LMFDB - Embedded newform 400.3.p.f.257.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,3,Mod(193,400)] 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("400.193"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	400.p (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,14,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$10.8992105744$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			257.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.257
Dual form			400.3.p.f.193.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{3} +(7.00000 + 7.00000i) q^{7} +7.00000i q^{9} +4.00000 q^{11} +(-12.0000 + 12.0000i) q^{13} +(-20.0000 - 20.0000i) q^{17} +28.0000i q^{19} +14.0000 q^{21} +(9.00000 - 9.00000i) q^{23} +(16.0000 + 16.0000i) q^{27} +34.0000i q^{29} +40.0000 q^{31} +(4.00000 - 4.00000i) q^{33} +(16.0000 + 16.0000i) q^{37} +24.0000i q^{39} +32.0000 q^{41} +(7.00000 - 7.00000i) q^{43} +(-31.0000 - 31.0000i) q^{47} +49.0000i q^{49} -40.0000 q^{51} +(52.0000 - 52.0000i) q^{53} +(28.0000 + 28.0000i) q^{57} +44.0000i q^{59} +(-49.0000 + 49.0000i) q^{63} +(-81.0000 - 81.0000i) q^{67} -18.0000i q^{69} +112.000 q^{71} +(-44.0000 + 44.0000i) q^{73} +(28.0000 + 28.0000i) q^{77} -72.0000i q^{79} -31.0000 q^{81} +(-49.0000 + 49.0000i) q^{83} +(34.0000 + 34.0000i) q^{87} -98.0000i q^{89} -168.000 q^{91} +(40.0000 - 40.0000i) q^{93} +(44.0000 + 44.0000i) q^{97} +28.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 14 q^{7} + 8 q^{11} - 24 q^{13} - 40 q^{17} + 28 q^{21} + 18 q^{23} + 32 q^{27} + 80 q^{31} + 8 q^{33} + 32 q^{37} + 64 q^{41} + 14 q^{43} - 62 q^{47} - 80 q^{51} + 104 q^{53} + 56 q^{57}+ \cdots + 88 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 14 * q^7 + 8 * q^11 - 24 * q^13 - 40 * q^17 + 28 * q^21 + 18 * q^23 + 32 * q^27 + 80 * q^31 + 8 * q^33 + 32 * q^37 + 64 * q^41 + 14 * q^43 - 62 * q^47 - 80 * q^51 + 104 * q^53 + 56 * q^57 - 98 * q^63 - 162 * q^67 + 224 * q^71 - 88 * q^73 + 56 * q^77 - 62 * q^81 - 98 * q^83 + 68 * q^87 - 336 * q^91 + 80 * q^93 + 88 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	1.00000	−	1.00000i	0.333333	−	0.333333i	−0.520518	−	0.853851i	$-0.674261\pi$
$3$	1.00000	−	1.00000i	0.333333	−	0.333333i	0.853851	+	0.520518i	$0.174261\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.00000	+	7.00000i	1.00000	+	1.00000i	1.00000			$0$
$7$	7.00000	+	7.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$8$		0			0
$9$			7.00000i			0.777778i
$10$		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	$-0.976370\pi$
$13$	−12.0000	+	12.0000i	−0.923077	+	0.923077i	0.0741688	+	0.997246i	$0.476370\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−20.0000	−	20.0000i	−1.17647	−	1.17647i	−0.980637	−	0.195833i	$-0.937259\pi$
$17$	−20.0000	−	20.0000i	−1.17647	−	1.17647i	−0.195833	−	0.980637i	$-0.562741\pi$
$18$		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
$21$	14.0000			0.666667
$22$		0			0
$23$	9.00000	−	9.00000i	0.391304	−	0.391304i	−0.483848	−	0.875152i	$-0.660761\pi$
$23$	9.00000	−	9.00000i	0.391304	−	0.391304i	0.875152	+	0.483848i	$0.160761\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	16.0000	+	16.0000i	0.592593	+	0.592593i
$28$		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	$-0.276790\pi$
$31$	40.0000			1.29032			0.645161	+	0.764046i	$0.276790\pi$
$32$		0			0
$33$	4.00000	−	4.00000i	0.121212	−	0.121212i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	16.0000	+	16.0000i	0.432432	+	0.432432i	0.889455	−	0.457023i	$-0.151084\pi$
$37$	16.0000	+	16.0000i	0.432432	+	0.432432i	−0.457023	+	0.889455i	$0.651084\pi$
$38$		0			0
$39$			24.0000i			0.615385i
$40$		0			0
$41$	32.0000			0.780488			0.390244	−	0.920712i	$-0.372391\pi$
$41$	32.0000			0.780488			0.390244	+	0.920712i	$0.372391\pi$
$42$		0			0
$43$	7.00000	−	7.00000i	0.162791	−	0.162791i	−0.621011	−	0.783802i	$-0.713278\pi$
$43$	7.00000	−	7.00000i	0.162791	−	0.162791i	0.783802	+	0.621011i	$0.213278\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−31.0000	−	31.0000i	−0.659574	−	0.659574i	0.295705	−	0.955279i	$-0.404446\pi$
$47$	−31.0000	−	31.0000i	−0.659574	−	0.659574i	−0.955279	+	0.295705i	$0.904446\pi$
$48$		0			0
$49$			49.0000i			1.00000i
$50$		0			0
$51$	−40.0000			−0.784314
$52$		0			0
$53$	52.0000	−	52.0000i	0.981132	−	0.981132i	−0.0186932	−	0.999825i	$-0.505951\pi$
$53$	52.0000	−	52.0000i	0.981132	−	0.981132i	0.999825	+	0.0186932i	$0.00595057\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	28.0000	+	28.0000i	0.491228	+	0.491228i
$58$		0			0
$59$			44.0000i			0.745763i	0.927879	+	0.372881i	$0.121630\pi$
$59$			44.0000i			0.745763i	−0.927879	+	0.372881i	$0.878370\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$	−49.0000	+	49.0000i	−0.777778	+	0.777778i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−81.0000	−	81.0000i	−1.20896	−	1.20896i	−0.971366	−	0.237590i	$-0.923643\pi$
$67$	−81.0000	−	81.0000i	−1.20896	−	1.20896i	−0.237590	−	0.971366i	$-0.576357\pi$
$68$		0			0
$69$		−	18.0000i		−	0.260870i
$70$		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	$-0.890148\pi$
$73$	−44.0000	+	44.0000i	−0.602740	+	0.602740i	0.338299	+	0.941039i	$0.390148\pi$
$74$		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
$81$	−31.0000			−0.382716
$82$		0			0
$83$	−49.0000	+	49.0000i	−0.590361	+	0.590361i	−0.937729	−	0.347368i	$-0.887076\pi$
$83$	−49.0000	+	49.0000i	−0.590361	+	0.590361i	0.347368	+	0.937729i	$0.387076\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	34.0000	+	34.0000i	0.390805	+	0.390805i
$88$		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
$93$	40.0000	−	40.0000i	0.430108	−	0.430108i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	44.0000	+	44.0000i	0.453608	+	0.453608i	0.896550	−	0.442942i	$-0.146065\pi$
$97$	44.0000	+	44.0000i	0.453608	+	0.453608i	−0.442942	+	0.896550i	$0.646065\pi$
$98$		0			0
$99$			28.0000i			0.282828i
$100$		0			0
$101$	−14.0000			−0.138614			−0.0693069	−	0.997595i	$-0.522079\pi$
$101$	−14.0000			−0.138614			−0.0693069	+	0.997595i	$0.522079\pi$
$102$		0			0
$103$	87.0000	−	87.0000i	0.844660	−	0.844660i	−0.144801	−	0.989461i	$-0.546254\pi$
$103$	87.0000	−	87.0000i	0.844660	−	0.844660i	0.989461	+	0.144801i	$0.0462541\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	105.000	+	105.000i	0.981308	+	0.981308i	0.999828	−	0.0185201i	$-0.00589546\pi$
$107$	105.000	+	105.000i	0.981308	+	0.981308i	−0.0185201	+	0.999828i	$0.505895\pi$
$108$		0			0
$109$		−	32.0000i		−	0.293578i	−0.989168	−	0.146789i	$-0.953106\pi$
$109$		−	32.0000i		−	0.293578i	0.989168	−	0.146789i	$-0.0468938\pi$
$110$		0			0
$111$	32.0000			0.288288
$112$		0			0
$113$	−8.00000	+	8.00000i	−0.0707965	+	0.0707965i	−0.741618	−	0.670822i	$-0.765941\pi$
$113$	−8.00000	+	8.00000i	−0.0707965	+	0.0707965i	0.670822	+	0.741618i	$0.265941\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−84.0000	−	84.0000i	−0.717949	−	0.717949i
$118$		0			0
$119$		−	280.000i		−	2.35294i
$120$		0			0
$121$	−105.000			−0.867769
$122$		0			0
$123$	32.0000	−	32.0000i	0.260163	−	0.260163i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−1.00000	−	1.00000i	−0.00787402	−	0.00787402i	0.703159	−	0.711033i	$-0.251772\pi$
$127$	−1.00000	−	1.00000i	−0.00787402	−	0.00787402i	−0.711033	+	0.703159i	$0.751772\pi$
$128$		0			0
$129$		−	14.0000i		−	0.108527i
$130$		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	−0.965362	−	0.260916i	$-0.915975\pi$
$137$	−168.000	−	168.000i	−1.22628	−	1.22628i	−0.260916	−	0.965362i	$-0.584025\pi$
$138$		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
$141$	−62.0000			−0.439716
$142$		0			0
$143$	−48.0000	+	48.0000i	−0.335664	+	0.335664i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	49.0000	+	49.0000i	0.333333	+	0.333333i
$148$		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	$-0.405882\pi$
$151$	88.0000			0.582781			0.291391	+	0.956604i	$0.405882\pi$
$152$		0			0
$153$	140.000	−	140.000i	0.915033	−	0.915033i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−56.0000	−	56.0000i	−0.356688	−	0.356688i	0.505903	−	0.862591i	$-0.331159\pi$
$157$	−56.0000	−	56.0000i	−0.356688	−	0.356688i	−0.862591	+	0.505903i	$0.831159\pi$
$158$		0			0
$159$		−	104.000i		−	0.654088i
$160$		0			0
$161$	126.000			0.782609
$162$		0			0
$163$	191.000	−	191.000i	1.17178	−	1.17178i	0.189994	−	0.981785i	$-0.439153\pi$
$163$	191.000	−	191.000i	1.17178	−	1.17178i	0.981785	−	0.189994i	$-0.0608468\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	23.0000	+	23.0000i	0.137725	+	0.137725i	0.772608	−	0.634883i	$-0.218952\pi$
$167$	23.0000	+	23.0000i	0.137725	+	0.137725i	−0.634883	+	0.772608i	$0.718952\pi$
$168$		0			0
$169$		−	119.000i		−	0.704142i
$170$		0			0
$171$	−196.000			−1.14620
$172$		0			0
$173$	−24.0000	+	24.0000i	−0.138728	+	0.138728i	−0.773061	−	0.634332i	$-0.781275\pi$
$173$	−24.0000	+	24.0000i	−0.138728	+	0.138728i	0.634332	+	0.773061i	$0.281275\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	44.0000	+	44.0000i	0.248588	+	0.248588i
$178$		0			0
$179$			28.0000i			0.156425i	0.996937	+	0.0782123i	$0.0249212\pi$
$179$			28.0000i			0.156425i	−0.996937	+	0.0782123i	$0.975079\pi$
$180$		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
$189$			224.000i			1.18519i
$190$		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	$-0.717288\pi$
$193$	28.0000	−	28.0000i	0.145078	−	0.145078i	0.775915	+	0.630837i	$0.217288\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	132.000	+	132.000i	0.670051	+	0.670051i	0.957728	−	0.287677i	$-0.0928829\pi$
$197$	132.000	+	132.000i	0.670051	+	0.670051i	−0.287677	+	0.957728i	$0.592883\pi$
$198$		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
$201$	−162.000			−0.805970
$202$		0			0
$203$	−238.000	+	238.000i	−1.17241	+	1.17241i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	63.0000	+	63.0000i	0.304348	+	0.304348i
$208$		0			0
$209$			112.000i			0.535885i
$210$		0			0
$211$	−244.000			−1.15640			−0.578199	−	0.815896i	$-0.696244\pi$
$211$	−244.000			−1.15640			−0.578199	+	0.815896i	$0.696244\pi$
$212$		0			0
$213$	112.000	−	112.000i	0.525822	−	0.525822i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	280.000	+	280.000i	1.29032	+	1.29032i
$218$		0			0
$219$			88.0000i			0.401826i
$220$		0			0
$221$	480.000			2.17195
$222$		0			0
$223$	33.0000	−	33.0000i	0.147982	−	0.147982i	−0.629234	−	0.777216i	$-0.716631\pi$
$223$	33.0000	−	33.0000i	0.147982	−	0.147982i	0.777216	+	0.629234i	$0.216631\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−127.000	−	127.000i	−0.559471	−	0.559471i	0.369686	−	0.929157i	$-0.379465\pi$
$227$	−127.000	−	127.000i	−0.559471	−	0.559471i	−0.929157	+	0.369686i	$0.879465\pi$
$228$		0			0
$229$			78.0000i			0.340611i	0.985391	+	0.170306i	$0.0544755\pi$
$229$			78.0000i			0.340611i	−0.985391	+	0.170306i	$0.945524\pi$
$230$		0			0
$231$	56.0000			0.242424
$232$		0			0
$233$	−28.0000	+	28.0000i	−0.120172	+	0.120172i	−0.764635	−	0.644463i	$-0.777081\pi$
$233$	−28.0000	+	28.0000i	−0.120172	+	0.120172i	0.644463	+	0.764635i	$0.277081\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−72.0000	−	72.0000i	−0.303797	−	0.303797i
$238$		0			0
$239$		−	8.00000i		−	0.0334728i	−0.999860	−	0.0167364i	$-0.994672\pi$
$239$		−	8.00000i		−	0.0334728i	0.999860	−	0.0167364i	$-0.00532761\pi$
$240$		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
$243$	−175.000	+	175.000i	−0.720165	+	0.720165i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−336.000	−	336.000i	−1.36032	−	1.36032i
$248$		0			0
$249$			98.0000i			0.393574i
$250$		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	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$			224.000i			0.864865i
$260$		0			0
$261$	−238.000			−0.911877
$262$		0			0
$263$	−105.000	+	105.000i	−0.399240	+	0.399240i	−0.877965	−	0.478725i	$-0.841099\pi$
$263$	−105.000	+	105.000i	−0.399240	+	0.399240i	0.478725	+	0.877965i	$0.341099\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−98.0000	−	98.0000i	−0.367041	−	0.367041i
$268$		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	$-0.467053\pi$
$271$	56.0000			0.206642			0.103321	+	0.994648i	$0.467053\pi$
$272$		0			0
$273$	−168.000	+	168.000i	−0.615385	+	0.615385i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	32.0000	+	32.0000i	0.115523	+	0.115523i	0.762505	−	0.646982i	$-0.223969\pi$
$277$	32.0000	+	32.0000i	0.115523	+	0.115523i	−0.646982	+	0.762505i	$0.723969\pi$
$278$		0			0
$279$			280.000i			1.00358i
$280$		0			0
$281$	48.0000			0.170819			0.0854093	−	0.996346i	$-0.472780\pi$
$281$	48.0000			0.170819			0.0854093	+	0.996346i	$0.472780\pi$
$282$		0			0
$283$	57.0000	−	57.0000i	0.201413	−	0.201413i	−0.599192	−	0.800605i	$-0.704511\pi$
$283$	57.0000	−	57.0000i	0.201413	−	0.201413i	0.800605	+	0.599192i	$0.204511\pi$
$284$		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
$291$	88.0000			0.302405
$292$		0			0
$293$	296.000	−	296.000i	1.01024	−	1.01024i	0.0102919	−	0.999947i	$-0.496724\pi$
$293$	296.000	−	296.000i	1.01024	−	1.01024i	0.999947	−	0.0102919i	$-0.00327606\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	64.0000	+	64.0000i	0.215488	+	0.215488i
$298$		0			0
$299$			216.000i			0.722408i
$300$		0			0
$301$	98.0000			0.325581
$302$		0			0
$303$	−14.0000	+	14.0000i	−0.0462046	+	0.0462046i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	383.000	+	383.000i	1.24756	+	1.24756i	0.956796	+	0.290761i	$0.0939086\pi$
$307$	383.000	+	383.000i	1.24756	+	1.24756i	0.290761	+	0.956796i	$0.406091\pi$
$308$		0			0
$309$		−	174.000i		−	0.563107i
$310$		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	$-0.855470\pi$
$313$	−144.000	+	144.000i	−0.460064	+	0.460064i	0.438612	+	0.898676i	$0.355470\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	28.0000	+	28.0000i	0.0883281	+	0.0883281i	0.749890	−	0.661562i	$-0.230106\pi$
$317$	28.0000	+	28.0000i	0.0883281	+	0.0883281i	−0.661562	+	0.749890i	$0.730106\pi$
$318$		0			0
$319$			136.000i			0.426332i
$320$		0			0
$321$	210.000			0.654206
$322$		0			0
$323$	560.000	−	560.000i	1.73375	−	1.73375i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−32.0000	−	32.0000i	−0.0978593	−	0.0978593i
$328$		0			0
$329$		−	434.000i		−	1.31915i
$330$		0			0
$331$	−28.0000			−0.0845921			−0.0422961	−	0.999105i	$-0.513467\pi$
$331$	−28.0000			−0.0845921			−0.0422961	+	0.999105i	$0.513467\pi$
$332$		0			0
$333$	−112.000	+	112.000i	−0.336336	+	0.336336i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−384.000	−	384.000i	−1.13947	−	1.13947i	−0.988546	−	0.150920i	$-0.951776\pi$
$337$	−384.000	−	384.000i	−1.13947	−	1.13947i	−0.150920	−	0.988546i	$-0.548224\pi$
$338$		0			0
$339$			16.0000i			0.0471976i
$340$		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	$-0.297527\pi$
$347$	−73.0000	−	73.0000i	−0.210375	−	0.210375i	−0.804427	+	0.594052i	$0.797527\pi$
$348$		0			0
$349$			574.000i			1.64470i	0.568983	+	0.822350i	$0.307337\pi$
$349$			574.000i			1.64470i	−0.568983	+	0.822350i	$0.692663\pi$
$350$		0			0
$351$	−384.000			−1.09402
$352$		0			0
$353$	160.000	−	160.000i	0.453258	−	0.453258i	−0.443177	−	0.896434i	$-0.646149\pi$
$353$	160.000	−	160.000i	0.453258	−	0.453258i	0.896434	+	0.443177i	$0.146149\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−280.000	−	280.000i	−0.784314	−	0.784314i
$358$		0			0
$359$		−	424.000i		−	1.18106i	−0.807016	−	0.590529i	$-0.798919\pi$
$359$		−	424.000i		−	1.18106i	0.807016	−	0.590529i	$-0.201081\pi$
$360$		0			0
$361$	−423.000			−1.17175
$362$		0			0
$363$	−105.000	+	105.000i	−0.289256	+	0.289256i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	225.000	+	225.000i	0.613079	+	0.613079i	0.943747	−	0.330668i	$-0.107274\pi$
$367$	225.000	+	225.000i	0.613079	+	0.613079i	−0.330668	+	0.943747i	$0.607274\pi$
$368$		0			0
$369$			224.000i			0.607046i
$370$		0			0
$371$	728.000			1.96226
$372$		0			0
$373$	−280.000	+	280.000i	−0.750670	+	0.750670i	−0.974604	−	0.223934i	$-0.928110\pi$
$373$	−280.000	+	280.000i	−0.750670	+	0.750670i	0.223934	+	0.974604i	$0.428110\pi$
$374$		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
$381$	−2.00000			−0.00524934
$382$		0			0
$383$	273.000	−	273.000i	0.712794	−	0.712794i	−0.254325	−	0.967119i	$-0.581853\pi$
$383$	273.000	−	273.000i	0.712794	−	0.712794i	0.967119	+	0.254325i	$0.0818533\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	49.0000	+	49.0000i	0.126615	+	0.126615i
$388$		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
$393$	−148.000	+	148.000i	−0.376590	+	0.376590i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	276.000	+	276.000i	0.695214	+	0.695214i	0.963374	−	0.268160i	$-0.0864157\pi$
$397$	276.000	+	276.000i	0.695214	+	0.695214i	−0.268160	+	0.963374i	$0.586416\pi$
$398$		0			0
$399$			392.000i			0.982456i
$400$		0			0
$401$	366.000			0.912718			0.456359	−	0.889796i	$-0.349153\pi$
$401$	366.000			0.912718			0.456359	+	0.889796i	$0.349153\pi$
$402$		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	$-0.693129\pi$
$409$		−	672.000i		−	1.64303i	0.570186	−	0.821516i	$-0.306871\pi$
$410$		0			0
$411$	−336.000			−0.817518
$412$		0			0
$413$	−308.000	+	308.000i	−0.745763	+	0.745763i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	52.0000	+	52.0000i	0.124700	+	0.124700i
$418$		0			0
$419$		−	564.000i		−	1.34606i	−0.739614	−	0.673031i	$-0.764992\pi$
$419$		−	564.000i		−	1.34606i	0.739614	−	0.673031i	$-0.235008\pi$
$420$		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
$423$	217.000	−	217.000i	0.513002	−	0.513002i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$			96.0000i			0.223776i
$430$		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	$-0.509419\pi$
$433$	420.000	−	420.000i	0.969977	−	0.969977i	0.999562	+	0.0295854i	$0.00941868\pi$
$434$		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
$441$	−343.000			−0.777778
$442$		0			0
$443$	7.00000	−	7.00000i	0.0158014	−	0.0158014i	−0.699162	−	0.714963i	$-0.746443\pi$
$443$	7.00000	−	7.00000i	0.0158014	−	0.0158014i	0.714963	+	0.699162i	$0.246443\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−224.000	−	224.000i	−0.501119	−	0.501119i
$448$		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
$453$	88.0000	−	88.0000i	0.194260	−	0.194260i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−16.0000	−	16.0000i	−0.0350109	−	0.0350109i	0.689385	−	0.724396i	$-0.257881\pi$
$457$	−16.0000	−	16.0000i	−0.0350109	−	0.0350109i	−0.724396	+	0.689385i	$0.757881\pi$
$458$		0			0
$459$		−	640.000i		−	1.39434i
$460$		0			0
$461$	−98.0000			−0.212581			−0.106291	−	0.994335i	$-0.533897\pi$
$461$	−98.0000			−0.212581			−0.106291	+	0.994335i	$0.533897\pi$
$462$		0			0
$463$	−273.000	+	273.000i	−0.589633	+	0.589633i	−0.937532	−	0.347899i	$-0.886895\pi$
$463$	−273.000	+	273.000i	−0.589633	+	0.589633i	0.347899	+	0.937532i	$0.386895\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−193.000	−	193.000i	−0.413276	−	0.413276i	0.469602	−	0.882878i	$-0.344397\pi$
$467$	−193.000	−	193.000i	−0.413276	−	0.413276i	−0.882878	+	0.469602i	$0.844397\pi$
$468$		0			0
$469$		−	1134.00i		−	2.41791i
$470$		0			0
$471$	−112.000			−0.237792
$472$		0			0
$473$	28.0000	−	28.0000i	0.0591966	−	0.0591966i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	364.000	+	364.000i	0.763103	+	0.763103i
$478$		0			0
$479$		−	464.000i		−	0.968685i	−0.874879	−	0.484342i	$-0.839059\pi$
$479$		−	464.000i		−	0.968685i	0.874879	−	0.484342i	$-0.160941\pi$
$480$		0			0
$481$	−384.000			−0.798337
$482$		0			0
$483$	126.000	−	126.000i	0.260870	−	0.260870i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	297.000	+	297.000i	0.609856	+	0.609856i	0.942908	−	0.333052i	$-0.108079\pi$
$487$	297.000	+	297.000i	0.609856	+	0.609856i	−0.333052	+	0.942908i	$0.608079\pi$
$488$		0			0
$489$		−	382.000i		−	0.781186i
$490$		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
$501$	46.0000			0.0918164
$502$		0			0
$503$	−537.000	+	537.000i	−1.06759	+	1.06759i	−0.0700510	+	0.997543i	$0.522316\pi$
$503$	−537.000	+	537.000i	−1.06759	+	1.06759i	−0.997543	+	0.0700510i	$0.977684\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−119.000	−	119.000i	−0.234714	−	0.234714i
$508$		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
$513$	−448.000	+	448.000i	−0.873294	+	0.873294i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−124.000	−	124.000i	−0.239845	−	0.239845i
$518$		0			0
$519$			48.0000i			0.0924855i
$520$		0			0
$521$	210.000			0.403071			0.201536	−	0.979481i	$-0.435407\pi$
$521$	210.000			0.403071			0.201536	+	0.979481i	$0.435407\pi$
$522$		0			0
$523$	105.000	−	105.000i	0.200765	−	0.200765i	−0.599563	−	0.800328i	$-0.704659\pi$
$523$	105.000	−	105.000i	0.200765	−	0.200765i	0.800328	+	0.599563i	$0.204659\pi$
$524$		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
$531$	−308.000			−0.580038
$532$		0			0
$533$	−384.000	+	384.000i	−0.720450	+	0.720450i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	28.0000	+	28.0000i	0.0521415	+	0.0521415i
$538$		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
$543$	−178.000	+	178.000i	−0.327808	+	0.327808i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−399.000	−	399.000i	−0.729433	−	0.729433i	0.241074	−	0.970507i	$-0.422500\pi$
$547$	−399.000	−	399.000i	−0.729433	−	0.729433i	−0.970507	+	0.241074i	$0.922500\pi$
$548$		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	−0.972165	−	0.234298i	$-0.924721\pi$
$557$	−672.000	−	672.000i	−1.20646	−	1.20646i	−0.234298	−	0.972165i	$-0.575279\pi$
$558$		0			0
$559$			168.000i			0.300537i
$560$		0			0
$561$	−160.000			−0.285205
$562$		0			0
$563$	−191.000	+	191.000i	−0.339254	+	0.339254i	−0.856087	−	0.516833i	$-0.827111\pi$
$563$	−191.000	+	191.000i	−0.339254	+	0.339254i	0.516833	+	0.856087i	$0.327111\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−217.000	−	217.000i	−0.382716	−	0.382716i
$568$		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	$-0.289979\pi$
$571$	700.000			1.22592			0.612960	+	0.790114i	$0.289979\pi$
$572$		0			0
$573$	320.000	−	320.000i	0.558464	−	0.558464i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−232.000	−	232.000i	−0.402080	−	0.402080i	0.476886	−	0.878965i	$-0.341766\pi$
$577$	−232.000	−	232.000i	−0.402080	−	0.402080i	−0.878965	+	0.476886i	$0.841766\pi$
$578$		0			0
$579$		−	56.0000i		−	0.0967185i
$580$		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	$-0.0179411\pi$
$587$	553.000	+	553.000i	0.942078	+	0.942078i	−0.0563336	+	0.998412i	$0.517941\pi$
$588$		0			0
$589$			1120.00i			1.90153i
$590$		0			0
$591$	264.000			0.446701
$592$		0			0
$593$	−312.000	+	312.000i	−0.526138	+	0.526138i	−0.919419	−	0.393280i	$-0.871340\pi$
$593$	−312.000	+	312.000i	−0.526138	+	0.526138i	0.393280	+	0.919419i	$0.371340\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	96.0000	+	96.0000i	0.160804	+	0.160804i
$598$		0			0
$599$			80.0000i			0.133556i	0.997768	+	0.0667780i	$0.0212719\pi$
$599$			80.0000i			0.133556i	−0.997768	+	0.0667780i	$0.978728\pi$
$600$		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
$603$	567.000	−	567.000i	0.940299	−	0.940299i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	305.000	+	305.000i	0.502471	+	0.502471i	0.912205	−	0.409734i	$-0.134378\pi$
$607$	305.000	+	305.000i	0.502471	+	0.502471i	−0.409734	+	0.912205i	$0.634378\pi$
$608$		0			0
$609$			476.000i			0.781609i
$610$		0			0
$611$	744.000			1.21768
$612$		0			0
$613$	420.000	−	420.000i	0.685155	−	0.685155i	−0.276002	−	0.961157i	$-0.589010\pi$
$613$	420.000	−	420.000i	0.685155	−	0.685155i	0.961157	+	0.276002i	$0.0890097\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	708.000	+	708.000i	1.14749	+	1.14749i	0.987045	+	0.160443i	$0.0512922\pi$
$617$	708.000	+	708.000i	1.14749	+	1.14749i	0.160443	+	0.987045i	$0.448708\pi$
$618$		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
$621$	288.000			0.463768
$622$		0			0
$623$	686.000	−	686.000i	1.10112	−	1.10112i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	112.000	+	112.000i	0.178628	+	0.178628i
$628$		0			0
$629$		−	640.000i		−	1.01749i
$630$		0			0
$631$	−440.000			−0.697306			−0.348653	−	0.937252i	$-0.613361\pi$
$631$	−440.000			−0.697306			−0.348653	+	0.937252i	$0.613361\pi$
$632$		0			0
$633$	−244.000	+	244.000i	−0.385466	+	0.385466i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−588.000	−	588.000i	−0.923077	−	0.923077i
$638$		0			0
$639$			784.000i			1.22692i
$640$		0			0
$641$	784.000			1.22309			0.611544	−	0.791210i	$-0.290549\pi$
$641$	784.000			1.22309			0.611544	+	0.791210i	$0.290549\pi$
$642$		0			0
$643$	−751.000	+	751.000i	−1.16796	+	1.16796i	−0.185276	+	0.982686i	$0.559318\pi$
$643$	−751.000	+	751.000i	−1.16796	+	1.16796i	−0.982686	+	0.185276i	$0.940682\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	777.000	+	777.000i	1.20093	+	1.20093i	0.973885	+	0.227043i	$0.0729056\pi$
$647$	777.000	+	777.000i	1.20093	+	1.20093i	0.227043	+	0.973885i	$0.427094\pi$
$648$		0			0
$649$			176.000i			0.271186i
$650$		0			0
$651$	560.000			0.860215
$652$		0			0
$653$	−588.000	+	588.000i	−0.900459	+	0.900459i	−0.995476	−	0.0950163i	$-0.969710\pi$
$653$	−588.000	+	588.000i	−0.900459	+	0.900459i	0.0950163	+	0.995476i	$0.469710\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−308.000	−	308.000i	−0.468798	−	0.468798i
$658$		0			0
$659$			972.000i			1.47496i	0.675368	+	0.737481i	$0.263985\pi$
$659$			972.000i			1.47496i	−0.675368	+	0.737481i	$0.736015\pi$
$660$		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
$663$	480.000	−	480.000i	0.723982	−	0.723982i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	306.000	+	306.000i	0.458771	+	0.458771i
$668$		0			0
$669$		−	66.0000i		−	0.0986547i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−252.000	+	252.000i	−0.374443	+	0.374443i	−0.869092	−	0.494650i	$-0.835296\pi$
$673$	−252.000	+	252.000i	−0.374443	+	0.374443i	0.494650	+	0.869092i	$0.335296\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−108.000	−	108.000i	−0.159527	−	0.159527i	0.622830	−	0.782357i	$-0.285983\pi$
$677$	−108.000	−	108.000i	−0.159527	−	0.159527i	−0.782357	+	0.622830i	$0.785983\pi$
$678$		0			0
$679$			616.000i			0.907216i
$680$		0			0
$681$	−254.000			−0.372981
$682$		0			0
$683$	231.000	−	231.000i	0.338214	−	0.338214i	−0.517481	−	0.855695i	$-0.673130\pi$
$683$	231.000	−	231.000i	0.338214	−	0.338214i	0.855695	+	0.517481i	$0.173130\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	78.0000	+	78.0000i	0.113537	+	0.113537i
$688$		0			0
$689$			1248.00i			1.81132i
$690$		0			0
$691$	84.0000			0.121563			0.0607815	−	0.998151i	$-0.480641\pi$
$691$	84.0000			0.121563			0.0607815	+	0.998151i	$0.480641\pi$
$692$		0			0
$693$	−196.000	+	196.000i	−0.282828	+	0.282828i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−640.000	−	640.000i	−0.918221	−	0.918221i
$698$		0			0
$699$			56.0000i			0.0801144i
$700$		0			0
$701$	512.000			0.730385			0.365193	−	0.930932i	$-0.381003\pi$
$701$	512.000			0.730385			0.365193	+	0.930932i	$0.381003\pi$
$702$		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	$0.0256181\pi$
$709$			114.000i			0.160790i	−0.996763	+	0.0803949i	$0.974382\pi$
$710$		0			0
$711$	504.000			0.708861
$712$		0			0
$713$	360.000	−	360.000i	0.504909	−	0.504909i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−8.00000	−	8.00000i	−0.0111576	−	0.0111576i
$718$		0			0
$719$		−	232.000i		−	0.322670i	−0.986900	−	0.161335i	$-0.948420\pi$
$719$		−	232.000i		−	0.322670i	0.986900	−	0.161335i	$-0.0515800\pi$
$720$		0			0
$721$	1218.00			1.68932
$722$		0			0
$723$	208.000	−	208.000i	0.287690	−	0.287690i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	423.000	+	423.000i	0.581843	+	0.581843i	0.935409	−	0.353566i	$-0.115031\pi$
$727$	423.000	+	423.000i	0.581843	+	0.581843i	−0.353566	+	0.935409i	$0.615031\pi$
$728$		0			0
$729$			71.0000i			0.0973937i
$730$		0			0
$731$	−280.000			−0.383037
$732$		0			0
$733$	−648.000	+	648.000i	−0.884038	+	0.884038i	−0.993942	−	0.109904i	$-0.964946\pi$
$733$	−648.000	+	648.000i	−0.884038	+	0.884038i	0.109904	+	0.993942i	$0.464946\pi$
$734$		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
$741$	−672.000			−0.906883
$742$		0			0
$743$	−55.0000	+	55.0000i	−0.0740242	+	0.0740242i	−0.743150	−	0.669125i	$-0.766669\pi$
$743$	−55.0000	+	55.0000i	−0.0740242	+	0.0740242i	0.669125	+	0.743150i	$0.266669\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−343.000	−	343.000i	−0.459170	−	0.459170i
$748$		0			0
$749$			1470.00i			1.96262i
$750$		0			0
$751$	656.000			0.873502			0.436751	−	0.899582i	$-0.356129\pi$
$751$	656.000			0.873502			0.436751	+	0.899582i	$0.356129\pi$
$752$		0			0
$753$	340.000	−	340.000i	0.451527	−	0.451527i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	168.000	+	168.000i	0.221929	+	0.221929i	0.809310	−	0.587382i	$-0.199841\pi$
$757$	168.000	+	168.000i	0.221929	+	0.221929i	−0.587382	+	0.809310i	$0.699841\pi$
$758$		0			0
$759$		−	72.0000i		−	0.0948617i
$760$		0			0
$761$	−1198.00			−1.57424			−0.787122	−	0.616797i	$-0.788430\pi$
$761$	−1198.00			−1.57424			−0.787122	+	0.616797i	$0.788430\pi$
$762$		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	$-0.997102\pi$
$769$		−	14.0000i		−	0.0182055i	0.999959	−	0.00910273i	$-0.00289753\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−580.000	+	580.000i	−0.750323	+	0.750323i	−0.974539	−	0.224216i	$-0.928018\pi$
$773$	−580.000	+	580.000i	−0.750323	+	0.750323i	0.224216	+	0.974539i	$0.428018\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	224.000	+	224.000i	0.288288	+	0.288288i
$778$		0			0
$779$			896.000i			1.15019i
$780$		0			0
$781$	448.000			0.573624
$782$		0			0
$783$	−544.000	+	544.000i	−0.694764	+	0.694764i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	113.000	+	113.000i	0.143583	+	0.143583i	0.775245	−	0.631661i	$-0.217627\pi$
$787$	113.000	+	113.000i	0.143583	+	0.143583i	−0.631661	+	0.775245i	$0.717627\pi$
$788$		0			0
$789$			210.000i			0.266160i
$790$		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	0.890879	−	0.454241i	$-0.150090\pi$
$797$	348.000	+	348.000i	0.436637	+	0.436637i	−0.454241	+	0.890879i	$0.650090\pi$
$798$		0			0
$799$			1240.00i			1.55194i
$800$		0			0
$801$	686.000			0.856429
$802$		0			0
$803$	−176.000	+	176.000i	−0.219178	+	0.219178i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	128.000	+	128.000i	0.158612	+	0.158612i
$808$		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	$-0.674197\pi$
$811$	−844.000			−1.04069			−0.520345	+	0.853956i	$0.674197\pi$
$812$		0			0
$813$	56.0000	−	56.0000i	0.0688807	−	0.0688807i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	196.000	+	196.000i	0.239902	+	0.239902i
$818$		0			0
$819$		−	1176.00i		−	1.43590i
$820$		0			0
$821$	992.000			1.20828			0.604141	−	0.796877i	$-0.293516\pi$
$821$	992.000			1.20828			0.604141	+	0.796877i	$0.293516\pi$
$822$		0			0
$823$	−711.000	+	711.000i	−0.863913	+	0.863913i	−0.991790	−	0.127877i	$-0.959184\pi$
$823$	−711.000	+	711.000i	−0.863913	+	0.863913i	0.127877	+	0.991790i	$0.459184\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−57.0000	−	57.0000i	−0.0689238	−	0.0689238i	0.671805	−	0.740728i	$-0.265519\pi$
$827$	−57.0000	−	57.0000i	−0.0689238	−	0.0689238i	−0.740728	+	0.671805i	$0.765519\pi$
$828$		0			0
$829$		−	800.000i		−	0.965018i	−0.875891	−	0.482509i	$-0.839726\pi$
$829$		−	800.000i		−	0.965018i	0.875891	−	0.482509i	$-0.160274\pi$
$830$		0			0
$831$	64.0000			0.0770156
$832$		0			0
$833$	980.000	−	980.000i	1.17647	−	1.17647i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	640.000	+	640.000i	0.764636	+	0.764636i
$838$		0			0
$839$		−	248.000i		−	0.295590i	−0.989018	−	0.147795i	$-0.952782\pi$
$839$		−	248.000i		−	0.295590i	0.989018	−	0.147795i	$-0.0472176\pi$
$840$		0			0
$841$	−315.000			−0.374554
$842$		0			0
$843$	48.0000	−	48.0000i	0.0569395	−	0.0569395i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−735.000	−	735.000i	−0.867769	−	0.867769i
$848$		0			0
$849$		−	114.000i		−	0.134276i
$850$		0			0
$851$	288.000			0.338425
$852$		0			0
$853$	1036.00	−	1036.00i	1.21454	−	1.21454i	0.245019	−	0.969518i	$-0.421206\pi$
$853$	1036.00	−	1036.00i	1.21454	−	1.21454i	0.969518	−	0.245019i	$-0.0787941\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	168.000	+	168.000i	0.196033	+	0.196033i	0.798297	−	0.602264i	$-0.205735\pi$
$857$	168.000	+	168.000i	0.196033	+	0.196033i	−0.602264	+	0.798297i	$0.705735\pi$
$858$		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
$861$	448.000			0.520325
$862$		0			0
$863$	929.000	−	929.000i	1.07648	−	1.07648i	0.0796549	−	0.996822i	$-0.474618\pi$
$863$	929.000	−	929.000i	1.07648	−	1.07648i	0.996822	−	0.0796549i	$-0.0253818\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	511.000	+	511.000i	0.589389	+	0.589389i
$868$		0			0
$869$		−	288.000i		−	0.331415i
$870$		0			0
$871$	1944.00			2.23192
$872$		0			0
$873$	−308.000	+	308.000i	−0.352806	+	0.352806i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1176.00	+	1176.00i	1.34094	+	1.34094i	0.895130	+	0.445805i	$0.147082\pi$
$877$	1176.00	+	1176.00i	1.34094	+	1.34094i	0.445805	+	0.895130i	$0.352918\pi$
$878$		0			0
$879$		−	592.000i		−	0.673493i
$880$		0			0
$881$	−1120.00			−1.27128			−0.635641	−	0.771985i	$-0.719264\pi$
$881$	−1120.00			−1.27128			−0.635641	+	0.771985i	$0.719264\pi$
$882$		0			0
$883$	385.000	−	385.000i	0.436014	−	0.436014i	−0.454654	−	0.890668i	$-0.650237\pi$
$883$	385.000	−	385.000i	0.436014	−	0.436014i	0.890668	+	0.454654i	$0.150237\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1177.00	−	1177.00i	−1.32694	−	1.32694i	−0.908022	−	0.418923i	$-0.862408\pi$
$887$	−1177.00	−	1177.00i	−1.32694	−	1.32694i	−0.418923	−	0.908022i	$-0.637592\pi$
$888$		0			0
$889$		−	14.0000i		−	0.0157480i
$890$		0			0
$891$	−124.000			−0.139169
$892$		0			0
$893$	868.000	−	868.000i	0.972004	−	0.972004i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	216.000	+	216.000i	0.240803	+	0.240803i
$898$		0			0
$899$			1360.00i			1.51279i
$900$		0			0
$901$	−2080.00			−2.30855
$902$		0			0
$903$	98.0000	−	98.0000i	0.108527	−	0.108527i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−633.000	−	633.000i	−0.697905	−	0.697905i	0.266053	−	0.963958i	$-0.414280\pi$
$907$	−633.000	−	633.000i	−0.697905	−	0.697905i	−0.963958	+	0.266053i	$0.914280\pi$
$908$		0			0
$909$		−	98.0000i		−	0.107811i
$910$		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
$921$	766.000			0.831705
$922$		0			0
$923$	−1344.00	+	1344.00i	−1.45612	+	1.45612i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	609.000	+	609.000i	0.656958	+	0.656958i
$928$		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
$933$	56.0000	−	56.0000i	0.0600214	−	0.0600214i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−180.000	−	180.000i	−0.192102	−	0.192102i	0.604502	−	0.796604i	$-0.293372\pi$
$937$	−180.000	−	180.000i	−0.192102	−	0.192102i	−0.796604	+	0.604502i	$0.793372\pi$
$938$		0			0
$939$			288.000i			0.306709i
$940$		0			0
$941$	−546.000			−0.580234			−0.290117	−	0.956991i	$-0.593694\pi$
$941$	−546.000			−0.580234			−0.290117	+	0.956991i	$0.593694\pi$
$942$		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	$-0.426462\pi$
$947$	−705.000	−	705.000i	−0.744456	−	0.744456i	−0.973432	+	0.228976i	$0.926462\pi$
$948$		0			0
$949$		−	1056.00i		−	1.11275i
$950$		0			0
$951$	56.0000			0.0588854
$952$		0			0
$953$	−888.000	+	888.000i	−0.931794	+	0.931794i	−0.997818	−	0.0660237i	$-0.978969\pi$
$953$	−888.000	+	888.000i	−0.931794	+	0.931794i	0.0660237	+	0.997818i	$0.478969\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	136.000	+	136.000i	0.142111	+	0.142111i
$958$		0			0
$959$		−	2352.00i		−	2.45255i
$960$		0			0
$961$	639.000			0.664932
$962$		0			0
$963$	−735.000	+	735.000i	−0.763240	+	0.763240i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−455.000	−	455.000i	−0.470527	−	0.470527i	0.431558	−	0.902085i	$-0.357964\pi$
$967$	−455.000	−	455.000i	−0.470527	−	0.470527i	−0.902085	+	0.431558i	$0.857964\pi$
$968$		0			0
$969$		−	1120.00i		−	1.15583i
$970$		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	0.566280	−	0.824213i	$-0.308382\pi$
$977$	−252.000	−	252.000i	−0.257932	−	0.257932i	−0.824213	+	0.566280i	$0.808382\pi$
$978$		0			0
$979$		−	392.000i		−	0.400409i
$980$		0			0
$981$	224.000			0.228338
$982$		0			0
$983$	−617.000	+	617.000i	−0.627670	+	0.627670i	−0.947481	−	0.319811i	$-0.896381\pi$
$983$	−617.000	+	617.000i	−0.627670	+	0.627670i	0.319811	+	0.947481i	$0.396381\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−434.000	−	434.000i	−0.439716	−	0.439716i
$988$		0			0
$989$		−	126.000i		−	0.127401i
$990$		0			0
$991$	1232.00			1.24319			0.621594	−	0.783339i	$-0.286485\pi$
$991$	1232.00			1.24319			0.621594	+	0.783339i	$0.286485\pi$
$992$		0			0
$993$	−28.0000	+	28.0000i	−0.0281974	+	0.0281974i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−428.000	−	428.000i	−0.429288	−	0.429288i	0.459098	−	0.888386i	$-0.348173\pi$
$997$	−428.000	−	428.000i	−0.429288	−	0.429288i	−0.888386	+	0.459098i	$0.848173\pi$
$998$		0			0
$999$			512.000i			0.512513i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.3.l.a.57.1	✓	2	4.3	odd	2
200.3.l.a.193.1	yes	2	20.3	even	4
200.3.l.c.57.1	yes	2	20.19	odd	2
200.3.l.c.193.1	yes	2	20.7	even	4
400.3.p.c.193.1		2	5.2	odd	4
400.3.p.c.257.1		2	5.4	even	2
400.3.p.f.193.1		2	5.3	odd	4	inner
400.3.p.f.257.1		2	1.1	even	1	trivial
1800.3.v.a.793.1		2	60.23	odd	4
1800.3.v.a.1657.1		2	12.11	even	2
1800.3.v.g.793.1		2	60.47	odd	4
1800.3.v.g.1657.1		2	60.59	even	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	400.p (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{3} +(7.00000 + 7.00000i) q^{7} +7.00000i q^{9} +4.00000 q^{11} +(-12.0000 + 12.0000i) q^{13} +(-20.0000 - 20.0000i) q^{17} +28.0000i q^{19} +14.0000 q^{21} +(9.00000 - 9.00000i) q^{23} +(16.0000 + 16.0000i) q^{27} +34.0000i q^{29} +40.0000 q^{31} +(4.00000 - 4.00000i) q^{33} +(16.0000 + 16.0000i) q^{37} +24.0000i q^{39} +32.0000 q^{41} +(7.00000 - 7.00000i) q^{43} +(-31.0000 - 31.0000i) q^{47} +49.0000i q^{49} -40.0000 q^{51} +(52.0000 - 52.0000i) q^{53} +(28.0000 + 28.0000i) q^{57} +44.0000i q^{59} +(-49.0000 + 49.0000i) q^{63} +(-81.0000 - 81.0000i) q^{67} -18.0000i q^{69} +112.000 q^{71} +(-44.0000 + 44.0000i) q^{73} +(28.0000 + 28.0000i) q^{77} -72.0000i q^{79} -31.0000 q^{81} +(-49.0000 + 49.0000i) q^{83} +(34.0000 + 34.0000i) q^{87} -98.0000i q^{89} -168.000 q^{91} +(40.0000 - 40.0000i) q^{93} +(44.0000 + 44.0000i) q^{97} +28.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 14 q^{7} + 8 q^{11} - 24 q^{13} - 40 q^{17} + 28 q^{21} + 18 q^{23} + 32 q^{27} + 80 q^{31} + 8 q^{33} + 32 q^{37} + 64 q^{41} + 14 q^{43} - 62 q^{47} - 80 q^{51} + 104 q^{53} + 56 q^{57}+ \cdots + 88 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 14 * q^7 + 8 * q^11 - 24 * q^13 - 40 * q^17 + 28 * q^21 + 18 * q^23 + 32 * q^27 + 80 * q^31 + 8 * q^33 + 32 * q^37 + 64 * q^41 + 14 * q^43 - 62 * q^47 - 80 * q^51 + 104 * q^53 + 56 * q^57 - 98 * q^63 - 162 * q^67 + 224 * q^71 - 88 * q^73 + 56 * q^77 - 62 * q^81 - 98 * q^83 + 68 * q^87 - 336 * q^91 + 80 * q^93 + 88 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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