LMFDB - Embedded newform 1800.3.v.g.793.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1800,3,Mod(793,1800)] 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("1800.793"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1800.v (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,14,0,0,0,8,0,24,0,0,0,-40,0,0,0,0,0,-18,0,0,0,0, 0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$49.0464475849$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			793.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1800.793
Dual form			1800.3.v.g.1657.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.00000 - 7.00000i) q^{7} +4.00000 q^{11} +(12.0000 + 12.0000i) q^{13} +(-20.0000 + 20.0000i) q^{17} +28.0000i q^{19} +(-9.00000 - 9.00000i) q^{23} +34.0000i q^{29} -40.0000 q^{31} +(-16.0000 + 16.0000i) q^{37} -32.0000 q^{41} +(7.00000 + 7.00000i) q^{43} +(31.0000 - 31.0000i) q^{47} -49.0000i q^{49} +(52.0000 + 52.0000i) q^{53} -44.0000i q^{59} +(-81.0000 + 81.0000i) q^{67} +112.000 q^{71} +(44.0000 + 44.0000i) q^{73} +(28.0000 - 28.0000i) q^{77} -72.0000i q^{79} +(49.0000 + 49.0000i) q^{83} -98.0000i q^{89} +168.000 q^{91} +(-44.0000 + 44.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{7} + 8 q^{11} + 24 q^{13} - 40 q^{17} - 18 q^{23} - 80 q^{31} - 32 q^{37} - 64 q^{41} + 14 q^{43} + 62 q^{47} + 104 q^{53} - 162 q^{67} + 224 q^{71} + 88 q^{73} + 56 q^{77} + 98 q^{83} + 336 q^{91}+ \cdots - 88 q^{97}+O(q^{100}) $ 2 * q + 14 * q^7 + 8 * q^11 + 24 * q^13 - 40 * q^17 - 18 * q^23 - 80 * q^31 - 32 * q^37 - 64 * q^41 + 14 * q^43 + 62 * q^47 + 104 * q^53 - 162 * q^67 + 224 * q^71 + 88 * q^73 + 56 * q^77 + 98 * q^83 + 336 * q^91 - 88 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.00000	−	7.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$7$	7.00000	−	7.00000i	1.00000	−	1.00000i	1.00000			$0$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.00000			0.363636			0.181818	−	0.983332i	$-0.441802\pi$
$11$	4.00000			0.363636			0.181818	+	0.983332i	$0.441802\pi$
$12$		0			0
$13$	12.0000	+	12.0000i	0.923077	+	0.923077i	0.997246	−	0.0741688i	$-0.0236304\pi$
$13$	12.0000	+	12.0000i	0.923077	+	0.923077i	−0.0741688	+	0.997246i	$0.523630\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−20.0000	+	20.0000i	−1.17647	+	1.17647i	−0.195833	+	0.980637i	$0.562741\pi$
$17$	−20.0000	+	20.0000i	−1.17647	+	1.17647i	−0.980637	+	0.195833i	$0.937259\pi$
$18$		0			0
$19$			28.0000i			1.47368i	0.676065	+	0.736842i	$0.263684\pi$
$19$			28.0000i			1.47368i	−0.676065	+	0.736842i	$0.736316\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−9.00000	−	9.00000i	−0.391304	−	0.391304i	0.483848	−	0.875152i	$-0.339239\pi$
$23$	−9.00000	−	9.00000i	−0.391304	−	0.391304i	−0.875152	+	0.483848i	$0.839239\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			34.0000i			1.17241i	0.810161	+	0.586207i	$0.199379\pi$
$29$			34.0000i			1.17241i	−0.810161	+	0.586207i	$0.800621\pi$
$30$		0			0
$31$	−40.0000			−1.29032			−0.645161	−	0.764046i	$-0.723210\pi$
$31$	−40.0000			−1.29032			−0.645161	+	0.764046i	$0.723210\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−16.0000	+	16.0000i	−0.432432	+	0.432432i	−0.889455	−	0.457023i	$-0.848916\pi$
$37$	−16.0000	+	16.0000i	−0.432432	+	0.432432i	0.457023	+	0.889455i	$0.348916\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−32.0000			−0.780488			−0.390244	−	0.920712i	$-0.627609\pi$
$41$	−32.0000			−0.780488			−0.390244	+	0.920712i	$0.627609\pi$
$42$		0			0
$43$	7.00000	+	7.00000i	0.162791	+	0.162791i	0.783802	−	0.621011i	$-0.213278\pi$
$43$	7.00000	+	7.00000i	0.162791	+	0.162791i	−0.621011	+	0.783802i	$0.713278\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	31.0000	−	31.0000i	0.659574	−	0.659574i	−0.295705	−	0.955279i	$-0.595554\pi$
$47$	31.0000	−	31.0000i	0.659574	−	0.659574i	0.955279	+	0.295705i	$0.0955545\pi$
$48$		0			0
$49$		−	49.0000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	52.0000	+	52.0000i	0.981132	+	0.981132i	0.999825	−	0.0186932i	$-0.00595057\pi$
$53$	52.0000	+	52.0000i	0.981132	+	0.981132i	−0.0186932	+	0.999825i	$0.505951\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	44.0000i		−	0.745763i	−0.927879	−	0.372881i	$-0.878370\pi$
$59$		−	44.0000i		−	0.745763i	0.927879	−	0.372881i	$-0.121630\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−81.0000	+	81.0000i	−1.20896	+	1.20896i	−0.237590	+	0.971366i	$0.576357\pi$
$67$	−81.0000	+	81.0000i	−1.20896	+	1.20896i	−0.971366	+	0.237590i	$0.923643\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	112.000			1.57746			0.788732	−	0.614737i	$-0.210738\pi$
$71$	112.000			1.57746			0.788732	+	0.614737i	$0.210738\pi$
$72$		0			0
$73$	44.0000	+	44.0000i	0.602740	+	0.602740i	0.941039	−	0.338299i	$-0.109852\pi$
$73$	44.0000	+	44.0000i	0.602740	+	0.602740i	−0.338299	+	0.941039i	$0.609852\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	28.0000	−	28.0000i	0.363636	−	0.363636i
$78$		0			0
$79$		−	72.0000i		−	0.911392i	−0.890135	−	0.455696i	$-0.849390\pi$
$79$		−	72.0000i		−	0.911392i	0.890135	−	0.455696i	$-0.150610\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	49.0000	+	49.0000i	0.590361	+	0.590361i	0.937729	−	0.347368i	$-0.112924\pi$
$83$	49.0000	+	49.0000i	0.590361	+	0.590361i	−0.347368	+	0.937729i	$0.612924\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	98.0000i		−	1.10112i	−0.834794	−	0.550562i	$-0.814414\pi$
$89$		−	98.0000i		−	1.10112i	0.834794	−	0.550562i	$-0.185586\pi$
$90$		0			0
$91$	168.000			1.84615
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−44.0000	+	44.0000i	−0.453608	+	0.453608i	−0.896550	−	0.442942i	$-0.853935\pi$
$97$	−44.0000	+	44.0000i	−0.453608	+	0.453608i	0.442942	+	0.896550i	$0.353935\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.0000			0.138614			0.0693069	−	0.997595i	$-0.477921\pi$
$101$	14.0000			0.138614			0.0693069	+	0.997595i	$0.477921\pi$
$102$		0			0
$103$	87.0000	+	87.0000i	0.844660	+	0.844660i	0.989461	−	0.144801i	$-0.0462541\pi$
$103$	87.0000	+	87.0000i	0.844660	+	0.844660i	−0.144801	+	0.989461i	$0.546254\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−105.000	+	105.000i	−0.981308	+	0.981308i	−0.999828	−	0.0185201i	$-0.994105\pi$
$107$	−105.000	+	105.000i	−0.981308	+	0.981308i	0.0185201	+	0.999828i	$0.494105\pi$
$108$		0			0
$109$			32.0000i			0.293578i	0.989168	+	0.146789i	$0.0468938\pi$
$109$			32.0000i			0.293578i	−0.989168	+	0.146789i	$0.953106\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.00000	−	8.00000i	−0.0707965	−	0.0707965i	0.670822	−	0.741618i	$-0.265941\pi$
$113$	−8.00000	−	8.00000i	−0.0707965	−	0.0707965i	−0.741618	+	0.670822i	$0.765941\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$			280.000i			2.35294i
$120$		0			0
$121$	−105.000			−0.867769
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−1.00000	+	1.00000i	−0.00787402	+	0.00787402i	−0.711033	−	0.703159i	$-0.751772\pi$
$127$	−1.00000	+	1.00000i	−0.00787402	+	0.00787402i	0.703159	+	0.711033i	$0.251772\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−148.000			−1.12977			−0.564885	−	0.825169i	$-0.691080\pi$
$131$	−148.000			−1.12977			−0.564885	+	0.825169i	$0.691080\pi$
$132$		0			0
$133$	196.000	+	196.000i	1.47368	+	1.47368i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−168.000	+	168.000i	−1.22628	+	1.22628i	−0.260916	+	0.965362i	$0.584025\pi$
$137$	−168.000	+	168.000i	−1.22628	+	1.22628i	−0.965362	+	0.260916i	$0.915975\pi$
$138$		0			0
$139$			52.0000i			0.374101i	0.982350	+	0.187050i	$0.0598928\pi$
$139$			52.0000i			0.374101i	−0.982350	+	0.187050i	$0.940107\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	48.0000	+	48.0000i	0.335664	+	0.335664i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	224.000i		−	1.50336i	−0.659530	−	0.751678i	$-0.729245\pi$
$149$		−	224.000i		−	1.50336i	0.659530	−	0.751678i	$-0.270755\pi$
$150$		0			0
$151$	−88.0000			−0.582781			−0.291391	−	0.956604i	$-0.594118\pi$
$151$	−88.0000			−0.582781			−0.291391	+	0.956604i	$0.594118\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	56.0000	−	56.0000i	0.356688	−	0.356688i	−0.505903	−	0.862591i	$-0.668841\pi$
$157$	56.0000	−	56.0000i	0.356688	−	0.356688i	0.862591	+	0.505903i	$0.168841\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−126.000			−0.782609
$162$		0			0
$163$	191.000	+	191.000i	1.17178	+	1.17178i	0.981785	+	0.189994i	$0.0608468\pi$
$163$	191.000	+	191.000i	1.17178	+	1.17178i	0.189994	+	0.981785i	$0.439153\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−23.0000	+	23.0000i	−0.137725	+	0.137725i	−0.772608	−	0.634883i	$-0.781048\pi$
$167$	−23.0000	+	23.0000i	−0.137725	+	0.137725i	0.634883	+	0.772608i	$0.281048\pi$
$168$		0			0
$169$			119.000i			0.704142i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−24.0000	−	24.0000i	−0.138728	−	0.138728i	0.634332	−	0.773061i	$-0.281275\pi$
$173$	−24.0000	−	24.0000i	−0.138728	−	0.138728i	−0.773061	+	0.634332i	$0.781275\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	28.0000i		−	0.156425i	−0.996937	−	0.0782123i	$-0.975079\pi$
$179$		−	28.0000i		−	0.156425i	0.996937	−	0.0782123i	$-0.0249212\pi$
$180$		0			0
$181$	−178.000			−0.983425			−0.491713	−	0.870757i	$-0.663629\pi$
$181$	−178.000			−0.983425			−0.491713	+	0.870757i	$0.663629\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−80.0000	+	80.0000i	−0.427807	+	0.427807i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	320.000			1.67539			0.837696	−	0.546136i	$-0.183902\pi$
$191$	320.000			1.67539			0.837696	+	0.546136i	$0.183902\pi$
$192$		0			0
$193$	−28.0000	−	28.0000i	−0.145078	−	0.145078i	0.630837	−	0.775915i	$-0.282712\pi$
$193$	−28.0000	−	28.0000i	−0.145078	−	0.145078i	−0.775915	+	0.630837i	$0.782712\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	132.000	−	132.000i	0.670051	−	0.670051i	−0.287677	−	0.957728i	$-0.592883\pi$
$197$	132.000	−	132.000i	0.670051	−	0.670051i	0.957728	+	0.287677i	$0.0928829\pi$
$198$		0			0
$199$			96.0000i			0.482412i	0.970474	+	0.241206i	$0.0775430\pi$
$199$			96.0000i			0.482412i	−0.970474	+	0.241206i	$0.922457\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	238.000	+	238.000i	1.17241	+	1.17241i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			112.000i			0.535885i
$210$		0			0
$211$	244.000			1.15640			0.578199	−	0.815896i	$-0.303756\pi$
$211$	244.000			1.15640			0.578199	+	0.815896i	$0.303756\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−280.000	+	280.000i	−1.29032	+	1.29032i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−480.000			−2.17195
$222$		0			0
$223$	33.0000	+	33.0000i	0.147982	+	0.147982i	0.777216	−	0.629234i	$-0.216631\pi$
$223$	33.0000	+	33.0000i	0.147982	+	0.147982i	−0.629234	+	0.777216i	$0.716631\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	127.000	−	127.000i	0.559471	−	0.559471i	−0.369686	−	0.929157i	$-0.620535\pi$
$227$	127.000	−	127.000i	0.559471	−	0.559471i	0.929157	+	0.369686i	$0.120535\pi$
$228$		0			0
$229$		−	78.0000i		−	0.340611i	−0.985391	−	0.170306i	$-0.945524\pi$
$229$		−	78.0000i		−	0.340611i	0.985391	−	0.170306i	$-0.0544755\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−28.0000	−	28.0000i	−0.120172	−	0.120172i	0.644463	−	0.764635i	$-0.277081\pi$
$233$	−28.0000	−	28.0000i	−0.120172	−	0.120172i	−0.764635	+	0.644463i	$0.777081\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			8.00000i			0.0334728i	0.999860	+	0.0167364i	$0.00532761\pi$
$239$			8.00000i			0.0334728i	−0.999860	+	0.0167364i	$0.994672\pi$
$240$		0			0
$241$	208.000			0.863071			0.431535	−	0.902096i	$-0.357972\pi$
$241$	208.000			0.863071			0.431535	+	0.902096i	$0.357972\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−336.000	+	336.000i	−1.36032	+	1.36032i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	340.000			1.35458			0.677291	−	0.735715i	$-0.263154\pi$
$251$	340.000			1.35458			0.677291	+	0.735715i	$0.263154\pi$
$252$		0			0
$253$	−36.0000	−	36.0000i	−0.142292	−	0.142292i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$			224.000i			0.864865i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	105.000	+	105.000i	0.399240	+	0.399240i	0.877965	−	0.478725i	$-0.158901\pi$
$263$	105.000	+	105.000i	0.399240	+	0.399240i	−0.478725	+	0.877965i	$0.658901\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			128.000i			0.475836i	0.971285	+	0.237918i	$0.0764650\pi$
$269$			128.000i			0.475836i	−0.971285	+	0.237918i	$0.923535\pi$
$270$		0			0
$271$	−56.0000			−0.206642			−0.103321	−	0.994648i	$-0.532947\pi$
$271$	−56.0000			−0.206642			−0.103321	+	0.994648i	$0.532947\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−32.0000	+	32.0000i	−0.115523	+	0.115523i	−0.762505	−	0.646982i	$-0.776031\pi$
$277$	−32.0000	+	32.0000i	−0.115523	+	0.115523i	0.646982	+	0.762505i	$0.276031\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−48.0000			−0.170819			−0.0854093	−	0.996346i	$-0.527220\pi$
$281$	−48.0000			−0.170819			−0.0854093	+	0.996346i	$0.527220\pi$
$282$		0			0
$283$	57.0000	+	57.0000i	0.201413	+	0.201413i	0.800605	−	0.599192i	$-0.204511\pi$
$283$	57.0000	+	57.0000i	0.201413	+	0.201413i	−0.599192	+	0.800605i	$0.704511\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−224.000	+	224.000i	−0.780488	+	0.780488i
$288$		0			0
$289$		−	511.000i		−	1.76817i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	296.000	+	296.000i	1.01024	+	1.01024i	0.999947	+	0.0102919i	$0.00327606\pi$
$293$	296.000	+	296.000i	1.01024	+	1.01024i	0.0102919	+	0.999947i	$0.496724\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	216.000i		−	0.722408i
$300$		0			0
$301$	98.0000			0.325581
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	383.000	−	383.000i	1.24756	−	1.24756i	0.290761	−	0.956796i	$-0.406091\pi$
$307$	383.000	−	383.000i	1.24756	−	1.24756i	0.956796	−	0.290761i	$-0.0939086\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	56.0000			0.180064			0.0900322	−	0.995939i	$-0.471303\pi$
$311$	56.0000			0.180064			0.0900322	+	0.995939i	$0.471303\pi$
$312$		0			0
$313$	144.000	+	144.000i	0.460064	+	0.460064i	0.898676	−	0.438612i	$-0.144530\pi$
$313$	144.000	+	144.000i	0.460064	+	0.460064i	−0.438612	+	0.898676i	$0.644530\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	28.0000	−	28.0000i	0.0883281	−	0.0883281i	−0.661562	−	0.749890i	$-0.730106\pi$
$317$	28.0000	−	28.0000i	0.0883281	−	0.0883281i	0.749890	+	0.661562i	$0.230106\pi$
$318$		0			0
$319$			136.000i			0.426332i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−560.000	−	560.000i	−1.73375	−	1.73375i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	434.000i		−	1.31915i
$330$		0			0
$331$	28.0000			0.0845921			0.0422961	−	0.999105i	$-0.486533\pi$
$331$	28.0000			0.0845921			0.0422961	+	0.999105i	$0.486533\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	384.000	−	384.000i	1.13947	−	1.13947i	0.150920	−	0.988546i	$-0.451776\pi$
$337$	384.000	−	384.000i	1.13947	−	1.13947i	0.988546	−	0.150920i	$-0.0482235\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−160.000			−0.469208
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	73.0000	−	73.0000i	0.210375	−	0.210375i	−0.594052	−	0.804427i	$-0.702473\pi$
$347$	73.0000	−	73.0000i	0.210375	−	0.210375i	0.804427	+	0.594052i	$0.202473\pi$
$348$		0			0
$349$		−	574.000i		−	1.64470i	−0.568983	−	0.822350i	$-0.692663\pi$
$349$		−	574.000i		−	1.64470i	0.568983	−	0.822350i	$-0.307337\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	160.000	+	160.000i	0.453258	+	0.453258i	0.896434	−	0.443177i	$-0.146149\pi$
$353$	160.000	+	160.000i	0.453258	+	0.453258i	−0.443177	+	0.896434i	$0.646149\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			424.000i			1.18106i	0.807016	+	0.590529i	$0.201081\pi$
$359$			424.000i			1.18106i	−0.807016	+	0.590529i	$0.798919\pi$
$360$		0			0
$361$	−423.000			−1.17175
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	225.000	−	225.000i	0.613079	−	0.613079i	−0.330668	−	0.943747i	$-0.607274\pi$
$367$	225.000	−	225.000i	0.613079	−	0.613079i	0.943747	+	0.330668i	$0.107274\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	728.000			1.96226
$372$		0			0
$373$	280.000	+	280.000i	0.750670	+	0.750670i	0.974604	−	0.223934i	$-0.0718900\pi$
$373$	280.000	+	280.000i	0.750670	+	0.750670i	−0.223934	+	0.974604i	$0.571890\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−408.000	+	408.000i	−1.08223	+	1.08223i
$378$		0			0
$379$		−	420.000i		−	1.10818i	−0.832457	−	0.554090i	$-0.813066\pi$
$379$		−	420.000i		−	1.10818i	0.832457	−	0.554090i	$-0.186934\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−273.000	−	273.000i	−0.712794	−	0.712794i	0.254325	−	0.967119i	$-0.418147\pi$
$383$	−273.000	−	273.000i	−0.712794	−	0.712794i	−0.967119	+	0.254325i	$0.918147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	480.000i		−	1.23393i	−0.786989	−	0.616967i	$-0.788361\pi$
$389$		−	480.000i		−	1.23393i	0.786989	−	0.616967i	$-0.211639\pi$
$390$		0			0
$391$	360.000			0.920716
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−276.000	+	276.000i	−0.695214	+	0.695214i	−0.963374	−	0.268160i	$-0.913584\pi$
$397$	−276.000	+	276.000i	−0.695214	+	0.695214i	0.268160	+	0.963374i	$0.413584\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−366.000			−0.912718			−0.456359	−	0.889796i	$-0.650847\pi$
$401$	−366.000			−0.912718			−0.456359	+	0.889796i	$0.650847\pi$
$402$		0			0
$403$	−480.000	−	480.000i	−1.19107	−	1.19107i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−64.0000	+	64.0000i	−0.157248	+	0.157248i
$408$		0			0
$409$			672.000i			1.64303i	0.570186	+	0.821516i	$0.306871\pi$
$409$			672.000i			1.64303i	−0.570186	+	0.821516i	$0.693129\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−308.000	−	308.000i	−0.745763	−	0.745763i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			564.000i			1.34606i	0.739614	+	0.673031i	$0.235008\pi$
$419$			564.000i			1.34606i	−0.739614	+	0.673031i	$0.764992\pi$
$420$		0			0
$421$	−512.000			−1.21615			−0.608076	−	0.793879i	$-0.708058\pi$
$421$	−512.000			−1.21615			−0.608076	+	0.793879i	$0.708058\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−536.000			−1.24362			−0.621810	−	0.783168i	$-0.713602\pi$
$431$	−536.000			−1.24362			−0.621810	+	0.783168i	$0.713602\pi$
$432$		0			0
$433$	−420.000	−	420.000i	−0.969977	−	0.969977i	0.0295854	−	0.999562i	$-0.490581\pi$
$433$	−420.000	−	420.000i	−0.969977	−	0.969977i	−0.999562	+	0.0295854i	$0.990581\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	252.000	−	252.000i	0.576659	−	0.576659i
$438$		0			0
$439$			568.000i			1.29385i	0.762554	+	0.646925i	$0.223945\pi$
$439$			568.000i			1.29385i	−0.762554	+	0.646925i	$0.776055\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.00000	−	7.00000i	−0.0158014	−	0.0158014i	0.699162	−	0.714963i	$-0.253557\pi$
$443$	−7.00000	−	7.00000i	−0.0158014	−	0.0158014i	−0.714963	+	0.699162i	$0.753557\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			272.000i			0.605791i	0.953024	+	0.302895i	$0.0979533\pi$
$449$			272.000i			0.605791i	−0.953024	+	0.302895i	$0.902047\pi$
$450$		0			0
$451$	−128.000			−0.283814
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	16.0000	−	16.0000i	0.0350109	−	0.0350109i	−0.689385	−	0.724396i	$-0.742119\pi$
$457$	16.0000	−	16.0000i	0.0350109	−	0.0350109i	0.724396	+	0.689385i	$0.242119\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	98.0000			0.212581			0.106291	−	0.994335i	$-0.466103\pi$
$461$	98.0000			0.212581			0.106291	+	0.994335i	$0.466103\pi$
$462$		0			0
$463$	−273.000	−	273.000i	−0.589633	−	0.589633i	0.347899	−	0.937532i	$-0.386895\pi$
$463$	−273.000	−	273.000i	−0.589633	−	0.589633i	−0.937532	+	0.347899i	$0.886895\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	193.000	−	193.000i	0.413276	−	0.413276i	−0.469602	−	0.882878i	$-0.655603\pi$
$467$	193.000	−	193.000i	0.413276	−	0.413276i	0.882878	+	0.469602i	$0.155603\pi$
$468$		0			0
$469$			1134.00i			2.41791i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	28.0000	+	28.0000i	0.0591966	+	0.0591966i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			464.000i			0.968685i	0.874879	+	0.484342i	$0.160941\pi$
$479$			464.000i			0.968685i	−0.874879	+	0.484342i	$0.839059\pi$
$480$		0			0
$481$	−384.000			−0.798337
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	297.000	−	297.000i	0.609856	−	0.609856i	−0.333052	−	0.942908i	$-0.608079\pi$
$487$	297.000	−	297.000i	0.609856	−	0.609856i	0.942908	+	0.333052i	$0.108079\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	764.000			1.55601			0.778004	−	0.628259i	$-0.216232\pi$
$491$	764.000			1.55601			0.778004	+	0.628259i	$0.216232\pi$
$492$		0			0
$493$	−680.000	−	680.000i	−1.37931	−	1.37931i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	784.000	−	784.000i	1.57746	−	1.57746i
$498$		0			0
$499$		−	572.000i		−	1.14629i	−0.819453	−	0.573146i	$-0.805723\pi$
$499$		−	572.000i		−	1.14629i	0.819453	−	0.573146i	$-0.194277\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	537.000	+	537.000i	1.06759	+	1.06759i	0.997543	+	0.0700510i	$0.0223162\pi$
$503$	537.000	+	537.000i	1.06759	+	1.06759i	0.0700510	+	0.997543i	$0.477684\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$			258.000i			0.506876i	0.967352	+	0.253438i	$0.0815614\pi$
$509$			258.000i			0.506876i	−0.967352	+	0.253438i	$0.918439\pi$
$510$		0			0
$511$	616.000			1.20548
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	124.000	−	124.000i	0.239845	−	0.239845i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−210.000			−0.403071			−0.201536	−	0.979481i	$-0.564593\pi$
$521$	−210.000			−0.403071			−0.201536	+	0.979481i	$0.564593\pi$
$522$		0			0
$523$	105.000	+	105.000i	0.200765	+	0.200765i	0.800328	−	0.599563i	$-0.204659\pi$
$523$	105.000	+	105.000i	0.200765	+	0.200765i	−0.599563	+	0.800328i	$0.704659\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	800.000	−	800.000i	1.51803	−	1.51803i
$528$		0			0
$529$		−	367.000i		−	0.693762i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−384.000	−	384.000i	−0.720450	−	0.720450i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	196.000i		−	0.363636i
$540$		0			0
$541$	478.000			0.883549			0.441774	−	0.897126i	$-0.354349\pi$
$541$	478.000			0.883549			0.441774	+	0.897126i	$0.354349\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−399.000	+	399.000i	−0.729433	+	0.729433i	−0.970507	−	0.241074i	$-0.922500\pi$
$547$	−399.000	+	399.000i	−0.729433	+	0.729433i	0.241074	+	0.970507i	$0.422500\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−952.000			−1.72777
$552$		0			0
$553$	−504.000	−	504.000i	−0.911392	−	0.911392i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−672.000	+	672.000i	−1.20646	+	1.20646i	−0.234298	+	0.972165i	$0.575279\pi$
$557$	−672.000	+	672.000i	−1.20646	+	1.20646i	−0.972165	+	0.234298i	$0.924721\pi$
$558$		0			0
$559$			168.000i			0.300537i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	191.000	+	191.000i	0.339254	+	0.339254i	0.856087	−	0.516833i	$-0.172889\pi$
$563$	191.000	+	191.000i	0.339254	+	0.339254i	−0.516833	+	0.856087i	$0.672889\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	896.000i		−	1.57469i	−0.616511	−	0.787346i	$-0.711454\pi$
$569$		−	896.000i		−	1.57469i	0.616511	−	0.787346i	$-0.288546\pi$
$570$		0			0
$571$	−700.000			−1.22592			−0.612960	−	0.790114i	$-0.710021\pi$
$571$	−700.000			−1.22592			−0.612960	+	0.790114i	$0.710021\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	232.000	−	232.000i	0.402080	−	0.402080i	−0.476886	−	0.878965i	$-0.658234\pi$
$577$	232.000	−	232.000i	0.402080	−	0.402080i	0.878965	+	0.476886i	$0.158234\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	686.000			1.18072
$582$		0			0
$583$	208.000	+	208.000i	0.356775	+	0.356775i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−553.000	+	553.000i	−0.942078	+	0.942078i	−0.998412	−	0.0563336i	$-0.982059\pi$
$587$	−553.000	+	553.000i	−0.942078	+	0.942078i	0.0563336	+	0.998412i	$0.482059\pi$
$588$		0			0
$589$		−	1120.00i		−	1.90153i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−312.000	−	312.000i	−0.526138	−	0.526138i	0.393280	−	0.919419i	$-0.371340\pi$
$593$	−312.000	−	312.000i	−0.526138	−	0.526138i	−0.919419	+	0.393280i	$0.871340\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	80.0000i		−	0.133556i	−0.997768	−	0.0667780i	$-0.978728\pi$
$599$		−	80.0000i		−	0.133556i	0.997768	−	0.0667780i	$-0.0212719\pi$
$600$		0			0
$601$	−112.000			−0.186356			−0.0931780	−	0.995649i	$-0.529703\pi$
$601$	−112.000			−0.186356			−0.0931780	+	0.995649i	$0.529703\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	305.000	−	305.000i	0.502471	−	0.502471i	−0.409734	−	0.912205i	$-0.634378\pi$
$607$	305.000	−	305.000i	0.502471	−	0.502471i	0.912205	+	0.409734i	$0.134378\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	744.000			1.21768
$612$		0			0
$613$	−420.000	−	420.000i	−0.685155	−	0.685155i	0.276002	−	0.961157i	$-0.410990\pi$
$613$	−420.000	−	420.000i	−0.685155	−	0.685155i	−0.961157	+	0.276002i	$0.910990\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	708.000	−	708.000i	1.14749	−	1.14749i	0.160443	−	0.987045i	$-0.448708\pi$
$617$	708.000	−	708.000i	1.14749	−	1.14749i	0.987045	−	0.160443i	$-0.0512922\pi$
$618$		0			0
$619$		−	348.000i		−	0.562197i	−0.959679	−	0.281099i	$-0.909301\pi$
$619$		−	348.000i		−	0.562197i	0.959679	−	0.281099i	$-0.0906988\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−686.000	−	686.000i	−1.10112	−	1.10112i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	640.000i		−	1.01749i
$630$		0			0
$631$	440.000			0.697306			0.348653	−	0.937252i	$-0.386639\pi$
$631$	440.000			0.697306			0.348653	+	0.937252i	$0.386639\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	588.000	−	588.000i	0.923077	−	0.923077i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−784.000			−1.22309			−0.611544	−	0.791210i	$-0.709451\pi$
$641$	−784.000			−1.22309			−0.611544	+	0.791210i	$0.709451\pi$
$642$		0			0
$643$	−751.000	−	751.000i	−1.16796	−	1.16796i	−0.982686	−	0.185276i	$-0.940682\pi$
$643$	−751.000	−	751.000i	−1.16796	−	1.16796i	−0.185276	−	0.982686i	$-0.559318\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−777.000	+	777.000i	−1.20093	+	1.20093i	−0.227043	+	0.973885i	$0.572906\pi$
$647$	−777.000	+	777.000i	−1.20093	+	1.20093i	−0.973885	+	0.227043i	$0.927094\pi$
$648$		0			0
$649$		−	176.000i		−	0.271186i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−588.000	−	588.000i	−0.900459	−	0.900459i	0.0950163	−	0.995476i	$-0.469710\pi$
$653$	−588.000	−	588.000i	−0.900459	−	0.900459i	−0.995476	+	0.0950163i	$0.969710\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	972.000i		−	1.47496i	−0.675368	−	0.737481i	$-0.736015\pi$
$659$		−	972.000i		−	1.47496i	0.675368	−	0.737481i	$-0.263985\pi$
$660$		0			0
$661$	224.000			0.338880			0.169440	−	0.985540i	$-0.445804\pi$
$661$	224.000			0.338880			0.169440	+	0.985540i	$0.445804\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	306.000	−	306.000i	0.458771	−	0.458771i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	252.000	+	252.000i	0.374443	+	0.374443i	0.869092	−	0.494650i	$-0.164704\pi$
$673$	252.000	+	252.000i	0.374443	+	0.374443i	−0.494650	+	0.869092i	$0.664704\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−108.000	+	108.000i	−0.159527	+	0.159527i	−0.782357	−	0.622830i	$-0.785983\pi$
$677$	−108.000	+	108.000i	−0.159527	+	0.159527i	0.622830	+	0.782357i	$0.285983\pi$
$678$		0			0
$679$			616.000i			0.907216i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−231.000	−	231.000i	−0.338214	−	0.338214i	0.517481	−	0.855695i	$-0.326870\pi$
$683$	−231.000	−	231.000i	−0.338214	−	0.338214i	−0.855695	+	0.517481i	$0.826870\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1248.00i			1.81132i
$690$		0			0
$691$	−84.0000			−0.121563			−0.0607815	−	0.998151i	$-0.519359\pi$
$691$	−84.0000			−0.121563			−0.0607815	+	0.998151i	$0.519359\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	640.000	−	640.000i	0.918221	−	0.918221i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−512.000			−0.730385			−0.365193	−	0.930932i	$-0.618997\pi$
$701$	−512.000			−0.730385			−0.365193	+	0.930932i	$0.618997\pi$
$702$		0			0
$703$	−448.000	−	448.000i	−0.637269	−	0.637269i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	98.0000	−	98.0000i	0.138614	−	0.138614i
$708$		0			0
$709$		−	114.000i		−	0.160790i	−0.996763	−	0.0803949i	$-0.974382\pi$
$709$		−	114.000i		−	0.160790i	0.996763	−	0.0803949i	$-0.0256181\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	360.000	+	360.000i	0.504909	+	0.504909i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			232.000i			0.322670i	0.986900	+	0.161335i	$0.0515800\pi$
$719$			232.000i			0.322670i	−0.986900	+	0.161335i	$0.948420\pi$
$720$		0			0
$721$	1218.00			1.68932
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	423.000	−	423.000i	0.581843	−	0.581843i	−0.353566	−	0.935409i	$-0.615031\pi$
$727$	423.000	−	423.000i	0.581843	−	0.581843i	0.935409	+	0.353566i	$0.115031\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−280.000			−0.383037
$732$		0			0
$733$	648.000	+	648.000i	0.884038	+	0.884038i	0.993942	−	0.109904i	$-0.0350543\pi$
$733$	648.000	+	648.000i	0.884038	+	0.884038i	−0.109904	+	0.993942i	$0.535054\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−324.000	+	324.000i	−0.439620	+	0.439620i
$738$		0			0
$739$		−	164.000i		−	0.221922i	−0.993825	−	0.110961i	$-0.964607\pi$
$739$		−	164.000i		−	0.221922i	0.993825	−	0.110961i	$-0.0353928\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	55.0000	+	55.0000i	0.0740242	+	0.0740242i	0.743150	−	0.669125i	$-0.233331\pi$
$743$	55.0000	+	55.0000i	0.0740242	+	0.0740242i	−0.669125	+	0.743150i	$0.733331\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1470.00i			1.96262i
$750$		0			0
$751$	−656.000			−0.873502			−0.436751	−	0.899582i	$-0.643871\pi$
$751$	−656.000			−0.873502			−0.436751	+	0.899582i	$0.643871\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−168.000	+	168.000i	−0.221929	+	0.221929i	−0.809310	−	0.587382i	$-0.800159\pi$
$757$	−168.000	+	168.000i	−0.221929	+	0.221929i	0.587382	+	0.809310i	$0.300159\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1198.00			1.57424			0.787122	−	0.616797i	$-0.211570\pi$
$761$	1198.00			1.57424			0.787122	+	0.616797i	$0.211570\pi$
$762$		0			0
$763$	224.000	+	224.000i	0.293578	+	0.293578i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	528.000	−	528.000i	0.688396	−	0.688396i
$768$		0			0
$769$			14.0000i			0.0182055i	0.999959	+	0.00910273i	$0.00289753\pi$
$769$			14.0000i			0.0182055i	−0.999959	+	0.00910273i	$0.997102\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−580.000	−	580.000i	−0.750323	−	0.750323i	0.224216	−	0.974539i	$-0.428018\pi$
$773$	−580.000	−	580.000i	−0.750323	−	0.750323i	−0.974539	+	0.224216i	$0.928018\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	896.000i		−	1.15019i
$780$		0			0
$781$	448.000			0.573624
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	113.000	−	113.000i	0.143583	−	0.143583i	−0.631661	−	0.775245i	$-0.717627\pi$
$787$	113.000	−	113.000i	0.143583	−	0.143583i	0.775245	+	0.631661i	$0.217627\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−112.000			−0.141593
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	348.000	−	348.000i	0.436637	−	0.436637i	−0.454241	−	0.890879i	$-0.650090\pi$
$797$	348.000	−	348.000i	0.436637	−	0.436637i	0.890879	+	0.454241i	$0.150090\pi$
$798$		0			0
$799$			1240.00i			1.55194i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	176.000	+	176.000i	0.219178	+	0.219178i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	494.000i		−	0.610630i	−0.952251	−	0.305315i	$-0.901238\pi$
$809$		−	494.000i		−	0.610630i	0.952251	−	0.305315i	$-0.0987618\pi$
$810$		0			0
$811$	844.000			1.04069			0.520345	−	0.853956i	$-0.325803\pi$
$811$	844.000			1.04069			0.520345	+	0.853956i	$0.325803\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−196.000	+	196.000i	−0.239902	+	0.239902i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−992.000			−1.20828			−0.604141	−	0.796877i	$-0.706484\pi$
$821$	−992.000			−1.20828			−0.604141	+	0.796877i	$0.706484\pi$
$822$		0			0
$823$	−711.000	−	711.000i	−0.863913	−	0.863913i	0.127877	−	0.991790i	$-0.459184\pi$
$823$	−711.000	−	711.000i	−0.863913	−	0.863913i	−0.991790	+	0.127877i	$0.959184\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	57.0000	−	57.0000i	0.0689238	−	0.0689238i	−0.671805	−	0.740728i	$-0.734481\pi$
$827$	57.0000	−	57.0000i	0.0689238	−	0.0689238i	0.740728	+	0.671805i	$0.234481\pi$
$828$		0			0
$829$			800.000i			0.965018i	0.875891	+	0.482509i	$0.160274\pi$
$829$			800.000i			0.965018i	−0.875891	+	0.482509i	$0.839726\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	980.000	+	980.000i	1.17647	+	1.17647i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			248.000i			0.295590i	0.989018	+	0.147795i	$0.0472176\pi$
$839$			248.000i			0.295590i	−0.989018	+	0.147795i	$0.952782\pi$
$840$		0			0
$841$	−315.000			−0.374554
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−735.000	+	735.000i	−0.867769	+	0.867769i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	288.000			0.338425
$852$		0			0
$853$	−1036.00	−	1036.00i	−1.21454	−	1.21454i	−0.969518	−	0.245019i	$-0.921206\pi$
$853$	−1036.00	−	1036.00i	−1.21454	−	1.21454i	−0.245019	−	0.969518i	$-0.578794\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	168.000	−	168.000i	0.196033	−	0.196033i	−0.602264	−	0.798297i	$-0.705735\pi$
$857$	168.000	−	168.000i	0.196033	−	0.196033i	0.798297	+	0.602264i	$0.205735\pi$
$858$		0			0
$859$			28.0000i			0.0325960i	0.999867	+	0.0162980i	$0.00518805\pi$
$859$			28.0000i			0.0325960i	−0.999867	+	0.0162980i	$0.994812\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−929.000	−	929.000i	−1.07648	−	1.07648i	−0.996822	−	0.0796549i	$-0.974618\pi$
$863$	−929.000	−	929.000i	−1.07648	−	1.07648i	−0.0796549	−	0.996822i	$-0.525382\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	288.000i		−	0.331415i
$870$		0			0
$871$	−1944.00			−2.23192
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1176.00	+	1176.00i	−1.34094	+	1.34094i	−0.445805	+	0.895130i	$0.647082\pi$
$877$	−1176.00	+	1176.00i	−1.34094	+	1.34094i	−0.895130	+	0.445805i	$0.852918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1120.00			1.27128			0.635641	−	0.771985i	$-0.280736\pi$
$881$	1120.00			1.27128			0.635641	+	0.771985i	$0.280736\pi$
$882$		0			0
$883$	385.000	+	385.000i	0.436014	+	0.436014i	0.890668	−	0.454654i	$-0.150237\pi$
$883$	385.000	+	385.000i	0.436014	+	0.436014i	−0.454654	+	0.890668i	$0.650237\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1177.00	−	1177.00i	1.32694	−	1.32694i	0.418923	−	0.908022i	$-0.362408\pi$
$887$	1177.00	−	1177.00i	1.32694	−	1.32694i	0.908022	−	0.418923i	$-0.137592\pi$
$888$		0			0
$889$			14.0000i			0.0157480i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	868.000	+	868.000i	0.972004	+	0.972004i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1360.00i		−	1.51279i
$900$		0			0
$901$	−2080.00			−2.30855
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−633.000	+	633.000i	−0.697905	+	0.697905i	−0.963958	−	0.266053i	$-0.914280\pi$
$907$	−633.000	+	633.000i	−0.697905	+	0.697905i	0.266053	+	0.963958i	$0.414280\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1528.00			−1.67728			−0.838639	−	0.544688i	$-0.816648\pi$
$911$	−1528.00			−1.67728			−0.838639	+	0.544688i	$0.816648\pi$
$912$		0			0
$913$	196.000	+	196.000i	0.214677	+	0.214677i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1036.00	+	1036.00i	−1.12977	+	1.12977i
$918$		0			0
$919$		−	1472.00i		−	1.60174i	−0.598838	−	0.800871i	$-0.704370\pi$
$919$		−	1472.00i		−	1.60174i	0.598838	−	0.800871i	$-0.295630\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1344.00	+	1344.00i	1.45612	+	1.45612i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			496.000i			0.533907i	0.963709	+	0.266954i	$0.0860171\pi$
$929$			496.000i			0.533907i	−0.963709	+	0.266954i	$0.913983\pi$
$930$		0			0
$931$	1372.00			1.47368
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	180.000	−	180.000i	0.192102	−	0.192102i	−0.604502	−	0.796604i	$-0.706628\pi$
$937$	180.000	−	180.000i	0.192102	−	0.192102i	0.796604	+	0.604502i	$0.206628\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	546.000			0.580234			0.290117	−	0.956991i	$-0.406306\pi$
$941$	546.000			0.580234			0.290117	+	0.956991i	$0.406306\pi$
$942$		0			0
$943$	288.000	+	288.000i	0.305408	+	0.305408i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	705.000	−	705.000i	0.744456	−	0.744456i	−0.228976	−	0.973432i	$-0.573538\pi$
$947$	705.000	−	705.000i	0.744456	−	0.744456i	0.973432	+	0.228976i	$0.0735377\pi$
$948$		0			0
$949$			1056.00i			1.11275i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−888.000	−	888.000i	−0.931794	−	0.931794i	0.0660237	−	0.997818i	$-0.478969\pi$
$953$	−888.000	−	888.000i	−0.931794	−	0.931794i	−0.997818	+	0.0660237i	$0.978969\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			2352.00i			2.45255i
$960$		0			0
$961$	639.000			0.664932
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−455.000	+	455.000i	−0.470527	+	0.470527i	−0.902085	−	0.431558i	$-0.857964\pi$
$967$	−455.000	+	455.000i	−0.470527	+	0.470527i	0.431558	+	0.902085i	$0.357964\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1324.00			−1.36354			−0.681771	−	0.731565i	$-0.738790\pi$
$971$	−1324.00			−1.36354			−0.681771	+	0.731565i	$0.738790\pi$
$972$		0			0
$973$	364.000	+	364.000i	0.374101	+	0.374101i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−252.000	+	252.000i	−0.257932	+	0.257932i	−0.824213	−	0.566280i	$-0.808382\pi$
$977$	−252.000	+	252.000i	−0.257932	+	0.257932i	0.566280	+	0.824213i	$0.308382\pi$
$978$		0			0
$979$		−	392.000i		−	0.400409i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	617.000	+	617.000i	0.627670	+	0.627670i	0.947481	−	0.319811i	$-0.103619\pi$
$983$	617.000	+	617.000i	0.627670	+	0.627670i	−0.319811	+	0.947481i	$0.603619\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	126.000i		−	0.127401i
$990$		0			0
$991$	−1232.00			−1.24319			−0.621594	−	0.783339i	$-0.713515\pi$
$991$	−1232.00			−1.24319			−0.621594	+	0.783339i	$0.713515\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	428.000	−	428.000i	0.429288	−	0.429288i	−0.459098	−	0.888386i	$-0.651827\pi$
$997$	428.000	−	428.000i	0.429288	−	0.429288i	0.888386	+	0.459098i	$0.151827\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.3.v.g.793.1		2
3.2	odd	2		200.3.l.c.193.1	yes	2
5.2	odd	4	inner	1800.3.v.g.1657.1		2
5.3	odd	4		1800.3.v.a.1657.1		2
5.4	even	2		1800.3.v.a.793.1		2
12.11	even	2		400.3.p.c.193.1		2
15.2	even	4		200.3.l.c.57.1	yes	2
15.8	even	4		200.3.l.a.57.1	✓	2
15.14	odd	2		200.3.l.a.193.1	yes	2
60.23	odd	4		400.3.p.f.257.1		2
60.47	odd	4		400.3.p.c.257.1		2
60.59	even	2		400.3.p.f.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.3.l.a.57.1	✓	2	15.8	even	4
200.3.l.a.193.1	yes	2	15.14	odd	2
200.3.l.c.57.1	yes	2	15.2	even	4
200.3.l.c.193.1	yes	2	3.2	odd	2
400.3.p.c.193.1		2	12.11	even	2
400.3.p.c.257.1		2	60.47	odd	4
400.3.p.f.193.1		2	60.59	even	2
400.3.p.f.257.1		2	60.23	odd	4
1800.3.v.a.793.1		2	5.4	even	2
1800.3.v.a.1657.1		2	5.3	odd	4
1800.3.v.g.793.1		2	1.1	even	1	trivial
1800.3.v.g.1657.1		2	5.2	odd	4	inner

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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1800.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(7.00000 - 7.00000i) q^{7} +4.00000 q^{11} +(12.0000 + 12.0000i) q^{13} +(-20.0000 + 20.0000i) q^{17} +28.0000i q^{19} +(-9.00000 - 9.00000i) q^{23} +34.0000i q^{29} -40.0000 q^{31} +(-16.0000 + 16.0000i) q^{37} -32.0000 q^{41} +(7.00000 + 7.00000i) q^{43} +(31.0000 - 31.0000i) q^{47} -49.0000i q^{49} +(52.0000 + 52.0000i) q^{53} -44.0000i q^{59} +(-81.0000 + 81.0000i) q^{67} +112.000 q^{71} +(44.0000 + 44.0000i) q^{73} +(28.0000 - 28.0000i) q^{77} -72.0000i q^{79} +(49.0000 + 49.0000i) q^{83} -98.0000i q^{89} +168.000 q^{91} +(-44.0000 + 44.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{7} + 8 q^{11} + 24 q^{13} - 40 q^{17} - 18 q^{23} - 80 q^{31} - 32 q^{37} - 64 q^{41} + 14 q^{43} + 62 q^{47} + 104 q^{53} - 162 q^{67} + 224 q^{71} + 88 q^{73} + 56 q^{77} + 98 q^{83} + 336 q^{91}+ \cdots - 88 q^{97}+O(q^{100}) \) 2 * q + 14 * q^7 + 8 * q^11 + 24 * q^13 - 40 * q^17 - 18 * q^23 - 80 * q^31 - 32 * q^37 - 64 * q^41 + 14 * q^43 + 62 * q^47 + 104 * q^53 - 162 * q^67 + 224 * q^71 + 88 * q^73 + 56 * q^77 + 98 * q^83 + 336 * q^91 - 88 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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