LMFDB - Embedded newform 504.3.e.b.251.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,3,Mod(251,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.251"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	504.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,124] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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

Self dual:	no
Analytic conductor:	$13.7330053238$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.6
Root			$0.581861 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	504.251
Dual form			504.3.e.b.251.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.125246 + 1.99607i) q^{2} +(-3.96863 + 0.500000i) q^{4} -7.00000i q^{7} +(-1.49509 - 7.85905i) q^{8} +10.7240i q^{11} +(13.9725 - 0.876721i) q^{14} +(15.5000 - 3.96863i) q^{16} +(-21.4059 + 1.34313i) q^{22} +42.6612 q^{23} +25.0000 q^{25} +(3.50000 + 27.7804i) q^{28} +53.1504 q^{29} +(9.86299 + 30.4421i) q^{32} +38.0000i q^{37} -63.4980 q^{43} +(-5.36199 - 42.5595i) q^{44} +(5.34313 + 85.1549i) q^{46} -49.0000 q^{49} +(3.13115 + 49.9019i) q^{50} +70.5905 q^{53} +(-55.0134 + 10.4656i) q^{56} +(6.65687 + 106.092i) q^{58} +(-59.5294 + 23.5000i) q^{64} +118.000 q^{67} -20.7437 q^{71} +(-75.8508 + 4.75934i) q^{74} +75.0679 q^{77} -126.996i q^{79} +(-7.95286 - 126.747i) q^{86} +(84.2804 - 16.0333i) q^{88} +(-169.306 + 21.3306i) q^{92} +(-6.13705 - 97.8077i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 124 q^{16} - 76 q^{22} + 200 q^{25} + 28 q^{28} - 116 q^{46} - 392 q^{49} + 212 q^{58} + 944 q^{67} + 452 q^{88}+O(q^{100}) $ 8 * q + 124 * q^16 - 76 * q^22 + 200 * q^25 + 28 * q^28 - 116 * q^46 - 392 * q^49 + 212 * q^58 + 944 * q^67 + 452 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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 $

$2$	0.125246	+	1.99607i	0.0626229	+	0.998037i
$2$	0.125246	+	1.99607i	0.0626229	+	0.998037i
$3$		0			0
$3$		0			0
$4$	−3.96863	+	0.500000i	−0.992157	+	0.125000i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	7.00000i		−	1.00000i
$7$		−	7.00000i		−	1.00000i
$8$	−1.49509	−	7.85905i	−0.186886	−	0.982382i
$9$		0			0
$10$		0			0
$11$			10.7240i			0.974908i	0.873149	+	0.487454i	$0.162074\pi$
$11$			10.7240i			0.974908i	−0.873149	+	0.487454i	$0.837926\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	13.9725	−	0.876721i	0.998037	−	0.0626229i
$15$		0			0
$16$	15.5000	−	3.96863i	0.968750	−	0.248039i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$	−21.4059	+	1.34313i	−0.972995	+	0.0610516i
$23$	42.6612			1.85483			0.927417	−	0.374029i	$-0.122024\pi$
$23$	42.6612			1.85483			0.927417	+	0.374029i	$0.122024\pi$
$24$		0			0
$25$	25.0000			1.00000
$26$		0			0
$27$		0			0
$28$	3.50000	+	27.7804i	0.125000	+	0.992157i
$29$	53.1504			1.83277			0.916386	−	0.400295i	$-0.131093\pi$
$29$	53.1504			1.83277			0.916386	+	0.400295i	$0.131093\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	9.86299	+	30.4421i	0.308218	+	0.951316i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			38.0000i			1.02703i	0.858082	+	0.513514i	$0.171656\pi$
$37$			38.0000i			1.02703i	−0.858082	+	0.513514i	$0.828344\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−63.4980			−1.47670			−0.738349	−	0.674419i	$-0.764394\pi$
$43$	−63.4980			−1.47670			−0.738349	+	0.674419i	$0.764394\pi$
$44$	−5.36199	−	42.5595i	−0.121864	−	0.967262i
$45$		0			0
$46$	5.34313	+	85.1549i	0.116155	+	1.85119i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−49.0000			−1.00000
$50$	3.13115	+	49.9019i	0.0626229	+	0.998037i
$51$		0			0
$52$		0			0
$53$	70.5905			1.33190			0.665948	−	0.745998i	$-0.268027\pi$
$53$	70.5905			1.33190			0.665948	+	0.745998i	$0.268027\pi$
$54$		0			0
$55$		0			0
$56$	−55.0134	+	10.4656i	−0.982382	+	0.186886i
$57$		0			0
$58$	6.65687	+	106.092i	0.114774	+	1.82918i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−59.5294	+	23.5000i	−0.930147	+	0.367188i
$65$		0			0
$66$		0			0
$67$	118.000			1.76119			0.880597	−	0.473866i	$-0.157142\pi$
$67$	118.000			1.76119			0.880597	+	0.473866i	$0.157142\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−20.7437			−0.292164			−0.146082	−	0.989272i	$-0.546666\pi$
$71$	−20.7437			−0.292164			−0.146082	+	0.989272i	$0.546666\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−75.8508	+	4.75934i	−1.02501	+	0.0643154i
$75$		0			0
$76$		0			0
$77$	75.0679			0.974908
$78$		0			0
$79$		−	126.996i		−	1.60755i	−0.594937	−	0.803773i	$-0.702823\pi$
$79$		−	126.996i		−	1.60755i	0.594937	−	0.803773i	$-0.297177\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−7.95286	−	126.747i	−0.0924752	−	1.47380i
$87$		0			0
$88$	84.2804	−	16.0333i	0.957732	−	0.182197i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−169.306	+	21.3306i	−1.84029	+	0.231854i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−6.13705	−	97.8077i	−0.0626229	−	0.998037i
$99$		0			0
$100$	−99.2157	+	12.5000i	−0.992157	+	0.125000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	8.84117	+	140.904i	0.0834072	+	1.32928i
$107$		−	56.6887i		−	0.529801i	−0.964276	−	0.264901i	$-0.914661\pi$
$107$		−	56.6887i		−	0.529801i	0.964276	−	0.264901i	$-0.0853391\pi$
$108$		0			0
$109$			106.000i			0.972477i	0.873826	+	0.486239i	$0.161631\pi$
$109$			106.000i			0.972477i	−0.873826	+	0.486239i	$0.838369\pi$
$110$		0			0
$111$		0			0
$112$	−27.7804	−	108.500i	−0.248039	−	0.968750i
$113$			186.911i			1.65408i	0.562144	+	0.827040i	$0.309977\pi$
$113$			186.911i			1.65408i	−0.562144	+	0.827040i	$0.690023\pi$
$114$		0			0
$115$		0			0
$116$	−210.934	+	26.5752i	−1.81840	+	0.229097i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.99606			0.0495542
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	253.992i		−	1.99994i	−0.00787402	−	0.999969i	$-0.502506\pi$
$127$		−	253.992i		−	1.99994i	0.00787402	−	0.999969i	$-0.497494\pi$
$128$	−54.3636	−	115.882i	−0.424715	−	0.905327i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	14.7790	+	235.537i	0.110291	+	1.75774i
$135$		0			0
$136$		0			0
$137$		−	26.6297i		−	0.194377i	−0.995266	−	0.0971887i	$-0.969015\pi$
$137$		−	26.6297i		−	0.194377i	0.995266	−	0.0971887i	$-0.0309850\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	−2.59806	−	41.4059i	−0.0182962	−	0.291591i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−19.0000	−	150.808i	−0.128378	−	1.01897i
$149$	296.155			1.98762			0.993808	−	0.111111i	$-0.0354409\pi$
$149$	296.155			1.98762			0.993808	+	0.111111i	$0.0354409\pi$
$150$		0			0
$151$		−	274.000i		−	1.81457i	−0.420517	−	0.907285i	$-0.638151\pi$
$151$		−	274.000i		−	1.81457i	0.420517	−	0.907285i	$-0.361849\pi$
$152$		0			0
$153$		0			0
$154$	9.40194	+	149.841i	0.0610516	+	0.972995i
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	253.494	−	15.9057i	1.60439	−	0.100669i
$159$		0			0
$160$		0			0
$161$		−	298.628i		−	1.85483i
$162$		0			0
$163$	74.0000			0.453988			0.226994	−	0.973896i	$-0.427110\pi$
$163$	74.0000			0.453988			0.226994	+	0.973896i	$0.427110\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−169.000			−1.00000
$170$		0			0
$171$		0			0
$172$	252.000	−	31.7490i	1.46512	−	0.184587i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	175.000i		−	1.00000i
$176$	42.5595	+	166.222i	0.241815	+	0.944442i
$177$		0			0
$178$		0			0
$179$			316.664i			1.76907i	0.466473	+	0.884535i	$0.345524\pi$
$179$			316.664i			1.76907i	−0.466473	+	0.884535i	$0.654476\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	−63.7824	−	335.276i	−0.346643	−	1.82215i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−374.526			−1.96087			−0.980435	−	0.196842i	$-0.936931\pi$
$191$	−374.526			−1.96087			−0.980435	+	0.196842i	$0.936931\pi$
$192$		0			0
$193$	380.988			1.97403			0.987016	−	0.160622i	$-0.0513499\pi$
$193$	380.988			1.97403			0.987016	+	0.160622i	$0.0513499\pi$
$194$		0			0
$195$		0			0
$196$	194.463	−	24.5000i	0.992157	−	0.125000i
$197$	4.83793			0.0245580			0.0122790	−	0.999925i	$-0.496091\pi$
$197$	4.83793			0.0245580			0.0122790	+	0.999925i	$0.496091\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−37.3773	−	196.476i	−0.186886	−	0.982382i
$201$		0			0
$202$		0			0
$203$		−	372.053i		−	1.83277i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−317.490			−1.50469			−0.752346	−	0.658768i	$-0.771078\pi$
$211$	−317.490			−1.50469			−0.752346	+	0.658768i	$0.771078\pi$
$212$	−280.147	+	35.2953i	−1.32145	+	0.166487i
$213$		0			0
$214$	113.155	−	7.10002i	0.528761	−	0.0331777i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−211.584	+	13.2761i	−0.970568	+	0.0608993i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	213.095	−	69.0409i	0.951316	−	0.308218i
$225$		0			0
$226$	−373.088	+	23.4098i	−1.65083	+	0.103583i
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	−79.4647	−	417.712i	−0.342520	−	1.80048i
$233$			341.994i			1.46778i	0.679266	+	0.733892i	$0.262298\pi$
$233$			341.994i			1.46778i	−0.679266	+	0.733892i	$0.737702\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−142.355			−0.595627			−0.297814	−	0.954624i	$-0.596257\pi$
$239$	−142.355			−0.595627			−0.297814	+	0.954624i	$0.596257\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	0.750982	+	11.9686i	0.00310323	+	0.0494570i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			457.498i			1.80829i
$254$	506.987	−	31.8115i	1.99601	−	0.125242i
$255$		0			0
$256$	224.500	−	123.027i	0.876953	−	0.480576i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	266.000			1.02703
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−471.872			−1.79419			−0.897095	−	0.441837i	$-0.854327\pi$
$263$	−471.872			−1.79419			−0.897095	+	0.441837i	$0.854327\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−468.298	+	59.0000i	−1.74738	+	0.220149i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	53.1549	−	3.33526i	0.193996	−	0.0121725i
$275$			268.100i			0.974908i
$276$		0			0
$277$			317.490i			1.14617i	0.819495	+	0.573087i	$0.194254\pi$
$277$			317.490i			1.14617i	−0.819495	+	0.573087i	$0.805746\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	308.522i		−	1.09794i	−0.835841	−	0.548972i	$-0.815019\pi$
$281$		−	308.522i		−	1.09794i	0.835841	−	0.548972i	$-0.184981\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	82.3238	−	10.3718i	0.289873	−	0.0365205i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−289.000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	298.644	−	56.8135i	1.00893	−	0.191937i
$297$		0			0
$298$	37.0922	+	591.147i	0.124470	+	1.98371i
$299$		0			0
$300$		0			0
$301$			444.486i			1.47670i
$302$	546.924	−	34.3174i	1.81101	−	0.113634i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−297.917	+	37.5340i	−0.967262	+	0.121864i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$	63.4980	+	504.000i	0.200943	+	1.59494i
$317$	−623.542			−1.96701			−0.983505	−	0.180879i	$-0.942106\pi$
$317$	−623.542			−1.96701			−0.983505	+	0.180879i	$0.942106\pi$
$318$		0			0
$319$			569.984i			1.78678i
$320$		0			0
$321$		0			0
$322$	596.084	−	37.4019i	1.85119	−	0.116155i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	9.26819	+	147.710i	0.0284300	+	0.453097i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	190.494			0.575511			0.287755	−	0.957704i	$-0.407091\pi$
$331$	190.494			0.575511			0.287755	+	0.957704i	$0.407091\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−634.980			−1.88421			−0.942107	−	0.335312i	$-0.891158\pi$
$337$	−634.980			−1.88421			−0.942107	+	0.335312i	$0.891158\pi$
$338$	−21.1665	−	337.337i	−0.0626229	−	0.998037i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			343.000i			1.00000i
$344$	94.9353	+	499.034i	0.275975	+	1.45068i
$345$		0			0
$346$		0			0
$347$		−	374.652i		−	1.07969i	−0.841765	−	0.539844i	$-0.818483\pi$
$347$		−	374.652i		−	1.07969i	0.841765	−	0.539844i	$-0.181517\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	349.313	−	21.9180i	0.998037	−	0.0626229i
$351$		0			0
$352$	−326.461	+	105.771i	−0.927445	+	0.300485i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−632.084	+	39.6608i	−1.76560	+	0.110784i
$359$	252.915			0.704499			0.352249	−	0.935906i	$-0.385417\pi$
$359$	252.915			0.704499			0.352249	+	0.935906i	$0.385417\pi$
$360$		0			0
$361$	361.000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	661.248	−	169.306i	1.79687	−	0.460072i
$369$		0			0
$370$		0			0
$371$		−	494.134i		−	1.33190i
$372$		0			0
$373$			698.478i			1.87260i	0.351206	+	0.936298i	$0.385772\pi$
$373$			698.478i			1.87260i	−0.351206	+	0.936298i	$0.614228\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−444.486			−1.17279			−0.586393	−	0.810026i	$-0.699453\pi$
$379$	−444.486			−1.17279			−0.586393	+	0.810026i	$0.699453\pi$
$380$		0			0
$381$		0			0
$382$	−46.9078	−	747.582i	−0.122795	−	1.95702i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	47.7172	+	760.481i	0.123620	+	1.97016i
$387$		0			0
$388$		0			0
$389$	186.567			0.479607			0.239804	−	0.970821i	$-0.422917\pi$
$389$	186.567			0.479607			0.239804	+	0.970821i	$0.422917\pi$
$390$		0			0
$391$		0			0
$392$	73.2595	+	385.094i	0.186886	+	0.982382i
$393$		0			0
$394$	0.605930	+	9.65687i	0.00153789	+	0.0245098i
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	387.500	−	99.2157i	0.968750	−	0.248039i
$401$			759.181i			1.89322i	0.322381	+	0.946610i	$0.395517\pi$
$401$			759.181i			1.89322i	−0.322381	+	0.946610i	$0.604483\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	742.645	−	46.5981i	1.82918	−	0.114774i
$407$	−407.512			−1.00126
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		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
$421$		−	825.474i		−	1.96075i	−0.197150	−	0.980373i	$-0.563168\pi$
$421$		−	825.474i		−	1.96075i	0.197150	−	0.980373i	$-0.436832\pi$
$422$	−39.7643	−	633.734i	−0.0942282	−	1.50174i
$423$		0			0
$424$	−105.539	−	554.774i	−0.248913	−	1.30843i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	28.3444	+	224.976i	0.0662251	+	0.525646i
$429$		0			0
$430$		0			0
$431$	484.114			1.12323			0.561617	−	0.827397i	$-0.310179\pi$
$431$	484.114			1.12323			0.561617	+	0.827397i	$0.310179\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−53.0000	−	420.674i	−0.121560	−	0.964850i
$437$		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
$443$			867.486i			1.95821i	0.203361	+	0.979104i	$0.434813\pi$
$443$			867.486i			1.95821i	−0.203361	+	0.979104i	$0.565187\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	164.500	+	416.706i	0.367188	+	0.930147i
$449$		−	572.287i		−	1.27458i	−0.770624	−	0.637291i	$-0.780055\pi$
$449$		−	572.287i		−	1.27458i	0.770624	−	0.637291i	$-0.219945\pi$
$450$		0			0
$451$		0			0
$452$	−93.4555	−	741.780i	−0.206760	−	1.64111i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	253.992			0.555781			0.277891	−	0.960613i	$-0.410365\pi$
$457$	253.992			0.555781			0.277891	+	0.960613i	$0.410365\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$			634.980i			1.37145i	0.727862	+	0.685724i	$0.240514\pi$
$463$			634.980i			1.37145i	−0.727862	+	0.685724i	$0.759486\pi$
$464$	823.831	−	210.934i	1.77550	−	0.454599i
$465$		0			0
$466$	−682.645	+	42.8333i	−1.46490	+	0.0919169i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		−	826.000i		−	1.76119i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	680.952i		−	1.43965i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−17.8294	−	284.151i	−0.0372999	−	0.594458i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−23.7961	+	2.99803i	−0.0491656	+	0.00619428i
$485$		0			0
$486$		0			0
$487$		−	398.000i		−	0.817248i	−0.912703	−	0.408624i	$-0.866009\pi$
$487$		−	398.000i		−	0.817248i	0.912703	−	0.408624i	$-0.133991\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			509.947i			1.03859i	0.854596	+	0.519294i	$0.173805\pi$
$491$			509.947i			1.03859i	−0.854596	+	0.519294i	$0.826195\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			145.206i			0.292164i
$498$		0			0
$499$	952.470			1.90876			0.954379	−	0.298597i	$-0.0965187\pi$
$499$	952.470			1.90876			0.954379	+	0.298597i	$0.0965187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$	−913.200	+	57.2997i	−1.80474	+	0.113241i
$507$		0			0
$508$	126.996	+	1008.00i	0.249992	+	1.98425i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	273.690	+	432.710i	0.534550	+	0.845137i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	33.3154	+	530.956i	0.0643154	+	1.02501i
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	−59.1000	−	941.892i	−0.112357	−	1.79067i
$527$		0			0
$528$		0			0
$529$	1290.98			2.44041
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−176.421	−	927.368i	−0.329143	−	1.73016i
$537$		0			0
$538$		0			0
$539$		−	525.475i		−	0.974908i
$540$		0			0
$541$		−	1079.47i		−	1.99532i	−0.0683919	−	0.997659i	$-0.521787\pi$
$541$		−	1079.47i		−	1.99532i	0.0683919	−	0.997659i	$-0.478213\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−842.000			−1.53931			−0.769653	−	0.638463i	$-0.779571\pi$
$547$	−842.000			−1.53931			−0.769653	+	0.638463i	$0.779571\pi$
$548$	13.3149	+	105.683i	0.0242972	+	0.192853i
$549$		0			0
$550$	−535.147	+	33.5784i	−0.972995	+	0.0610516i
$551$		0			0
$552$		0			0
$553$	−888.972			−1.60755
$554$	−633.734	+	39.7643i	−1.14392	+	0.0717767i
$555$		0			0
$556$		0			0
$557$	−1052.75			−1.89004			−0.945021	−	0.327009i	$-0.893959\pi$
$557$	−1052.75			−1.89004			−0.945021	+	0.327009i	$0.893959\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	615.833	−	38.6411i	1.09579	−	0.0687564i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	31.0137	+	163.025i	0.0546015	+	0.287017i
$569$			1120.98i			1.97009i	0.172306	+	0.985044i	$0.444878\pi$
$569$			1120.98i			1.97009i	−0.172306	+	0.985044i	$0.555122\pi$
$570$		0			0
$571$	−1126.00			−1.97198			−0.985989	−	0.166807i	$-0.946654\pi$
$571$	−1126.00			−1.97198			−0.985989	+	0.166807i	$0.946654\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1066.53			1.85483
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−36.1960	−	576.866i	−0.0626229	−	0.998037i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			757.012i			1.29848i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	150.808	+	589.000i	0.254743	+	0.994932i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−1175.33	+	148.077i	−1.97203	+	0.248452i
$597$		0			0
$598$		0			0
$599$	715.095			1.19381			0.596907	−	0.802310i	$-0.296396\pi$
$599$	715.095			1.19381			0.596907	+	0.802310i	$0.296396\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−887.228	+	55.6700i	−1.47380	+	0.0924752i
$603$		0			0
$604$	137.000	+	1087.40i	0.226821	+	1.80034i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	1206.46i		−	1.96813i	−0.177814	−	0.984064i	$-0.556903\pi$
$613$		−	1206.46i		−	1.96813i	0.177814	−	0.984064i	$-0.443097\pi$
$614$		0			0
$615$		0			0
$616$	−112.233	−	589.963i	−0.182197	−	0.957732i
$617$		−	383.699i		−	0.621879i	−0.950430	−	0.310939i	$-0.899356\pi$
$617$		−	383.699i		−	0.621879i	0.950430	−	0.310939i	$-0.100644\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	625.000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	761.976i		−	1.20757i	−0.797147	−	0.603785i	$-0.793659\pi$
$631$		−	761.976i		−	1.20757i	0.797147	−	0.603785i	$-0.206341\pi$
$632$	−998.069	+	189.871i	−1.57922	+	0.300428i
$633$		0			0
$634$	−78.0961	−	1244.64i	−0.123180	−	1.96315i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−1137.73	+	71.3881i	−1.78328	+	0.111894i
$639$		0			0
$640$		0			0
$641$		−	1278.19i		−	1.99406i	−0.0770186	−	0.997030i	$-0.524540\pi$
$641$		−	1278.19i		−	1.99406i	0.0770186	−	0.997030i	$-0.475460\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	149.314	+	1185.14i	0.231854	+	1.84029i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−293.678	+	37.0000i	−0.450427	+	0.0567485i
$653$	−470.120			−0.719938			−0.359969	−	0.932964i	$-0.617213\pi$
$653$	−470.120			−0.719938			−0.359969	+	0.932964i	$0.617213\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1260.16i		−	1.91223i	−0.292999	−	0.956113i	$-0.594653\pi$
$659$		−	1260.16i		−	1.91223i	0.292999	−	0.956113i	$-0.405347\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	23.8586	+	380.240i	0.0360402	+	0.574381i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2267.46			3.39949
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1269.96			−1.88701			−0.943507	−	0.331352i	$-0.892495\pi$
$673$	−1269.96			−1.88701			−0.943507	+	0.331352i	$0.892495\pi$
$674$	−79.5286	−	1267.47i	−0.117995	−	1.88052i
$675$		0			0
$676$	670.698	−	84.5000i	0.992157	−	0.125000i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1140.68i			1.67010i	0.550178	+	0.835048i	$0.314560\pi$
$683$			1140.68i			1.67010i	−0.550178	+	0.835048i	$0.685440\pi$
$684$		0			0
$685$		0			0
$686$	−684.654	+	42.9593i	−0.998037	+	0.0626229i
$687$		0			0
$688$	−984.219	+	252.000i	−1.43055	+	0.366279i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	747.833	−	46.9236i	1.07757	−	0.0676133i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	87.5000	+	694.510i	0.125000	+	0.992157i
$701$	−913.702			−1.30343			−0.651713	−	0.758465i	$-0.725949\pi$
$701$	−913.702			−1.30343			−0.651713	+	0.758465i	$0.725949\pi$
$702$		0			0
$703$		0			0
$704$	−252.014	−	638.393i	−0.357974	−	0.906808i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	1382.00i		−	1.94922i	−0.223900	−	0.974612i	$-0.571879\pi$
$709$		−	1382.00i		−	1.94922i	0.223900	−	0.974612i	$-0.428121\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−158.332	−	1256.72i	−0.221134	−	1.75520i
$717$		0			0
$718$	31.6766	+	504.837i	0.0441178	+	0.703116i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	45.2137	+	720.583i	0.0626229	+	0.998037i
$723$		0			0
$724$		0			0
$725$	1328.76			1.83277
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	420.767	+	1298.70i	0.571694	+	1.76453i
$737$			1265.43i			1.71700i
$738$		0			0
$739$	−1226.00			−1.65900			−0.829499	−	0.558508i	$-0.811374\pi$
$739$	−1226.00			−1.65900			−0.829499	+	0.558508i	$0.811374\pi$
$740$		0			0
$741$		0			0
$742$	986.327	−	61.8882i	1.32928	−	0.0834072i
$743$	967.054			1.30155			0.650777	−	0.759269i	$-0.274443\pi$
$743$	967.054			1.30155			0.650777	+	0.759269i	$0.274443\pi$
$744$		0			0
$745$		0			0
$746$	−1394.21	+	87.4815i	−1.86892	+	0.117267i
$747$		0			0
$748$		0			0
$749$	−396.821			−0.529801
$750$		0			0
$751$			802.000i			1.06791i	0.845513	+	0.533955i	$0.179295\pi$
$751$			802.000i			1.06791i	−0.845513	+	0.533955i	$0.820705\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	1402.00i		−	1.85205i	−0.377465	−	0.926024i	$-0.623204\pi$
$757$		−	1402.00i		−	1.85205i	0.377465	−	0.926024i	$-0.376796\pi$
$758$	−55.6700	−	887.228i	−0.0734433	−	1.17048i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	742.000			0.972477
$764$	1486.35	−	187.263i	1.94549	−	0.245109i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−1512.00	+	190.494i	−1.95855	+	0.246754i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	23.3668	+	372.402i	0.0300344	+	0.478666i
$779$		0			0
$780$		0			0
$781$		−	222.455i		−	0.284833i
$782$		0			0
$783$		0			0
$784$	−759.500	+	194.463i	−0.968750	+	0.248039i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−19.1999	+	2.41896i	−0.0243654	+	0.00306975i
$789$		0			0
$790$		0			0
$791$	1308.38			1.65408
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	246.575	+	761.053i	0.308218	+	0.951316i
$801$		0			0
$802$	−1515.38	+	95.0843i	−1.88950	+	0.118559i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1014.43i			1.25393i	0.779048	+	0.626964i	$0.215703\pi$
$809$			1014.43i			1.25393i	−0.779048	+	0.626964i	$0.784297\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	186.026	+	1476.54i	0.229097	+	1.81840i
$813$		0			0
$814$	−51.0391	−	813.423i	−0.0627016	−	0.999292i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1641.99			1.99999			0.999997	−	0.00264006i	$-0.000840357\pi$
$821$	1641.99			1.99999			0.999997	+	0.00264006i	$0.000840357\pi$
$822$		0			0
$823$			1523.95i			1.85170i	0.377886	+	0.925852i	$0.376651\pi$
$823$			1523.95i			1.85170i	−0.377886	+	0.925852i	$0.623349\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	953.026i		−	1.15239i	−0.817312	−	0.576195i	$-0.804537\pi$
$827$		−	953.026i		−	1.15239i	0.817312	−	0.576195i	$-0.195463\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		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		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1983.96			2.35905
$842$	1647.71	−	103.387i	1.95690	−	0.122788i
$843$		0			0
$844$	1260.00	−	158.745i	1.49289	−	0.188087i
$845$		0			0
$846$		0			0
$847$		−	41.9724i		−	0.0495542i
$848$	1094.15	−	280.147i	1.29027	−	0.330362i
$849$		0			0
$850$		0			0
$851$			1621.12i			1.90496i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−445.520	+	84.7548i	−0.520467	+	0.0990126i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	60.6332	+	966.327i	0.0703402	+	1.12103i
$863$	1504.48			1.74331			0.871655	−	0.490119i	$-0.163047\pi$
$863$	1504.48			1.74331			0.871655	+	0.490119i	$0.163047\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1361.90			1.56721
$870$		0			0
$871$		0			0
$872$	833.060	−	158.480i	0.955344	−	0.181743i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			1587.45i			1.81009i	0.425314	+	0.905046i	$0.360164\pi$
$877$			1587.45i			1.81009i	−0.425314	+	0.905046i	$0.639836\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−698.478			−0.791029			−0.395514	−	0.918460i	$-0.629434\pi$
$883$	−698.478			−0.791029			−0.395514	+	0.918460i	$0.629434\pi$
$884$		0			0
$885$		0			0
$886$	−1731.57	+	108.649i	−1.95436	+	0.122629i
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−1777.94			−1.99994
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−811.173	+	380.545i	−0.905327	+	0.424715i
$897$		0			0
$898$	1142.33	−	71.6766i	1.27208	−	0.0798180i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1468.94	−	279.449i	1.62494	−	0.309125i
$905$		0			0
$906$		0			0
$907$	−317.490			−0.350044			−0.175022	−	0.984564i	$-0.556000\pi$
$907$	−317.490			−0.350044			−0.175022	+	0.984564i	$0.556000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−448.798			−0.492643			−0.246321	−	0.969188i	$-0.579222\pi$
$911$	−448.798			−0.492643			−0.246321	+	0.969188i	$0.579222\pi$
$912$		0			0
$913$		0			0
$914$	31.8115	+	506.987i	0.0348047	+	0.554691i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	466.000i		−	0.507073i	−0.967326	−	0.253536i	$-0.918406\pi$
$919$		−	466.000i		−	0.507073i	0.967326	−	0.253536i	$-0.0815938\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$			950.000i			1.02703i
$926$	−1267.47	+	79.5286i	−1.36876	+	0.0858841i
$927$		0			0
$928$	524.222	+	1618.01i	0.564894	+	1.74354i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$	−170.997	−	1357.25i	−0.183473	−	1.45627i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	1648.76	−	103.453i	1.75774	−	0.110291i
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	1359.23	−	85.2864i	1.43682	−	0.0901548i
$947$		−	233.253i		−	0.246307i	−0.992388	−	0.123154i	$-0.960699\pi$
$947$		−	233.253i		−	0.246307i	0.992388	−	0.123154i	$-0.0393008\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1899.03i		−	1.99268i	−0.0854655	−	0.996341i	$-0.527238\pi$
$953$		−	1899.03i		−	1.99268i	0.0854655	−	0.996341i	$-0.472762\pi$
$954$		0			0
$955$		0			0
$956$	564.953	−	71.1774i	0.590955	−	0.0744534i
$957$		0			0
$958$		0			0
$959$	−186.408			−0.194377
$960$		0			0
$961$	−961.000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1904.94i			1.96995i	0.172699	+	0.984975i	$0.444751\pi$
$967$			1904.94i			1.96995i	−0.172699	+	0.984975i	$0.555249\pi$
$968$	−8.96466	−	47.1234i	−0.00926101	−	0.0486812i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$	794.438	−	49.8478i	0.815644	−	0.0511785i
$975$		0			0
$976$		0			0
$977$		−	1262.38i		−	1.29210i	−0.763296	−	0.646048i	$-0.776420\pi$
$977$		−	1262.38i		−	1.29210i	0.763296	−	0.646048i	$-0.223580\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−1017.89	+	63.8687i	−1.03655	+	0.0650394i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2708.90			−2.73903
$990$		0			0
$991$		−	1406.00i		−	1.41877i	−0.704822	−	0.709384i	$-0.748973\pi$
$991$		−	1406.00i		−	1.41877i	0.704822	−	0.709384i	$-0.251027\pi$
$992$		0			0
$993$		0			0
$994$	−289.841	+	18.1864i	−0.291591	+	0.0182962i
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	119.293	+	1901.20i	0.119532	+	1.90501i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.3.e.b.251.6	yes	8
3.2	odd	2	inner	504.3.e.b.251.3	✓	8
4.3	odd	2		2016.3.e.b.1007.6		8
7.6	odd	2	CM	504.3.e.b.251.6	yes	8
8.3	odd	2	inner	504.3.e.b.251.4	yes	8
8.5	even	2		2016.3.e.b.1007.2		8
12.11	even	2		2016.3.e.b.1007.7		8
21.20	even	2	inner	504.3.e.b.251.3	✓	8
24.5	odd	2		2016.3.e.b.1007.3		8
24.11	even	2	inner	504.3.e.b.251.5	yes	8
28.27	even	2		2016.3.e.b.1007.6		8
56.13	odd	2		2016.3.e.b.1007.2		8
56.27	even	2	inner	504.3.e.b.251.4	yes	8
84.83	odd	2		2016.3.e.b.1007.7		8
168.83	odd	2	inner	504.3.e.b.251.5	yes	8
168.125	even	2		2016.3.e.b.1007.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.e.b.251.3	✓	8	3.2	odd	2	inner
504.3.e.b.251.3	✓	8	21.20	even	2	inner
504.3.e.b.251.4	yes	8	8.3	odd	2	inner
504.3.e.b.251.4	yes	8	56.27	even	2	inner
504.3.e.b.251.5	yes	8	24.11	even	2	inner
504.3.e.b.251.5	yes	8	168.83	odd	2	inner
504.3.e.b.251.6	yes	8	1.1	even	1	trivial
504.3.e.b.251.6	yes	8	7.6	odd	2	CM
2016.3.e.b.1007.2		8	8.5	even	2
2016.3.e.b.1007.2		8	56.13	odd	2
2016.3.e.b.1007.3		8	24.5	odd	2
2016.3.e.b.1007.3		8	168.125	even	2
2016.3.e.b.1007.6		8	4.3	odd	2
2016.3.e.b.1007.6		8	28.27	even	2
2016.3.e.b.1007.7		8	12.11	even	2
2016.3.e.b.1007.7		8	84.83	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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	504.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.125246 + 1.99607i) q^{2} +(-3.96863 + 0.500000i) q^{4} -7.00000i q^{7} +(-1.49509 - 7.85905i) q^{8} +10.7240i q^{11} +(13.9725 - 0.876721i) q^{14} +(15.5000 - 3.96863i) q^{16} +(-21.4059 + 1.34313i) q^{22} +42.6612 q^{23} +25.0000 q^{25} +(3.50000 + 27.7804i) q^{28} +53.1504 q^{29} +(9.86299 + 30.4421i) q^{32} +38.0000i q^{37} -63.4980 q^{43} +(-5.36199 - 42.5595i) q^{44} +(5.34313 + 85.1549i) q^{46} -49.0000 q^{49} +(3.13115 + 49.9019i) q^{50} +70.5905 q^{53} +(-55.0134 + 10.4656i) q^{56} +(6.65687 + 106.092i) q^{58} +(-59.5294 + 23.5000i) q^{64} +118.000 q^{67} -20.7437 q^{71} +(-75.8508 + 4.75934i) q^{74} +75.0679 q^{77} -126.996i q^{79} +(-7.95286 - 126.747i) q^{86} +(84.2804 - 16.0333i) q^{88} +(-169.306 + 21.3306i) q^{92} +(-6.13705 - 97.8077i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 124 q^{16} - 76 q^{22} + 200 q^{25} + 28 q^{28} - 116 q^{46} - 392 q^{49} + 212 q^{58} + 944 q^{67} + 452 q^{88}+O(q^{100}) \) 8 * q + 124 * q^16 - 76 * q^22 + 200 * q^25 + 28 * q^28 - 116 * q^46 - 392 * q^49 + 212 * q^58 + 944 * q^67 + 452 * q^88

Introduction

L-functions

Modular forms

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