LMFDB - Embedded newform 504.3.l.e.181.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,3,Mod(181,504)] 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("504.181"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$13.7330053238$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			181.2
Root			$1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	504.181
Dual form			504.3.l.e.181.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32288 + 1.50000i) q^{2} +(-0.500000 - 3.96863i) q^{4} -7.00000 q^{7} +(6.61438 + 4.50000i) q^{8} +6.00000i q^{11} +(9.26013 - 10.5000i) q^{14} +(-15.5000 + 3.96863i) q^{16} +(-9.00000 - 7.93725i) q^{22} +42.3320 q^{23} -25.0000 q^{25} +(3.50000 + 27.7804i) q^{28} -54.0000i q^{29} +(14.5516 - 28.5000i) q^{32} -63.4980i q^{37} -63.4980i q^{43} +(23.8118 - 3.00000i) q^{44} +(-56.0000 + 63.4980i) q^{46} +49.0000 q^{49} +(33.0719 - 37.5000i) q^{50} -6.00000i q^{53} +(-46.3006 - 31.5000i) q^{56} +(81.0000 + 71.4353i) q^{58} +(23.5000 + 59.5294i) q^{64} -63.4980i q^{67} +84.6640 q^{71} +(95.2470 + 84.0000i) q^{74} -42.0000i q^{77} +94.0000 q^{79} +(95.2470 + 84.0000i) q^{86} +(-27.0000 + 39.6863i) q^{88} +(-21.1660 - 168.000i) q^{92} +(-64.8209 + 73.5000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} - 28 q^{7} - 62 q^{16} - 36 q^{22} - 100 q^{25} + 14 q^{28} - 224 q^{46} + 196 q^{49} + 324 q^{58} + 94 q^{64} + 376 q^{79} - 108 q^{88}+O(q^{100}) $ 4 * q - 2 * q^4 - 28 * q^7 - 62 * q^16 - 36 * q^22 - 100 * q^25 + 14 * q^28 - 224 * q^46 + 196 * q^49 + 324 * q^58 + 94 * q^64 + 376 * q^79 - 108 * 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 $

$2$	−1.32288	+	1.50000i	−0.661438	+	0.750000i
$2$	−1.32288	+	1.50000i	−0.661438	+	0.750000i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	3.96863i	−0.125000	−	0.992157i
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−7.00000			−1.00000
$7$	−7.00000			−1.00000
$8$	6.61438	+	4.50000i	0.826797	+	0.562500i
$9$		0			0
$10$		0			0
$11$			6.00000i			0.545455i	0.962091	+	0.272727i	$0.0879257\pi$
$11$			6.00000i			0.545455i	−0.962091	+	0.272727i	$0.912074\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	9.26013	−	10.5000i	0.661438	−	0.750000i
$15$		0			0
$16$	−15.5000	+	3.96863i	−0.968750	+	0.248039i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	−9.00000	−	7.93725i	−0.409091	−	0.360784i
$23$	42.3320			1.84052			0.920261	−	0.391304i	$-0.127976\pi$
$23$	42.3320			1.84052			0.920261	+	0.391304i	$0.127976\pi$
$24$		0			0
$25$	−25.0000			−1.00000
$26$		0			0
$27$		0			0
$28$	3.50000	+	27.7804i	0.125000	+	0.992157i
$29$		−	54.0000i		−	1.86207i	−0.364931	−	0.931034i	$-0.618907\pi$
$29$		−	54.0000i		−	1.86207i	0.364931	−	0.931034i	$-0.381093\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	14.5516	−	28.5000i	0.454739	−	0.890625i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	63.4980i		−	1.71616i	−0.513514	−	0.858082i	$-0.671656\pi$
$37$		−	63.4980i		−	1.71616i	0.513514	−	0.858082i	$-0.328344\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	63.4980i		−	1.47670i	−0.674419	−	0.738349i	$-0.735606\pi$
$43$		−	63.4980i		−	1.47670i	0.674419	−	0.738349i	$-0.264394\pi$
$44$	23.8118	−	3.00000i	0.541176	−	0.0681818i
$45$		0			0
$46$	−56.0000	+	63.4980i	−1.21739	+	1.38039i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	49.0000			1.00000
$50$	33.0719	−	37.5000i	0.661438	−	0.750000i
$51$		0			0
$52$		0			0
$53$		−	6.00000i		−	0.113208i	−0.998397	−	0.0566038i	$-0.981973\pi$
$53$		−	6.00000i		−	0.113208i	0.998397	−	0.0566038i	$-0.0180272\pi$
$54$		0			0
$55$		0			0
$56$	−46.3006	−	31.5000i	−0.826797	−	0.562500i
$57$		0			0
$58$	81.0000	+	71.4353i	1.39655	+	1.23164i
$59$		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
$64$	23.5000	+	59.5294i	0.367188	+	0.930147i
$65$		0			0
$66$		0			0
$67$		−	63.4980i		−	0.947732i	−0.880597	−	0.473866i	$-0.842858\pi$
$67$		−	63.4980i		−	0.947732i	0.880597	−	0.473866i	$-0.157142\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	84.6640			1.19245			0.596226	−	0.802817i	$-0.296666\pi$
$71$	84.6640			1.19245			0.596226	+	0.802817i	$0.296666\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	95.2470	+	84.0000i	1.28712	+	1.13514i
$75$		0			0
$76$		0			0
$77$		−	42.0000i		−	0.545455i
$78$		0			0
$79$	94.0000			1.18987			0.594937	−	0.803773i	$-0.297177\pi$
$79$	94.0000			1.18987			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
$86$	95.2470	+	84.0000i	1.10752	+	0.976744i
$87$		0			0
$88$	−27.0000	+	39.6863i	−0.306818	+	0.450980i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−21.1660	−	168.000i	−0.230065	−	1.82609i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−64.8209	+	73.5000i	−0.661438	+	0.750000i
$99$		0			0
$100$	12.5000	+	99.2157i	0.125000	+	0.992157i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	9.00000	+	7.93725i	0.0849057	+	0.0748798i
$107$			186.000i			1.73832i	0.494533	+	0.869159i	$0.335339\pi$
$107$			186.000i			1.73832i	−0.494533	+	0.869159i	$0.664661\pi$
$108$		0			0
$109$		−	190.494i		−	1.74765i	−0.486239	−	0.873826i	$-0.661631\pi$
$109$		−	190.494i		−	1.74765i	0.486239	−	0.873826i	$-0.338369\pi$
$110$		0			0
$111$		0			0
$112$	108.500	−	27.7804i	0.968750	−	0.248039i
$113$	−42.3320			−0.374620			−0.187310	−	0.982301i	$-0.559977\pi$
$113$	−42.3320			−0.374620			−0.187310	+	0.982301i	$0.559977\pi$
$114$		0			0
$115$		0			0
$116$	−214.306	+	27.0000i	−1.84746	+	0.232759i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	85.0000			0.702479
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.00000			0.0157480			0.00787402	−	0.999969i	$-0.497494\pi$
$127$	2.00000			0.0157480			0.00787402	+	0.999969i	$0.497494\pi$
$128$	−120.382	−	43.5000i	−0.940482	−	0.339844i
$129$		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
$134$	95.2470	+	84.0000i	0.710799	+	0.626866i
$135$		0			0
$136$		0			0
$137$	−211.660			−1.54496			−0.772482	−	0.635036i	$-0.780985\pi$
$137$	−211.660			−1.54496			−0.772482	+	0.635036i	$0.780985\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−112.000	+	126.996i	−0.788732	+	0.894338i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−252.000	+	31.7490i	−1.70270	+	0.214520i
$149$		−	186.000i		−	1.24832i	−0.781296	−	0.624161i	$-0.785441\pi$
$149$		−	186.000i		−	1.24832i	0.781296	−	0.624161i	$-0.214559\pi$
$150$		0			0
$151$	−274.000			−1.81457			−0.907285	−	0.420517i	$-0.861849\pi$
$151$	−274.000			−1.81457			−0.907285	+	0.420517i	$0.861849\pi$
$152$		0			0
$153$		0			0
$154$	63.0000	+	55.5608i	0.409091	+	0.360784i
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−124.350	+	141.000i	−0.787027	+	0.892405i
$159$		0			0
$160$		0			0
$161$	−296.324			−1.84052
$162$		0			0
$163$		−	317.490i		−	1.94779i	−0.226994	−	0.973896i	$-0.572890\pi$
$163$		−	317.490i		−	1.94779i	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			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	175.000			1.00000
$176$	−23.8118	−	93.0000i	−0.135294	−	0.528409i
$177$		0			0
$178$		0			0
$179$			342.000i			1.91061i	0.295615	+	0.955307i	$0.404476\pi$
$179$			342.000i			1.91061i	−0.295615	+	0.955307i	$0.595524\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	280.000	+	190.494i	1.52174	+	1.03529i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	211.660			1.10817			0.554084	−	0.832461i	$-0.313069\pi$
$191$	211.660			1.10817			0.554084	+	0.832461i	$0.313069\pi$
$192$		0			0
$193$	−62.0000			−0.321244			−0.160622	−	0.987016i	$-0.551350\pi$
$193$	−62.0000			−0.321244			−0.160622	+	0.987016i	$0.551350\pi$
$194$		0			0
$195$		0			0
$196$	−24.5000	−	194.463i	−0.125000	−	0.992157i
$197$		−	282.000i		−	1.43147i	−0.698371	−	0.715736i	$-0.746091\pi$
$197$		−	282.000i		−	1.43147i	0.698371	−	0.715736i	$-0.253909\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−165.359	−	112.500i	−0.826797	−	0.562500i
$201$		0			0
$202$		0			0
$203$			378.000i			1.86207i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			317.490i			1.50469i	0.658768	+	0.752346i	$0.271078\pi$
$211$			317.490i			1.50469i	−0.658768	+	0.752346i	$0.728922\pi$
$212$	−23.8118	+	3.00000i	−0.112320	+	0.0141509i
$213$		0			0
$214$	−279.000	−	246.055i	−1.30374	−	1.14979i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	285.741	+	252.000i	1.31074	+	1.15596i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−101.861	+	199.500i	−0.454739	+	0.890625i
$225$		0			0
$226$	56.0000	−	63.4980i	0.247788	−	0.280965i
$227$		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
$232$	243.000	−	357.176i	1.04741	−	1.53955i
$233$	−465.652			−1.99851			−0.999254	−	0.0386266i	$-0.987702\pi$
$233$	−465.652			−1.99851			−0.999254	+	0.0386266i	$0.987702\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	423.320			1.77121			0.885607	−	0.464435i	$-0.153743\pi$
$239$	423.320			1.77121			0.885607	+	0.464435i	$0.153743\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−112.444	+	127.500i	−0.464646	+	0.526860i
$243$		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
$253$			253.992i			1.00392i
$254$	−2.64575	+	3.00000i	−0.0104163	+	0.0118110i
$255$		0			0
$256$	224.500	−	123.027i	0.876953	−	0.480576i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$			444.486i			1.71616i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−169.328			−0.643833			−0.321917	−	0.946768i	$-0.604327\pi$
$263$	−169.328			−0.643833			−0.321917	+	0.946768i	$0.604327\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−252.000	+	31.7490i	−0.940299	+	0.118466i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	280.000	−	317.490i	1.02190	−	1.15872i
$275$		−	150.000i		−	0.545455i
$276$		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
$281$	550.316			1.95842			0.979210	−	0.202847i	$-0.0650194\pi$
$281$	550.316			1.95842			0.979210	+	0.202847i	$0.0650194\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	−42.3320	−	336.000i	−0.149056	−	1.18310i
$285$		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			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	285.741	−	420.000i	0.965342	−	1.41892i
$297$		0			0
$298$	279.000	+	246.055i	0.936242	+	0.825687i
$299$		0			0
$300$		0			0
$301$			444.486i			1.47670i
$302$	362.468	−	411.000i	1.20022	−	1.36093i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−166.682	+	21.0000i	−0.541176	+	0.0681818i
$309$		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
$316$	−47.0000	−	373.051i	−0.148734	−	1.18054i
$317$		−	522.000i		−	1.64669i	−0.567543	−	0.823344i	$-0.692106\pi$
$317$		−	522.000i		−	1.64669i	0.567543	−	0.823344i	$-0.307894\pi$
$318$		0			0
$319$	324.000			1.01567
$320$		0			0
$321$		0			0
$322$	392.000	−	444.486i	1.21739	−	1.38039i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	476.235	+	420.000i	1.46084	+	1.28834i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			190.494i			0.575511i	0.957704	+	0.287755i	$0.0929090\pi$
$331$			190.494i			0.575511i	−0.957704	+	0.287755i	$0.907091\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	226.000			0.670623			0.335312	−	0.942107i	$-0.391158\pi$
$337$	226.000			0.670623			0.335312	+	0.942107i	$0.391158\pi$
$338$	223.566	−	253.500i	0.661438	−	0.750000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−343.000			−1.00000
$344$	285.741	−	420.000i	0.830643	−	1.22093i
$345$		0			0
$346$		0			0
$347$		−	678.000i		−	1.95389i	−0.213490	−	0.976945i	$-0.568483\pi$
$347$		−	678.000i		−	1.95389i	0.213490	−	0.976945i	$-0.431517\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−231.503	+	262.500i	−0.661438	+	0.750000i
$351$		0			0
$352$	171.000	+	87.3098i	0.485795	+	0.248039i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−513.000	−	452.423i	−1.43296	−	1.26375i
$359$	296.324			0.825415			0.412708	−	0.910864i	$-0.364583\pi$
$359$	296.324			0.825415			0.412708	+	0.910864i	$0.364583\pi$
$360$		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		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	−656.146	+	168.000i	−1.78301	+	0.456522i
$369$		0			0
$370$		0			0
$371$			42.0000i			0.113208i
$372$		0			0
$373$		−	698.478i		−	1.87260i	−0.351206	−	0.936298i	$-0.614228\pi$
$373$		−	698.478i		−	1.87260i	0.351206	−	0.936298i	$-0.385772\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			444.486i			1.17279i	0.810026	+	0.586393i	$0.199453\pi$
$379$			444.486i			1.17279i	−0.810026	+	0.586393i	$0.800547\pi$
$380$		0			0
$381$		0			0
$382$	−280.000	+	317.490i	−0.732984	+	0.831126i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	82.0183	−	93.0000i	0.212483	−	0.240933i
$387$		0			0
$388$		0			0
$389$			666.000i			1.71208i	0.516908	+	0.856041i	$0.327083\pi$
$389$			666.000i			1.71208i	−0.516908	+	0.856041i	$0.672917\pi$
$390$		0			0
$391$		0			0
$392$	324.105	+	220.500i	0.826797	+	0.562500i
$393$		0			0
$394$	423.000	+	373.051i	1.07360	+	0.946830i
$395$		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
$401$	719.644			1.79462			0.897312	−	0.441397i	$-0.145517\pi$
$401$	719.644			1.79462			0.897312	+	0.441397i	$0.145517\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−567.000	−	500.047i	−1.39655	−	1.23164i
$407$	380.988			0.936089
$408$		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	$0.436832\pi$
$421$			825.474i			1.96075i	−0.197150	+	0.980373i	$0.563168\pi$
$422$	−476.235	−	420.000i	−1.12852	−	0.995261i
$423$		0			0
$424$	27.0000	−	39.6863i	0.0636792	−	0.0935997i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	738.165	−	93.0000i	1.72468	−	0.217290i
$429$		0			0
$430$		0			0
$431$	−846.640			−1.96436			−0.982181	−	0.187935i	$-0.939821\pi$
$431$	−846.640			−1.96436			−0.982181	+	0.187935i	$0.939821\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−756.000	+	95.2470i	−1.73394	+	0.218457i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			486.000i			1.09707i	0.836129	+	0.548533i	$0.184813\pi$
$443$			486.000i			1.09707i	−0.836129	+	0.548533i	$0.815187\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−164.500	−	416.706i	−0.367188	−	0.930147i
$449$	−84.6640			−0.188561			−0.0942807	−	0.995546i	$-0.530055\pi$
$449$	−84.6640			−0.188561			−0.0942807	+	0.995546i	$0.530055\pi$
$450$		0			0
$451$		0			0
$452$	21.1660	+	168.000i	0.0468275	+	0.371681i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−878.000			−1.92123			−0.960613	−	0.277891i	$-0.910365\pi$
$457$	−878.000			−1.92123			−0.960613	+	0.277891i	$0.910365\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−674.000			−1.45572			−0.727862	−	0.685724i	$-0.759486\pi$
$463$	−674.000			−1.45572			−0.727862	+	0.685724i	$0.759486\pi$
$464$	214.306	+	837.000i	0.461866	+	1.80388i
$465$		0			0
$466$	616.000	−	698.478i	1.32189	−	1.49888i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$			444.486i			0.947732i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	380.988			0.805472
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−560.000	+	634.980i	−1.17155	+	1.32841i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−42.5000	−	337.333i	−0.0878099	−	0.696970i
$485$		0			0
$486$		0			0
$487$	398.000			0.817248			0.408624	−	0.912703i	$-0.366009\pi$
$487$	398.000			0.817248			0.408624	+	0.912703i	$0.366009\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	954.000i		−	1.94297i	−0.237094	−	0.971487i	$-0.576195\pi$
$491$		−	954.000i		−	1.94297i	0.237094	−	0.971487i	$-0.423805\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−592.648			−1.19245
$498$		0			0
$499$		−	952.470i		−	1.90876i	−0.298597	−	0.954379i	$-0.596519\pi$
$499$		−	952.470i		−	1.90876i	0.298597	−	0.954379i	$-0.403481\pi$
$500$		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
$506$	−380.988	−	336.000i	−0.752941	−	0.664032i
$507$		0			0
$508$	−1.00000	−	7.93725i	−0.00196850	−	0.0156245i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−112.444	+	499.500i	−0.219618	+	0.975586i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−666.729	−	588.000i	−1.28712	−	1.13514i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	224.000	−	253.992i	0.425856	−	0.482875i
$527$		0			0
$528$		0			0
$529$	1263.00			2.38752
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	285.741	−	420.000i	0.533099	−	0.783582i
$537$		0			0
$538$		0			0
$539$			294.000i			0.545455i
$540$		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
$547$			698.478i			1.27693i	0.769653	+	0.638463i	$0.220429\pi$
$547$			698.478i			1.27693i	−0.769653	+	0.638463i	$0.779571\pi$
$548$	105.830	+	840.000i	0.193121	+	1.53285i
$549$		0			0
$550$	225.000	+	198.431i	0.409091	+	0.360784i
$551$		0			0
$552$		0			0
$553$	−658.000			−1.18987
$554$	−476.235	−	420.000i	−0.859630	−	0.758123i
$555$		0			0
$556$		0			0
$557$			1002.00i			1.79892i	0.437000	+	0.899461i	$0.356041\pi$
$557$			1002.00i			1.79892i	−0.437000	+	0.899461i	$0.643959\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−728.000	+	825.474i	−1.29537	+	1.46882i
$563$		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
$568$	560.000	+	380.988i	0.985915	+	0.670754i
$569$	−931.304			−1.63674			−0.818369	−	0.574692i	$-0.805122\pi$
$569$	−931.304			−1.63674			−0.818369	+	0.574692i	$0.805122\pi$
$570$		0			0
$571$		−	190.494i		−	0.333615i	−0.985989	−	0.166807i	$-0.946654\pi$
$571$		−	190.494i		−	0.333615i	0.985989	−	0.166807i	$-0.0533458\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1058.30			−1.84052
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−382.311	+	433.500i	−0.661438	+	0.750000i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	36.0000			0.0617496
$584$		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
$592$	252.000	+	984.219i	0.425676	+	1.66253i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	−738.165	+	93.0000i	−1.23853	+	0.156040i
$597$		0			0
$598$		0			0
$599$	−1185.30			−1.97879			−0.989396	−	0.145242i	$-0.953604\pi$
$599$	−1185.30			−1.97879			−0.989396	+	0.145242i	$0.953604\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−666.729	−	588.000i	−1.10752	−	0.976744i
$603$		0			0
$604$	137.000	+	1087.40i	0.226821	+	1.80034i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		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
$616$	189.000	−	277.804i	0.306818	−	0.450980i
$617$	1100.63			1.78385			0.891923	−	0.452188i	$-0.149356\pi$
$617$	1100.63			1.78385			0.891923	+	0.452188i	$0.149356\pi$
$618$		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
$623$		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
$631$	1006.00			1.59429			0.797147	−	0.603785i	$-0.206341\pi$
$631$	1006.00			1.59429			0.797147	+	0.603785i	$0.206341\pi$
$632$	621.752	+	423.000i	0.983784	+	0.669304i
$633$		0			0
$634$	783.000	+	690.541i	1.23502	+	1.08918i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−428.612	+	486.000i	−0.671805	+	0.761755i
$639$		0			0
$640$		0			0
$641$	973.636			1.51893			0.759467	−	0.650546i	$-0.225460\pi$
$641$	973.636			1.51893			0.759467	+	0.650546i	$0.225460\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	148.162	+	1176.00i	0.230065	+	1.82609i
$645$		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
$652$	−1260.00	+	158.745i	−1.93252	+	0.243474i
$653$			1194.00i			1.82848i	0.405169	+	0.914242i	$0.367213\pi$
$653$			1194.00i			1.82848i	−0.405169	+	0.914242i	$0.632787\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	618.000i		−	0.937785i	−0.883255	−	0.468892i	$-0.844653\pi$
$659$		−	618.000i		−	0.937785i	0.883255	−	0.468892i	$-0.155347\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−285.741	−	252.000i	−0.431633	−	0.380665i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	2285.93i		−	3.42718i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	446.000			0.662704			0.331352	−	0.943507i	$-0.392495\pi$
$673$	446.000			0.662704			0.331352	+	0.943507i	$0.392495\pi$
$674$	−298.970	+	339.000i	−0.443576	+	0.502967i
$675$		0			0
$676$	84.5000	+	670.698i	0.125000	+	0.992157i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1338.00i		−	1.95900i	−0.201433	−	0.979502i	$-0.564560\pi$
$683$		−	1338.00i		−	1.95900i	0.201433	−	0.979502i	$-0.435440\pi$
$684$		0			0
$685$		0			0
$686$	453.746	−	514.500i	0.661438	−	0.750000i
$687$		0			0
$688$	252.000	+	984.219i	0.366279	+	1.43055i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$	1017.00	+	896.910i	1.46542	+	1.29238i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−87.5000	−	694.510i	−0.125000	−	0.992157i
$701$			1398.00i			1.99429i	0.0754851	+	0.997147i	$0.475949\pi$
$701$			1398.00i			1.99429i	−0.0754851	+	0.997147i	$0.524051\pi$
$702$		0			0
$703$		0			0
$704$	−357.176	+	141.000i	−0.507353	+	0.200284i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	317.490i		−	0.447800i	−0.974612	−	0.223900i	$-0.928121\pi$
$709$		−	317.490i		−	0.447800i	0.974612	−	0.223900i	$-0.0718789\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1357.27	−	171.000i	1.89563	−	0.238827i
$717$		0			0
$718$	−392.000	+	444.486i	−0.545961	+	0.619062i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	477.558	−	541.500i	0.661438	−	0.750000i
$723$		0			0
$724$		0			0
$725$			1350.00i			1.86207i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		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
$736$	616.000	−	1206.46i	0.836957	−	1.63922i
$737$	380.988			0.516945
$738$		0			0
$739$		−	825.474i		−	1.11702i	−0.829499	−	0.558508i	$-0.811374\pi$
$739$		−	825.474i		−	1.11702i	0.829499	−	0.558508i	$-0.188626\pi$
$740$		0			0
$741$		0			0
$742$	−63.0000	−	55.5608i	−0.0849057	−	0.0748798i
$743$	−1481.62			−1.99411			−0.997053	−	0.0767160i	$-0.975557\pi$
$743$	−1481.62			−1.99411			−0.997053	+	0.0767160i	$0.975557\pi$
$744$		0			0
$745$		0			0
$746$	1047.72	+	924.000i	1.40445	+	1.23861i
$747$		0			0
$748$		0			0
$749$		−	1302.00i		−	1.73832i
$750$		0			0
$751$	−802.000			−1.06791			−0.533955	−	0.845513i	$-0.679295\pi$
$751$	−802.000			−1.06791			−0.533955	+	0.845513i	$0.679295\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	571.482i		−	0.754930i	−0.926024	−	0.377465i	$-0.876796\pi$
$757$		−	571.482i		−	0.754930i	0.926024	−	0.377465i	$-0.123204\pi$
$758$	−666.729	−	588.000i	−0.879590	−	0.775726i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$			1333.46i			1.74765i
$764$	−105.830	−	840.000i	−0.138521	−	1.09948i
$765$		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
$772$	31.0000	+	246.055i	0.0401554	+	0.318724i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−999.000	−	881.035i	−1.28406	−	1.13244i
$779$		0			0
$780$		0			0
$781$			507.984i			0.650428i
$782$		0			0
$783$		0			0
$784$	−759.500	+	194.463i	−0.968750	+	0.248039i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−1119.15	+	141.000i	−1.42024	+	0.178934i
$789$		0			0
$790$		0			0
$791$	296.324			0.374620
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−363.791	+	712.500i	−0.454739	+	0.890625i
$801$		0			0
$802$	−952.000	+	1079.47i	−1.18703	+	1.34597i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1608.62			1.98840			0.994201	−	0.107540i	$-0.0342974\pi$
$809$	1608.62			1.98840			0.994201	+	0.107540i	$0.0342974\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	1500.14	−	189.000i	1.84746	−	0.232759i
$813$		0			0
$814$	−504.000	+	571.482i	−0.619165	+	0.702067i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1158.00i		−	1.41048i	−0.708971	−	0.705238i	$-0.750840\pi$
$821$		−	1158.00i		−	1.41048i	0.708971	−	0.705238i	$-0.249160\pi$
$822$		0			0
$823$	622.000			0.755772			0.377886	−	0.925852i	$-0.376651\pi$
$823$	622.000			0.755772			0.377886	+	0.925852i	$0.376651\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			282.000i			0.340992i	0.985358	+	0.170496i	$0.0545369\pi$
$827$			282.000i			0.340992i	−0.985358	+	0.170496i	$0.945463\pi$
$828$		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
$841$	−2075.00			−2.46730
$842$	−1238.21	−	1092.00i	−1.47056	−	1.29691i
$843$		0			0
$844$	1260.00	−	158.745i	1.49289	−	0.188087i
$845$		0			0
$846$		0			0
$847$	−595.000			−0.702479
$848$	23.8118	+	93.0000i	0.0280799	+	0.109670i
$849$		0			0
$850$		0			0
$851$		−	2688.00i		−	3.15864i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−837.000	+	1230.27i	−0.977804	+	1.43724i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	1120.00	−	1269.96i	1.29930	−	1.47327i
$863$	465.652			0.539574			0.269787	−	0.962920i	$-0.413047\pi$
$863$	465.652			0.539574			0.269787	+	0.962920i	$0.413047\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			564.000i			0.649022i
$870$		0			0
$871$		0			0
$872$	857.223	−	1260.00i	0.983054	−	1.44495i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1587.45i		−	1.81009i	−0.425314	−	0.905046i	$-0.639836\pi$
$877$		−	1587.45i		−	1.81009i	0.425314	−	0.905046i	$-0.360164\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		−	698.478i		−	0.791029i	−0.918460	−	0.395514i	$-0.870566\pi$
$883$		−	698.478i		−	0.791029i	0.918460	−	0.395514i	$-0.129434\pi$
$884$		0			0
$885$		0			0
$886$	−729.000	−	642.918i	−0.822799	−	0.725641i
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−14.0000			−0.0157480
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	842.672	+	304.500i	0.940482	+	0.339844i
$897$		0			0
$898$	112.000	−	126.996i	0.124722	−	0.141421i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−280.000	−	190.494i	−0.309735	−	0.210724i
$905$		0			0
$906$		0			0
$907$		−	317.490i		−	0.350044i	−0.984564	−	0.175022i	$-0.944000\pi$
$907$		−	317.490i		−	0.350044i	0.984564	−	0.175022i	$-0.0559997\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	931.304			1.02229			0.511144	−	0.859495i	$-0.329222\pi$
$911$	931.304			1.02229			0.511144	+	0.859495i	$0.329222\pi$
$912$		0			0
$913$		0			0
$914$	1161.48	−	1317.00i	1.27077	−	1.44092i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	466.000			0.507073			0.253536	−	0.967326i	$-0.418406\pi$
$919$	466.000			0.507073			0.253536	+	0.967326i	$0.418406\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$			1587.45i			1.71616i
$926$	891.618	−	1011.00i	0.962871	−	1.09179i
$927$		0			0
$928$	−1539.00	−	785.788i	−1.65841	−	0.846754i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	232.826	+	1848.00i	0.249813	+	1.98283i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−666.729	−	588.000i	−0.710799	−	0.626866i
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−504.000	+	571.482i	−0.532770	+	0.604104i
$947$			1494.00i			1.57761i	0.614641	+	0.788807i	$0.289301\pi$
$947$			1494.00i			1.57761i	−0.614641	+	0.788807i	$0.710699\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1227.63			−1.28817			−0.644086	−	0.764953i	$-0.722762\pi$
$953$	−1227.63			−1.28817			−0.644086	+	0.764953i	$0.722762\pi$
$954$		0			0
$955$		0			0
$956$	−211.660	−	1680.00i	−0.221402	−	1.75732i
$957$		0			0
$958$		0			0
$959$	1481.62			1.54496
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−334.000			−0.345398			−0.172699	−	0.984975i	$-0.555249\pi$
$967$	−334.000			−0.345398			−0.172699	+	0.984975i	$0.555249\pi$
$968$	562.222	+	382.500i	0.580808	+	0.395145i
$969$		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
$974$	−526.505	+	597.000i	−0.540559	+	0.612936i
$975$		0			0
$976$		0			0
$977$	1947.27			1.99311			0.996557	−	0.0829069i	$-0.0264204\pi$
$977$	1947.27			1.99311			0.996557	+	0.0829069i	$0.0264204\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	1431.00	+	1262.02i	1.45723	+	1.28516i
$983$		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
$989$		−	2688.00i		−	2.71790i
$990$		0			0
$991$	−1406.00			−1.41877			−0.709384	−	0.704822i	$-0.751027\pi$
$991$	−1406.00			−1.41877			−0.709384	+	0.704822i	$0.751027\pi$
$992$		0			0
$993$		0			0
$994$	784.000	−	888.972i	0.788732	−	0.894338i
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	1428.71	+	1260.00i	1.43157	+	1.26253i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.3.l.e.181.2	yes	4
3.2	odd	2	inner	504.3.l.e.181.3	yes	4
4.3	odd	2		2016.3.l.e.433.1		4
7.6	odd	2	CM	504.3.l.e.181.2	yes	4
8.3	odd	2		2016.3.l.e.433.3		4
8.5	even	2	inner	504.3.l.e.181.1	✓	4
12.11	even	2		2016.3.l.e.433.4		4
21.20	even	2	inner	504.3.l.e.181.3	yes	4
24.5	odd	2	inner	504.3.l.e.181.4	yes	4
24.11	even	2		2016.3.l.e.433.2		4
28.27	even	2		2016.3.l.e.433.1		4
56.13	odd	2	inner	504.3.l.e.181.1	✓	4
56.27	even	2		2016.3.l.e.433.3		4
84.83	odd	2		2016.3.l.e.433.4		4
168.83	odd	2		2016.3.l.e.433.2		4
168.125	even	2	inner	504.3.l.e.181.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.l.e.181.1	✓	4	8.5	even	2	inner
504.3.l.e.181.1	✓	4	56.13	odd	2	inner
504.3.l.e.181.2	yes	4	1.1	even	1	trivial
504.3.l.e.181.2	yes	4	7.6	odd	2	CM
504.3.l.e.181.3	yes	4	3.2	odd	2	inner
504.3.l.e.181.3	yes	4	21.20	even	2	inner
504.3.l.e.181.4	yes	4	24.5	odd	2	inner
504.3.l.e.181.4	yes	4	168.125	even	2	inner
2016.3.l.e.433.1		4	4.3	odd	2
2016.3.l.e.433.1		4	28.27	even	2
2016.3.l.e.433.2		4	24.11	even	2
2016.3.l.e.433.2		4	168.83	odd	2
2016.3.l.e.433.3		4	8.3	odd	2
2016.3.l.e.433.3		4	56.27	even	2
2016.3.l.e.433.4		4	12.11	even	2
2016.3.l.e.433.4		4	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.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.32288 + 1.50000i) q^{2} +(-0.500000 - 3.96863i) q^{4} -7.00000 q^{7} +(6.61438 + 4.50000i) q^{8} +6.00000i q^{11} +(9.26013 - 10.5000i) q^{14} +(-15.5000 + 3.96863i) q^{16} +(-9.00000 - 7.93725i) q^{22} +42.3320 q^{23} -25.0000 q^{25} +(3.50000 + 27.7804i) q^{28} -54.0000i q^{29} +(14.5516 - 28.5000i) q^{32} -63.4980i q^{37} -63.4980i q^{43} +(23.8118 - 3.00000i) q^{44} +(-56.0000 + 63.4980i) q^{46} +49.0000 q^{49} +(33.0719 - 37.5000i) q^{50} -6.00000i q^{53} +(-46.3006 - 31.5000i) q^{56} +(81.0000 + 71.4353i) q^{58} +(23.5000 + 59.5294i) q^{64} -63.4980i q^{67} +84.6640 q^{71} +(95.2470 + 84.0000i) q^{74} -42.0000i q^{77} +94.0000 q^{79} +(95.2470 + 84.0000i) q^{86} +(-27.0000 + 39.6863i) q^{88} +(-21.1660 - 168.000i) q^{92} +(-64.8209 + 73.5000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} - 28 q^{7} - 62 q^{16} - 36 q^{22} - 100 q^{25} + 14 q^{28} - 224 q^{46} + 196 q^{49} + 324 q^{58} + 94 q^{64} + 376 q^{79} - 108 q^{88}+O(q^{100}) \) 4 * q - 2 * q^4 - 28 * q^7 - 62 * q^16 - 36 * q^22 - 100 * q^25 + 14 * q^28 - 224 * q^46 + 196 * q^49 + 324 * q^58 + 94 * q^64 + 376 * q^79 - 108 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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