{46} - 14 q^{47} - 24 q^{56} + 12 q^{58} - 16 q^{61} - 6 q^{67} + 24 q^{82} - 22 q^{83} + 36 q^{86}+ \cdots + 22 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 6 * q^7 + 4 * q^8 - 8 * q^16 - 2 * q^23 + 12 * q^28 + 8 * q^32 - 24 * q^41 - 18 * q^43 + 4 * q^46 - 14 * q^47 - 24 * q^56 + 12 * q^58 - 16 * q^61 - 6 * q^67 + 24 * q^82 - 22 * q^83 + 36 * q^86 - 4 * q^92 + 22 * q^98

Character values

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

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$3$		0			0
$3$		0			0
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.144370	+	0.989524i	$0.546115\pi$
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.989524	+	0.144370i	$0.953885\pi$
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$		0			0
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		−	6.00000i		−	1.60357i
$15$		0			0
$16$	−4.00000			−1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00000	−	1.00000i	−0.208514	−	0.208514i	0.595121	−	0.803636i	$-0.297104\pi$
$23$	−1.00000	−	1.00000i	−0.208514	−	0.208514i	−0.803636	+	0.595121i	$0.797104\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	6.00000	+	6.00000i	1.13389	+	1.13389i
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.0000			−1.87409			−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000			−1.87409			−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$	−9.00000	−	9.00000i	−1.37249	−	1.37249i	−0.856742	−	0.515745i	$-0.827515\pi$
$43$	−9.00000	−	9.00000i	−1.37249	−	1.37249i	−0.515745	−	0.856742i	$-0.672485\pi$
$44$		0			0
$45$		0			0
$46$	2.00000			0.294884
$47$	−7.00000	+	7.00000i	−1.02105	+	1.02105i	−0.0212814	+	0.999774i	$0.506775\pi$
$47$	−7.00000	+	7.00000i	−1.02105	+	1.02105i	−0.999774	+	0.0212814i	$0.993225\pi$
$48$		0			0
$49$		−	11.0000i		−	1.57143i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$	−12.0000			−1.60357
$57$		0			0
$58$	6.00000	+	6.00000i	0.787839	+	0.787839i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$		0			0
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	−0.866202	−	0.499694i	$-0.833446\pi$
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	0.499694	+	0.866202i	$0.333446\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$	12.0000	−	12.0000i	1.32518	−	1.32518i
$83$	−11.0000	−	11.0000i	−1.20741	−	1.20741i	−0.971864	−	0.235543i	$-0.924313\pi$
$83$	−11.0000	−	11.0000i	−1.20741	−	1.20741i	−0.235543	−	0.971864i	$-0.575687\pi$
$84$		0			0
$85$		0			0
$86$	18.0000			1.94099
$87$		0			0
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$		0			0
$92$	−2.00000	+	2.00000i	−0.208514	+	0.208514i
$93$		0			0
$94$		−	14.0000i		−	1.44399i
$95$		0			0
$96$		0			0
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	11.0000	+	11.0000i	1.11117	+	1.11117i
$99$		0			0
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$	−9.00000	−	9.00000i	−0.886796	−	0.886796i	0.107418	−	0.994214i	$-0.465742\pi$
$103$	−9.00000	−	9.00000i	−0.886796	−	0.886796i	−0.994214	+	0.107418i	$0.965742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.0000	−	13.0000i	1.25676	−	1.25676i	0.304125	−	0.952632i	$-0.401636\pi$
$107$	13.0000	−	13.0000i	1.25676	−	1.25676i	0.952632	−	0.304125i	$-0.0983642\pi$
$108$		0			0
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$		0			0
$112$	12.0000	−	12.0000i	1.13389	−	1.13389i
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$	−12.0000			−1.11417
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	8.00000	−	8.00000i	0.724286	−	0.724286i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.00000	+	3.00000i	−0.266207	+	0.266207i	−0.827570	−	0.561363i	$-0.810277\pi$
$127$	−3.00000	+	3.00000i	−0.266207	+	0.266207i	0.561363	+	0.827570i	$0.310277\pi$
$128$	−8.00000	−	8.00000i	−0.707107	−	0.707107i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		−	6.00000i		−	0.518321i
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			24.0000i			1.96616i	0.183186	+	0.983078i	$0.441359\pi$
$149$			24.0000i			1.96616i	−0.183186	+	0.983078i	$0.558641\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	0.260531	−	0.965465i	$-0.416102\pi$
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	−0.965465	+	0.260531i	$0.916102\pi$
$164$			24.0000i			1.87409i
$165$		0			0
$166$	22.0000			1.70753
$167$	−17.0000	+	17.0000i	−1.31550	+	1.31550i	−0.398202	+	0.917298i	$0.630366\pi$
$167$	−17.0000	+	17.0000i	−1.31550	+	1.31550i	−0.917298	+	0.398202i	$0.869634\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$		0			0
$172$	−18.0000	+	18.0000i	−1.37249	+	1.37249i
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	6.00000	+	6.00000i	0.449719	+	0.449719i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		−	4.00000i		−	0.294884i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	14.0000	+	14.0000i	1.02105	+	1.02105i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	−22.0000			−1.57143
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$	−18.0000	+	18.0000i	−1.26648	+	1.26648i
$203$	18.0000	+	18.0000i	1.26335	+	1.26335i
$204$		0			0
$205$		0			0
$206$	18.0000			1.25412
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$			26.0000i			1.77732i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−16.0000	−	16.0000i	−1.08366	−	1.08366i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	21.0000	+	21.0000i	1.40626	+	1.40626i	0.778001	+	0.628263i	$0.216234\pi$
$223$	21.0000	+	21.0000i	1.40626	+	1.40626i	0.628263	+	0.778001i	$0.283766\pi$
$224$			24.0000i			1.60357i
$225$		0			0
$226$		0			0
$227$	−7.00000	+	7.00000i	−0.464606	+	0.464606i	−0.900162	−	0.435556i	$-0.856552\pi$
$227$	−7.00000	+	7.00000i	−0.464606	+	0.464606i	0.435556	+	0.900162i	$0.356552\pi$
$228$		0			0
$229$		−	14.0000i		−	0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$		−	14.0000i		−	0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$		0			0
$231$		0			0
$232$	12.0000	−	12.0000i	0.787839	−	0.787839i
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−28.0000			−1.80364			−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000			−1.80364			−0.901819	+	0.432113i	$0.857768\pi$
$242$	−11.0000	+	11.0000i	−0.707107	+	0.707107i
$243$		0			0
$244$			16.0000i			1.02430i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		−	6.00000i		−	0.376473i
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.0000	−	11.0000i	−0.678289	−	0.678289i	0.281324	−	0.959613i	$-0.409226\pi$
$263$	−11.0000	−	11.0000i	−0.678289	−	0.678289i	−0.959613	+	0.281324i	$0.909226\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	6.00000	+	6.00000i	0.366508	+	0.366508i
$269$			24.0000i			1.46331i	0.681677	+	0.731653i	$0.261251\pi$
$269$			24.0000i			1.46331i	−0.681677	+	0.731653i	$0.738749\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$	21.0000	+	21.0000i	1.24832	+	1.24832i	0.956461	+	0.291859i	$0.0942738\pi$
$283$	21.0000	+	21.0000i	1.24832	+	1.24832i	0.291859	+	0.956461i	$0.405726\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	36.0000	−	36.0000i	2.12501	−	2.12501i
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−24.0000	−	24.0000i	−1.39028	−	1.39028i
$299$		0			0
$300$		0			0
$301$	54.0000			3.11251
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−3.00000	+	3.00000i	−0.171219	+	0.171219i	−0.787515	−	0.616296i	$-0.788633\pi$
$307$	−3.00000	+	3.00000i	−0.171219	+	0.171219i	0.616296	+	0.787515i	$0.288633\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−6.00000	+	6.00000i	−0.334367	+	0.334367i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	18.0000			0.996928
$327$		0			0
$328$	−24.0000	−	24.0000i	−1.32518	−	1.32518i
$329$		−	42.0000i		−	2.31553i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−22.0000	+	22.0000i	−1.20741	+	1.20741i
$333$		0			0
$334$		−	34.0000i		−	1.86040i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−13.0000	−	13.0000i	−0.707107	−	0.707107i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$		−	36.0000i		−	1.94099i
$345$		0			0
$346$		0			0
$347$	−17.0000	+	17.0000i	−0.912608	+	0.912608i	−0.996477	−	0.0838690i	$-0.973272\pi$
$347$	−17.0000	+	17.0000i	−0.912608	+	0.912608i	0.0838690	+	0.996477i	$0.473272\pi$
$348$		0			0
$349$			26.0000i			1.39175i	0.718164	+	0.695874i	$0.244983\pi$
$349$			26.0000i			1.39175i	−0.718164	+	0.695874i	$0.755017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$	−12.0000			−0.635999
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$	−2.00000	+	2.00000i	−0.105118	+	0.105118i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	27.0000	−	27.0000i	1.40939	−	1.40939i	0.646333	−	0.763055i	$-0.276302\pi$
$367$	27.0000	−	27.0000i	1.40939	−	1.40939i	0.763055	−	0.646333i	$-0.223698\pi$
$368$	4.00000	+	4.00000i	0.208514	+	0.208514i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$	−28.0000			−1.44399
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	19.0000	+	19.0000i	0.970855	+	0.970855i	0.999587	−	0.0287325i	$-0.00914709\pi$
$383$	19.0000	+	19.0000i	0.970855	+	0.970855i	−0.0287325	+	0.999587i	$0.509147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			24.0000i			1.21685i	0.793612	+	0.608424i	$0.208198\pi$
$389$			24.0000i			1.21685i	−0.793612	+	0.608424i	$0.791802\pi$
$390$		0			0
$391$		0			0
$392$	22.0000	−	22.0000i	1.11117	−	1.11117i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$		0			0
$404$		−	36.0000i		−	1.79107i
$405$		0			0
$406$	−36.0000			−1.78665
$407$		0			0
$408$		0			0
$409$		−	4.00000i		−	0.197787i	−0.995098	−	0.0988936i	$-0.968470\pi$
$409$		−	4.00000i		−	0.197787i	0.995098	−	0.0988936i	$-0.0315304\pi$
$410$		0			0
$411$		0			0
$412$	−18.0000	+	18.0000i	−0.886796	+	0.886796i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	24.0000	−	24.0000i	1.16144	−	1.16144i
$428$	−26.0000	−	26.0000i	−1.25676	−	1.25676i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$	32.0000			1.53252
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	29.0000	+	29.0000i	1.37783	+	1.37783i	0.848274	+	0.529558i	$0.177642\pi$
$443$	29.0000	+	29.0000i	1.37783	+	1.37783i	0.529558	+	0.848274i	$0.322358\pi$
$444$		0			0
$445$		0			0
$446$	−42.0000			−1.98876
$447$		0			0
$448$	−24.0000	−	24.0000i	−1.13389	−	1.13389i
$449$		−	36.0000i		−	1.69895i	−0.527633	−	0.849473i	$-0.676920\pi$
$449$		−	36.0000i		−	1.69895i	0.527633	−	0.849473i	$-0.323080\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	14.0000i		−	0.657053i
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$	14.0000	+	14.0000i	0.654177	+	0.654177i
$459$		0			0
$460$		0			0
$461$	−42.0000			−1.95614			−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000			−1.95614			−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$	−9.00000	−	9.00000i	−0.418265	−	0.418265i	0.466340	−	0.884606i	$-0.345572\pi$
$463$	−9.00000	−	9.00000i	−0.418265	−	0.418265i	−0.884606	+	0.466340i	$0.845572\pi$
$464$			24.0000i			1.11417i
$465$		0			0
$466$		0			0
$467$	23.0000	−	23.0000i	1.06431	−	1.06431i	0.0665285	−	0.997785i	$-0.478808\pi$
$467$	23.0000	−	23.0000i	1.06431	−	1.06431i	0.997785	−	0.0665285i	$-0.0211923\pi$
$468$		0			0
$469$		−	18.0000i		−	0.831163i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	28.0000	−	28.0000i	1.27537	−	1.27537i
$483$		0			0
$484$		−	22.0000i		−	1.00000i
$485$		0			0
$486$		0			0
$487$	27.0000	−	27.0000i	1.22349	−	1.22349i	0.257103	−	0.966384i	$-0.417232\pi$
$487$	27.0000	−	27.0000i	1.22349	−	1.22349i	0.966384	−	0.257103i	$-0.0827679\pi$
$488$	−16.0000	−	16.0000i	−0.724286	−	0.724286i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−31.0000	−	31.0000i	−1.38222	−	1.38222i	−0.840663	−	0.541559i	$-0.817834\pi$
$503$	−31.0000	−	31.0000i	−1.38222	−	1.38222i	−0.541559	−	0.840663i	$-0.682166\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	6.00000	+	6.00000i	0.266207	+	0.266207i
$509$		−	6.00000i		−	0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$		−	6.00000i		−	0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−42.0000			−1.84005			−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000			−1.84005			−0.920027	+	0.391856i	$0.871833\pi$
$522$		0			0
$523$	21.0000	+	21.0000i	0.918266	+	0.918266i	0.996903	−	0.0786374i	$-0.0250569\pi$
$523$	21.0000	+	21.0000i	0.918266	+	0.918266i	−0.0786374	+	0.996903i	$0.525057\pi$
$524$		0			0
$525$		0			0
$526$	22.0000			0.959246
$527$		0			0
$528$		0			0
$529$		−	21.0000i		−	0.913043i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−12.0000			−0.518321
$537$		0			0
$538$	−24.0000	−	24.0000i	−1.03471	−	1.03471i
$539$		0			0
$540$		0			0
$541$	−38.0000			−1.63375			−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000			−1.63375			−0.816874	+	0.576816i	$0.804295\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−33.0000	+	33.0000i	−1.41098	+	1.41098i	−0.657685	+	0.753293i	$0.728464\pi$
$547$	−33.0000	+	33.0000i	−1.41098	+	1.41098i	−0.753293	+	0.657685i	$0.771536\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	12.0000	−	12.0000i	0.506189	−	0.506189i
$563$	−1.00000	−	1.00000i	−0.0421450	−	0.0421450i	0.685720	−	0.727865i	$-0.259487\pi$
$563$	−1.00000	−	1.00000i	−0.0421450	−	0.0421450i	−0.727865	+	0.685720i	$0.759487\pi$
$564$		0			0
$565$		0			0
$566$	−42.0000			−1.76539
$567$		0			0
$568$		0			0
$569$		−	36.0000i		−	1.50920i	−0.656186	−	0.754599i	$-0.727831\pi$
$569$		−	36.0000i		−	1.50920i	0.656186	−	0.754599i	$-0.272169\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$			72.0000i			3.00522i
$575$		0			0
$576$		0			0
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	17.0000	+	17.0000i	0.707107	+	0.707107i
$579$		0			0
$580$		0			0
$581$	66.0000			2.73814
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.00000	+	7.00000i	−0.288921	+	0.288921i	−0.836653	−	0.547733i	$-0.815491\pi$
$587$	−7.00000	+	7.00000i	−0.288921	+	0.288921i	0.547733	+	0.836653i	$0.315491\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$	48.0000			1.96616
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−28.0000			−1.14214			−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000			−1.14214			−0.571072	+	0.820900i	$0.693472\pi$
$602$	−54.0000	+	54.0000i	−2.20088	+	2.20088i
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−33.0000	+	33.0000i	−1.33943	+	1.33943i	−0.442816	+	0.896612i	$0.646021\pi$
$607$	−33.0000	+	33.0000i	−1.33943	+	1.33943i	−0.896612	+	0.442816i	$0.853979\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		−	6.00000i		−	0.242140i
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.0000	+	18.0000i	0.721155	+	0.721155i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0000			−0.473972			−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000			−0.473972			−0.236986	+	0.971513i	$0.576159\pi$
$642$		0			0
$643$	21.0000	+	21.0000i	0.828159	+	0.828159i	0.987262	−	0.159103i	$-0.0508601\pi$
$643$	21.0000	+	21.0000i	0.828159	+	0.828159i	−0.159103	+	0.987262i	$0.550860\pi$
$644$		−	12.0000i		−	0.472866i
$645$		0			0
$646$		0			0
$647$	13.0000	−	13.0000i	0.511083	−	0.511083i	−0.403775	−	0.914858i	$-0.632302\pi$
$647$	13.0000	−	13.0000i	0.511083	−	0.511083i	0.914858	+	0.403775i	$0.132302\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−18.0000	+	18.0000i	−0.704934	+	0.704934i
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$	48.0000			1.87409
$657$		0			0
$658$	42.0000	+	42.0000i	1.63733	+	1.63733i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	32.0000			1.24466			0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000			1.24466			0.622328	+	0.782757i	$0.286187\pi$
$662$		0			0
$663$		0			0
$664$		−	44.0000i		−	1.70753i
$665$		0			0
$666$		0			0
$667$	−6.00000	+	6.00000i	−0.232321	+	0.232321i
$668$	34.0000	+	34.0000i	1.31550	+	1.31550i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$	26.0000			1.00000
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−31.0000	−	31.0000i	−1.18618	−	1.18618i	−0.978114	−	0.208068i	$-0.933283\pi$
$683$	−31.0000	−	31.0000i	−1.18618	−	1.18618i	−0.208068	−	0.978114i	$-0.566717\pi$
$684$		0			0
$685$		0			0
$686$	−24.0000			−0.916324
$687$		0			0
$688$	36.0000	+	36.0000i	1.37249	+	1.37249i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		−	34.0000i		−	1.29062i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−26.0000	−	26.0000i	−0.984115	−	0.984115i
$699$		0			0
$700$		0			0
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−54.0000	+	54.0000i	−2.03088	+	2.03088i
$708$		0			0
$709$			46.0000i			1.72757i	0.503864	+	0.863783i	$0.331911\pi$
$709$			46.0000i			1.72757i	−0.503864	+	0.863783i	$0.668089\pi$
$710$		0			0
$711$		0			0
$712$	12.0000	−	12.0000i	0.449719	−	0.449719i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	54.0000			2.01107
$722$	19.0000	−	19.0000i	0.707107	−	0.707107i
$723$		0			0
$724$		−	4.00000i		−	0.148659i
$725$		0			0
$726$		0			0
$727$	−3.00000	+	3.00000i	−0.111264	+	0.111264i	−0.760547	−	0.649283i	$-0.775069\pi$
$727$	−3.00000	+	3.00000i	−0.111264	+	0.111264i	0.649283	+	0.760547i	$0.275069\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$			54.0000i			1.99318i
$735$		0			0
$736$	−8.00000			−0.294884
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	19.0000	+	19.0000i	0.697042	+	0.697042i	0.963772	−	0.266729i	$-0.0859429\pi$
$743$	19.0000	+	19.0000i	0.697042	+	0.697042i	−0.266729	+	0.963772i	$0.585943\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			78.0000i			2.85006i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	28.0000	−	28.0000i	1.02105	−	1.02105i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$	−48.0000	−	48.0000i	−1.73772	−	1.73772i
$764$		0			0
$765$		0			0
$766$	−38.0000			−1.37300
$767$		0			0
$768$		0			0
$769$		−	14.0000i		−	0.504853i	−0.967616	−	0.252426i	$-0.918771\pi$
$769$		−	14.0000i		−	0.504853i	0.967616	−	0.252426i	$-0.0812286\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−24.0000	−	24.0000i	−0.860442	−	0.860442i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			44.0000i			1.57143i
$785$		0			0
$786$		0			0
$787$	27.0000	−	27.0000i	0.962446	−	0.962446i	−0.0368739	−	0.999320i	$-0.511740\pi$
$787$	27.0000	−	27.0000i	0.962446	−	0.962446i	0.999320	+	0.0368739i	$0.0117400\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	−18.0000	+	18.0000i	−0.635602	+	0.635602i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	36.0000	+	36.0000i	1.26648	+	1.26648i
$809$			54.0000i			1.89854i	0.314464	+	0.949269i	$0.398175\pi$
$809$			54.0000i			1.89854i	−0.314464	+	0.949269i	$0.601825\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	36.0000	−	36.0000i	1.26335	−	1.26335i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	4.00000	+	4.00000i	0.139857	+	0.139857i
$819$		0			0
$820$		0			0
$821$	48.0000			1.67521			0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000			1.67521			0.837606	+	0.546275i	$0.183955\pi$
$822$		0			0
$823$	−39.0000	−	39.0000i	−1.35945	−	1.35945i	−0.874588	−	0.484866i	$-0.838868\pi$
$823$	−39.0000	−	39.0000i	−1.35945	−	1.35945i	−0.484866	−	0.874588i	$-0.661132\pi$
$824$		−	36.0000i		−	1.25412i
$825$		0			0
$826$		0			0
$827$	23.0000	−	23.0000i	0.799788	−	0.799788i	−0.183274	−	0.983062i	$-0.558669\pi$
$827$	23.0000	−	23.0000i	0.799788	−	0.799788i	0.983062	+	0.183274i	$0.0586694\pi$
$828$		0			0
$829$			56.0000i			1.94496i	0.232986	+	0.972480i	$0.425151\pi$
$829$			56.0000i			1.94496i	−0.232986	+	0.972480i	$0.574849\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$	8.00000	−	8.00000i	0.275698	−	0.275698i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−33.0000	+	33.0000i	−1.13389	+	1.13389i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$			48.0000i			1.64253i
$855$		0			0
$856$	52.0000			1.77732
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−41.0000	−	41.0000i	−1.39566	−	1.39566i	−0.812003	−	0.583653i	$-0.801623\pi$
$863$	−41.0000	−	41.0000i	−1.39566	−	1.39566i	−0.583653	−	0.812003i	$-0.698377\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−32.0000	+	32.0000i	−1.08366	+	1.08366i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−12.0000			−0.404290			−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000			−0.404290			−0.202145	+	0.979356i	$0.564791\pi$
$882$		0			0
$883$	−39.0000	−	39.0000i	−1.31245	−	1.31245i	−0.919601	−	0.392853i	$-0.871488\pi$
$883$	−39.0000	−	39.0000i	−1.31245	−	1.31245i	−0.392853	−	0.919601i	$-0.628512\pi$
$884$		0			0
$885$		0			0
$886$	−58.0000			−1.94855
$887$	−37.0000	+	37.0000i	−1.24234	+	1.24234i	−0.283310	+	0.959028i	$0.591433\pi$
$887$	−37.0000	+	37.0000i	−1.24234	+	1.24234i	−0.959028	+	0.283310i	$0.908567\pi$
$888$		0			0
$889$		−	18.0000i		−	0.603701i
$890$		0			0
$891$		0			0
$892$	42.0000	−	42.0000i	1.40626	−	1.40626i
$893$		0			0
$894$		0			0
$895$		0			0
$896$	48.0000			1.60357
$897$		0			0
$898$	36.0000	+	36.0000i	1.20134	+	1.20134i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−3.00000	+	3.00000i	−0.0996134	+	0.0996134i	−0.755157	−	0.655544i	$-0.772439\pi$
$907$	−3.00000	+	3.00000i	−0.0996134	+	0.0996134i	0.655544	+	0.755157i	$0.272439\pi$
$908$	14.0000	+	14.0000i	0.464606	+	0.464606i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−28.0000			−0.925146
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	42.0000	−	42.0000i	1.38320	−	1.38320i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	18.0000			0.591517
$927$		0			0
$928$	−24.0000	−	24.0000i	−0.787839	−	0.787839i
$929$		−	36.0000i		−	1.18112i	−0.806993	−	0.590561i	$-0.798907\pi$
$929$		−	36.0000i		−	1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$			46.0000i			1.50517i
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$	18.0000	+	18.0000i	0.587721	+	0.587721i
$939$		0			0
$940$		0			0
$941$	−42.0000			−1.36916			−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000			−1.36916			−0.684580	+	0.728937i	$0.740015\pi$
$942$		0			0
$943$	12.0000	+	12.0000i	0.390774	+	0.390774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	43.0000	−	43.0000i	1.39731	−	1.39731i	0.589662	−	0.807650i	$-0.299261\pi$
$947$	43.0000	−	43.0000i	1.39731	−	1.39731i	0.807650	−	0.589662i	$-0.200739\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$			56.0000i			1.80364i
$965$		0			0
$966$		0			0
$967$	−33.0000	+	33.0000i	−1.06121	+	1.06121i	−0.0632081	+	0.998000i	$0.520133\pi$
$967$	−33.0000	+	33.0000i	−1.06121	+	1.06121i	−0.998000	+	0.0632081i	$0.979867\pi$
$968$	22.0000	+	22.0000i	0.707107	+	0.707107i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$			54.0000i			1.73027i
$975$		0			0
$976$	32.0000			1.02430
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	29.0000	+	29.0000i	0.924956	+	0.924956i	0.997374	−	0.0724180i	$-0.0230716\pi$
$983$	29.0000	+	29.0000i	0.924956	+	0.924956i	−0.0724180	+	0.997374i	$0.523072\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			18.0000i			0.572367i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.2.k.a.307.1		2
3.2	odd	2		100.2.e.c.7.1	yes	2
4.3	odd	2		900.2.k.e.307.1		2
5.2	odd	4	inner	900.2.k.a.343.1		2
5.3	odd	4		900.2.k.e.343.1		2
5.4	even	2		900.2.k.e.307.1		2
12.11	even	2		100.2.e.a.7.1	✓	2
15.2	even	4		100.2.e.c.43.1	yes	2
15.8	even	4		100.2.e.a.43.1	yes	2
15.14	odd	2		100.2.e.a.7.1	✓	2
20.3	even	4	inner	900.2.k.a.343.1		2
20.7	even	4		900.2.k.e.343.1		2
20.19	odd	2	CM	900.2.k.a.307.1		2
24.5	odd	2		1600.2.n.b.1407.1		2
24.11	even	2		1600.2.n.l.1407.1		2
60.23	odd	4		100.2.e.c.43.1	yes	2
60.47	odd	4		100.2.e.a.43.1	yes	2
60.59	even	2		100.2.e.c.7.1	yes	2
120.29	odd	2		1600.2.n.l.1407.1		2
120.53	even	4		1600.2.n.l.1343.1		2
120.59	even	2		1600.2.n.b.1407.1		2
120.77	even	4		1600.2.n.b.1343.1		2
120.83	odd	4		1600.2.n.b.1343.1		2
120.107	odd	4		1600.2.n.l.1343.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.e.a.7.1	✓	2	12.11	even	2
100.2.e.a.7.1	✓	2	15.14	odd	2
100.2.e.a.43.1	yes	2	15.8	even	4
100.2.e.a.43.1	yes	2	60.47	odd	4
100.2.e.c.7.1	yes	2	3.2	odd	2
100.2.e.c.7.1	yes	2	60.59	even	2
100.2.e.c.43.1	yes	2	15.2	even	4
100.2.e.c.43.1	yes	2	60.23	odd	4
900.2.k.a.307.1		2	1.1	even	1	trivial
900.2.k.a.307.1		2	20.19	odd	2	CM
900.2.k.a.343.1		2	5.2	odd	4	inner
900.2.k.a.343.1		2	20.3	even	4	inner
900.2.k.e.307.1		2	4.3	odd	2
900.2.k.e.307.1		2	5.4	even	2
900.2.k.e.343.1		2	5.3	odd	4
900.2.k.e.343.1		2	20.7	even	4
1600.2.n.b.1343.1		2	120.77	even	4
1600.2.n.b.1343.1		2	120.83	odd	4
1600.2.n.b.1407.1		2	24.5	odd	2
1600.2.n.b.1407.1		2	120.59	even	2
1600.2.n.l.1343.1		2	120.53	even	4
1600.2.n.l.1343.1		2	120.107	odd	4
1600.2.n.l.1407.1		2	24.11	even	2
1600.2.n.l.1407.1		2	120.29	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(-3.00000 + 3.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} -6.00000i q^{14} -4.00000 q^{16} +(-1.00000 - 1.00000i) q^{23} +(6.00000 + 6.00000i) q^{28} -6.00000i q^{29} +(4.00000 - 4.00000i) q^{32} -12.0000 q^{41} +(-9.00000 - 9.00000i) q^{43} +2.00000 q^{46} +(-7.00000 + 7.00000i) q^{47} -11.0000i q^{49} -12.0000 q^{56} +(6.00000 + 6.00000i) q^{58} -8.00000 q^{61} +8.00000i q^{64} +(-3.00000 + 3.00000i) q^{67} +(12.0000 - 12.0000i) q^{82} +(-11.0000 - 11.0000i) q^{83} +18.0000 q^{86} -6.00000i q^{89} +(-2.00000 + 2.00000i) q^{92} -14.0000i q^{94} +(11.0000 + 11.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 6 q^{7} + 4 q^{8} - 8 q^{16} - 2 q^{23} + 12 q^{28} + 8 q^{32} - 24 q^{41} - 18 q^{43} + 4 q^{46} - 14 q^{47} - 24 q^{56} + 12 q^{58} - 16 q^{61} - 6 q^{67} + 24 q^{82} - 22 q^{83} + 36 q^{86}+ \cdots + 22 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 6 * q^7 + 4 * q^8 - 8 * q^16 - 2 * q^23 + 12 * q^28 + 8 * q^32 - 24 * q^41 - 18 * q^43 + 4 * q^46 - 14 * q^47 - 24 * q^56 + 12 * q^58 - 16 * q^61 - 6 * q^67 + 24 * q^82 - 22 * q^83 + 36 * q^86 - 4 * q^92 + 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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