LMFDB - Embedded newform 20.2.e.a.7.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [20,2,Mod(3,20)] 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("20.3"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	20.e (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$0.159700804043$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			7.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	20.7
Dual form			20.2.e.a.3.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(-2.00000 + 1.00000i) q^{5} +(2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +(3.00000 + 1.00000i) q^{10} +(-1.00000 + 1.00000i) q^{13} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{17} +(-3.00000 + 3.00000i) q^{18} +(-2.00000 - 4.00000i) q^{20} +(3.00000 - 4.00000i) q^{25} +2.00000 q^{26} +4.00000i q^{29} +(4.00000 + 4.00000i) q^{32} -6.00000i q^{34} +6.00000 q^{36} +(-7.00000 - 7.00000i) q^{37} +(-2.00000 + 6.00000i) q^{40} -8.00000 q^{41} +(3.00000 + 6.00000i) q^{45} +7.00000i q^{49} +(-7.00000 + 1.00000i) q^{50} +(-2.00000 - 2.00000i) q^{52} +(9.00000 - 9.00000i) q^{53} +(4.00000 - 4.00000i) q^{58} +12.0000 q^{61} -8.00000i q^{64} +(1.00000 - 3.00000i) q^{65} +(-6.00000 + 6.00000i) q^{68} +(-6.00000 - 6.00000i) q^{72} +(-11.0000 + 11.0000i) q^{73} +14.0000i q^{74} +(8.00000 - 4.00000i) q^{80} -9.00000 q^{81} +(8.00000 + 8.00000i) q^{82} +(-9.00000 - 3.00000i) q^{85} -16.0000i q^{89} +(3.00000 - 9.00000i) q^{90} +(13.0000 + 13.0000i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{5} + 4 q^{8} + 6 q^{10} - 2 q^{13} - 8 q^{16} + 6 q^{17} - 6 q^{18} - 4 q^{20} + 6 q^{25} + 4 q^{26} + 8 q^{32} + 12 q^{36} - 14 q^{37} - 4 q^{40} - 16 q^{41} + 6 q^{45} - 14 q^{50}+ \cdots + 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 + 6 * q^10 - 2 * q^13 - 8 * q^16 + 6 * q^17 - 6 * q^18 - 4 * q^20 + 6 * q^25 + 4 * q^26 + 8 * q^32 + 12 * q^36 - 14 * q^37 - 4 * q^40 - 16 * q^41 + 6 * q^45 - 14 * q^50 - 4 * q^52 + 18 * q^53 + 8 * q^58 + 24 * q^61 + 2 * q^65 - 12 * q^68 - 12 * q^72 - 22 * q^73 + 16 * q^80 - 18 * q^81 + 16 * q^82 - 18 * q^85 + 6 * q^90 + 26 * q^97 + 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$			2.00000i			1.00000i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$		−	3.00000i		−	1.00000i
$10$	3.00000	+	1.00000i	0.948683	+	0.316228i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−1.00000	+	1.00000i	−0.277350	+	0.277350i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−1.00000	+	1.00000i	−0.277350	+	0.277350i	0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	3.00000	+	3.00000i	0.727607	+	0.727607i	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	3.00000	+	3.00000i	0.727607	+	0.727607i	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−3.00000	+	3.00000i	−0.707107	+	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−2.00000	−	4.00000i	−0.447214	−	0.894427i
$21$		0			0
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	2.00000			0.392232
$27$		0			0
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$		0			0
$34$		−	6.00000i		−	1.02899i
$35$		0			0
$36$	6.00000			1.00000
$37$	−7.00000	−	7.00000i	−1.15079	−	1.15079i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−7.00000	−	7.00000i	−1.15079	−	1.15079i	−0.164399	−	0.986394i	$-0.552568\pi$
$38$		0			0
$39$		0			0
$40$	−2.00000	+	6.00000i	−0.316228	+	0.948683i
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	3.00000	+	6.00000i	0.447214	+	0.894427i
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$	−7.00000	+	1.00000i	−0.989949	+	0.141421i
$51$		0			0
$52$	−2.00000	−	2.00000i	−0.277350	−	0.277350i
$53$	9.00000	−	9.00000i	1.23625	−	1.23625i	0.274721	−	0.961524i	$-0.411414\pi$
$53$	9.00000	−	9.00000i	1.23625	−	1.23625i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	4.00000	−	4.00000i	0.525226	−	0.525226i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	12.0000			1.53644			0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000			1.53644			0.768221	+	0.640184i	$0.221142\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	1.00000	−	3.00000i	0.124035	−	0.372104i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	−6.00000	+	6.00000i	−0.727607	+	0.727607i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−6.00000	−	6.00000i	−0.707107	−	0.707107i
$73$	−11.0000	+	11.0000i	−1.28745	+	1.28745i	−0.351123	+	0.936329i	$0.614200\pi$
$73$	−11.0000	+	11.0000i	−1.28745	+	1.28745i	−0.936329	+	0.351123i	$0.885800\pi$
$74$			14.0000i			1.62747i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	8.00000	−	4.00000i	0.894427	−	0.447214i
$81$	−9.00000			−1.00000
$82$	8.00000	+	8.00000i	0.883452	+	0.883452i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	−9.00000	−	3.00000i	−0.976187	−	0.325396i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	16.0000i		−	1.69600i	−0.529999	−	0.847998i	$-0.677808\pi$
$89$		−	16.0000i		−	1.69600i	0.529999	−	0.847998i	$-0.322192\pi$
$90$	3.00000	−	9.00000i	0.316228	−	0.948683i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.0000	+	13.0000i	1.31995	+	1.31995i	0.913812	+	0.406138i	$0.133125\pi$
$97$	13.0000	+	13.0000i	1.31995	+	1.31995i	0.406138	+	0.913812i	$0.366875\pi$
$98$	7.00000	−	7.00000i	0.707107	−	0.707107i
$99$		0			0
$100$	8.00000	+	6.00000i	0.800000	+	0.600000i
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$			4.00000i			0.392232i
$105$		0			0
$106$	−18.0000			−1.74831
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.00000	+	1.00000i	−0.0940721	+	0.0940721i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−1.00000	+	1.00000i	−0.0940721	+	0.0940721i	0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		0			0
$116$	−8.00000			−0.742781
$117$	3.00000	+	3.00000i	0.277350	+	0.277350i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	−12.0000	−	12.0000i	−1.08643	−	1.08643i
$123$		0			0
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$129$		0			0
$130$	−4.00000	+	2.00000i	−0.350823	+	0.175412i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	12.0000			1.02899
$137$	−7.00000	−	7.00000i	−0.598050	−	0.598050i	0.341743	−	0.939793i	$-0.388983\pi$
$137$	−7.00000	−	7.00000i	−0.598050	−	0.598050i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			12.0000i			1.00000i
$145$	−4.00000	−	8.00000i	−0.332182	−	0.664364i
$146$	22.0000			1.82073
$147$		0			0
$148$	14.0000	−	14.0000i	1.15079	−	1.15079i
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	9.00000	−	9.00000i	0.727607	−	0.727607i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−17.0000	−	17.0000i	−1.35675	−	1.35675i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−17.0000	−	17.0000i	−1.35675	−	1.35675i	−0.478852	−	0.877896i	$-0.658947\pi$
$158$		0			0
$159$		0			0
$160$	−12.0000	−	4.00000i	−0.948683	−	0.316228i
$161$		0			0
$162$	9.00000	+	9.00000i	0.707107	+	0.707107i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		−	16.0000i		−	1.24939i
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$			11.0000i			0.846154i
$170$	6.00000	+	12.0000i	0.460179	+	0.920358i
$171$		0			0
$172$		0			0
$173$	−11.0000	+	11.0000i	−0.836315	+	0.836315i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−11.0000	+	11.0000i	−0.836315	+	0.836315i	0.152057	+	0.988372i	$0.451410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−16.0000	+	16.0000i	−1.19925	+	1.19925i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−12.0000	+	6.00000i	−0.894427	+	0.447214i
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	21.0000	+	7.00000i	1.54395	+	0.514650i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	19.0000	−	19.0000i	1.36765	−	1.36765i	0.503871	−	0.863779i	$-0.331909\pi$
$193$	19.0000	−	19.0000i	1.36765	−	1.36765i	0.863779	−	0.503871i	$-0.168091\pi$
$194$		−	26.0000i		−	1.86669i
$195$		0			0
$196$	−14.0000			−1.00000
$197$	13.0000	+	13.0000i	0.926212	+	0.926212i	0.997459	−	0.0712470i	$-0.0226979\pi$
$197$	13.0000	+	13.0000i	0.926212	+	0.926212i	−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−2.00000	−	14.0000i	−0.141421	−	0.989949i
$201$		0			0
$202$	−2.00000	−	2.00000i	−0.140720	−	0.140720i
$203$		0			0
$204$		0			0
$205$	16.0000	−	8.00000i	1.11749	−	0.558744i
$206$		0			0
$207$		0			0
$208$	4.00000	−	4.00000i	0.277350	−	0.277350i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	18.0000	+	18.0000i	1.23625	+	1.23625i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−6.00000	+	6.00000i	−0.406371	+	0.406371i
$219$		0			0
$220$		0			0
$221$	−6.00000			−0.403604
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$	−12.0000	−	9.00000i	−0.800000	−	0.600000i
$226$	2.00000			0.133038
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$		0			0
$232$	8.00000	+	8.00000i	0.525226	+	0.525226i
$233$	−21.0000	+	21.0000i	−1.37576	+	1.37576i	−0.524097	+	0.851658i	$0.675597\pi$
$233$	−21.0000	+	21.0000i	−1.37576	+	1.37576i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		−	6.00000i		−	0.392232i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$	−11.0000	−	11.0000i	−0.707107	−	0.707107i
$243$		0			0
$244$			24.0000i			1.53644i
$245$	−7.00000	−	14.0000i	−0.447214	−	0.894427i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	13.0000	−	9.00000i	0.822192	−	0.569210i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−17.0000	−	17.0000i	−1.06043	−	1.06043i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	−17.0000	−	17.0000i	−1.06043	−	1.06043i	−0.0623783	−	0.998053i	$-0.519869\pi$
$258$		0			0
$259$		0			0
$260$	6.00000	+	2.00000i	0.372104	+	0.124035i
$261$	12.0000			0.742781
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	−9.00000	+	27.0000i	−0.552866	+	1.65860i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	26.0000i		−	1.58525i	−0.609711	−	0.792624i	$-0.708714\pi$
$269$		−	26.0000i		−	1.58525i	0.609711	−	0.792624i	$-0.291286\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−12.0000	−	12.0000i	−0.727607	−	0.727607i
$273$		0			0
$274$			14.0000i			0.845771i
$275$		0			0
$276$		0			0
$277$	23.0000	+	23.0000i	1.38194	+	1.38194i	0.841178	+	0.540758i	$0.181862\pi$
$277$	23.0000	+	23.0000i	1.38194	+	1.38194i	0.540758	+	0.841178i	$0.318138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	32.0000			1.90896			0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000			1.90896			0.954480	+	0.298275i	$0.0964112\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	12.0000	−	12.0000i	0.707107	−	0.707107i
$289$			1.00000i			0.0588235i
$290$	−4.00000	+	12.0000i	−0.234888	+	0.704664i
$291$		0			0
$292$	−22.0000	−	22.0000i	−1.28745	−	1.28745i
$293$	19.0000	−	19.0000i	1.10999	−	1.10999i	0.116841	−	0.993151i	$-0.462723\pi$
$293$	19.0000	−	19.0000i	1.10999	−	1.10999i	0.993151	−	0.116841i	$-0.0372769\pi$
$294$		0			0
$295$		0			0
$296$	−28.0000			−1.62747
$297$		0			0
$298$	14.0000	−	14.0000i	0.810998	−	0.810998i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−24.0000	+	12.0000i	−1.37424	+	0.687118i
$306$	−18.0000			−1.02899
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−1.00000	+	1.00000i	−0.0565233	+	0.0565233i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−1.00000	+	1.00000i	−0.0565233	+	0.0565233i	0.678280	+	0.734803i	$0.262726\pi$
$314$			34.0000i			1.91873i
$315$		0			0
$316$		0			0
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	0.786318	−	0.617822i	$-0.211985\pi$
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	−0.617822	+	0.786318i	$0.711985\pi$
$318$		0			0
$319$		0			0
$320$	8.00000	+	16.0000i	0.447214	+	0.894427i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	18.0000i		−	1.00000i
$325$	1.00000	+	7.00000i	0.0554700	+	0.388290i
$326$		0			0
$327$		0			0
$328$	−16.0000	+	16.0000i	−0.883452	+	0.883452i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−21.0000	+	21.0000i	−1.15079	+	1.15079i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.00000	−	7.00000i	−0.381314	−	0.381314i	0.490261	−	0.871576i	$-0.336901\pi$
$337$	−7.00000	−	7.00000i	−0.381314	−	0.381314i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	11.0000	−	11.0000i	0.598321	−	0.598321i
$339$		0			0
$340$	6.00000	−	18.0000i	0.325396	−	0.976187i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	22.0000			1.18273
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		−	36.0000i		−	1.92704i	−0.267644	−	0.963518i	$-0.586245\pi$
$349$		−	36.0000i		−	1.92704i	0.267644	−	0.963518i	$-0.413755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.00000	−	9.00000i	0.479022	−	0.479022i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	9.00000	−	9.00000i	0.479022	−	0.479022i	0.904819	+	0.425797i	$0.140006\pi$
$354$		0			0
$355$		0			0
$356$	32.0000			1.69600
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	18.0000	+	6.00000i	0.948683	+	0.316228i
$361$	−19.0000			−1.00000
$362$	18.0000	+	18.0000i	0.946059	+	0.946059i
$363$		0			0
$364$		0			0
$365$	11.0000	−	33.0000i	0.575766	−	1.72730i
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$			24.0000i			1.24939i
$370$	−14.0000	−	28.0000i	−0.727825	−	1.45565i
$371$		0			0
$372$		0			0
$373$	−11.0000	+	11.0000i	−0.569558	+	0.569558i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−11.0000	+	11.0000i	−0.569558	+	0.569558i	0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.00000	−	4.00000i	−0.206010	−	0.206010i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	−38.0000			−1.93415
$387$		0			0
$388$	−26.0000	+	26.0000i	−1.31995	+	1.31995i
$389$			34.0000i			1.72387i	0.507020	+	0.861934i	$0.330747\pi$
$389$			34.0000i			1.72387i	−0.507020	+	0.861934i	$0.669253\pi$
$390$		0			0
$391$		0			0
$392$	14.0000	+	14.0000i	0.707107	+	0.707107i
$393$		0			0
$394$		−	26.0000i		−	1.30986i
$395$		0			0
$396$		0			0
$397$	13.0000	+	13.0000i	0.652451	+	0.652451i	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	13.0000	+	13.0000i	0.652451	+	0.652451i	−0.301131	+	0.953583i	$0.597364\pi$
$398$		0			0
$399$		0			0
$400$	−12.0000	+	16.0000i	−0.600000	+	0.800000i
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$		0			0
$404$			4.00000i			0.199007i
$405$	18.0000	−	9.00000i	0.894427	−	0.447214i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	6.00000i		−	0.296681i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$		−	6.00000i		−	0.296681i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$	−24.0000	−	8.00000i	−1.18528	−	0.395092i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−8.00000			−0.392232
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$		0			0
$424$		−	36.0000i		−	1.74831i
$425$	21.0000	−	3.00000i	1.01865	−	0.145521i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	29.0000	−	29.0000i	1.39365	−	1.39365i	0.576683	−	0.816968i	$-0.304347\pi$
$433$	29.0000	−	29.0000i	1.39365	−	1.39365i	0.816968	−	0.576683i	$-0.195653\pi$
$434$		0			0
$435$		0			0
$436$	12.0000			0.574696
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$	6.00000	+	6.00000i	0.285391	+	0.285391i
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	16.0000	+	32.0000i	0.758473	+	1.51695i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			14.0000i			0.660701i	0.943858	+	0.330350i	$0.107167\pi$
$449$			14.0000i			0.660701i	−0.943858	+	0.330350i	$0.892833\pi$
$450$	3.00000	+	21.0000i	0.141421	+	0.989949i
$451$		0			0
$452$	−2.00000	−	2.00000i	−0.0940721	−	0.0940721i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	4.00000	−	4.00000i	0.186908	−	0.186908i
$459$		0			0
$460$		0			0
$461$	−38.0000			−1.76984			−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000			−1.76984			−0.884918	+	0.465746i	$0.845786\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		−	16.0000i		−	0.742781i
$465$		0			0
$466$	42.0000			1.94561
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$	−6.00000	+	6.00000i	−0.277350	+	0.277350i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−27.0000	−	27.0000i	−1.23625	−	1.23625i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	14.0000			0.638345
$482$	8.00000	+	8.00000i	0.364390	+	0.364390i
$483$		0			0
$484$			22.0000i			1.00000i
$485$	−39.0000	−	13.0000i	−1.77090	−	0.590300i
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	24.0000	−	24.0000i	1.08643	−	1.08643i
$489$		0			0
$490$	−7.00000	+	21.0000i	−0.316228	+	0.948683i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−12.0000	+	12.0000i	−0.540453	+	0.540453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−22.0000	−	4.00000i	−0.983870	−	0.178885i
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	−4.00000	+	2.00000i	−0.177998	+	0.0889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			44.0000i			1.95027i	0.221621	+	0.975133i	$0.428865\pi$
$509$			44.0000i			1.95027i	−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$			34.0000i			1.49968i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−4.00000	−	8.00000i	−0.175412	−	0.350823i
$521$	22.0000			0.963837			0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000			0.963837			0.481919	+	0.876216i	$0.339940\pi$
$522$	−12.0000	−	12.0000i	−0.525226	−	0.525226i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	23.0000i		−	1.00000i
$530$	36.0000	−	18.0000i	1.56374	−	0.781870i
$531$		0			0
$532$		0			0
$533$	8.00000	−	8.00000i	0.346518	−	0.346518i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−26.0000	+	26.0000i	−1.12094	+	1.12094i
$539$		0			0
$540$		0			0
$541$	42.0000			1.80572			0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000			1.80572			0.902861	+	0.429934i	$0.141463\pi$
$542$		0			0
$543$		0			0
$544$			24.0000i			1.02899i
$545$	6.00000	+	12.0000i	0.257012	+	0.514024i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	14.0000	−	14.0000i	0.598050	−	0.598050i
$549$		−	36.0000i		−	1.53644i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	46.0000i		−	1.95435i
$555$		0			0
$556$		0			0
$557$	33.0000	+	33.0000i	1.39825	+	1.39825i	0.805056	+	0.593199i	$0.202135\pi$
$557$	33.0000	+	33.0000i	1.39825	+	1.39825i	0.593199	+	0.805056i	$0.297865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−32.0000	−	32.0000i	−1.34984	−	1.34984i
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	1.00000	−	3.00000i	0.0420703	−	0.126211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26.0000i		−	1.08998i	−0.838444	−	0.544988i	$-0.816534\pi$
$569$		−	26.0000i		−	1.08998i	0.838444	−	0.544988i	$-0.183466\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	0.999133	−	0.0416305i	$-0.0132552\pi$
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	1.00000	−	1.00000i	0.0415945	−	0.0415945i
$579$		0			0
$580$	16.0000	−	8.00000i	0.664364	−	0.332182i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			44.0000i			1.82073i
$585$	−9.00000	−	3.00000i	−0.372104	−	0.124035i
$586$	−38.0000			−1.56977
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	28.0000	+	28.0000i	1.15079	+	1.15079i
$593$	−31.0000	+	31.0000i	−1.27302	+	1.27302i	−0.328521	+	0.944497i	$0.606550\pi$
$593$	−31.0000	+	31.0000i	−1.27302	+	1.27302i	−0.944497	+	0.328521i	$0.893450\pi$
$594$		0			0
$595$		0			0
$596$	−28.0000			−1.14692
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−48.0000			−1.95796			−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−48.0000			−1.95796			−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.0000	+	11.0000i	−0.894427	+	0.447214i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$	36.0000	+	12.0000i	1.45760	+	0.485866i
$611$		0			0
$612$	18.0000	+	18.0000i	0.727607	+	0.727607i
$613$	−1.00000	+	1.00000i	−0.0403896	+	0.0403896i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	−1.00000	+	1.00000i	−0.0403896	+	0.0403896i	0.686624	+	0.727013i	$0.259092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	+	3.00000i	0.120775	+	0.120775i	0.764911	−	0.644136i	$-0.222783\pi$
$617$	3.00000	+	3.00000i	0.120775	+	0.120775i	−0.644136	+	0.764911i	$0.722783\pi$
$618$		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
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	2.00000			0.0799361
$627$		0			0
$628$	34.0000	−	34.0000i	1.35675	−	1.35675i
$629$		−	42.0000i		−	1.67465i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		−	6.00000i		−	0.238290i
$635$		0			0
$636$		0			0
$637$	−7.00000	−	7.00000i	−0.277350	−	0.277350i
$638$		0			0
$639$		0			0
$640$	8.00000	−	24.0000i	0.316228	−	0.948683i
$641$	−8.00000			−0.315981			−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000			−0.315981			−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	−18.0000	+	18.0000i	−0.707107	+	0.707107i
$649$		0			0
$650$	6.00000	−	8.00000i	0.235339	−	0.313786i
$651$		0			0
$652$		0			0
$653$	9.00000	−	9.00000i	0.352197	−	0.352197i	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	9.00000	−	9.00000i	0.352197	−	0.352197i	0.860927	+	0.508729i	$0.169885\pi$
$654$		0			0
$655$		0			0
$656$	32.0000			1.24939
$657$	33.0000	+	33.0000i	1.28745	+	1.28745i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	12.0000			0.466746			0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000			0.466746			0.233373	+	0.972387i	$0.425024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	42.0000			1.62747
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−11.0000	+	11.0000i	−0.424019	+	0.424019i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−11.0000	+	11.0000i	−0.424019	+	0.424019i	0.462566	+	0.886585i	$0.346929\pi$
$674$			14.0000i			0.539260i
$675$		0			0
$676$	−22.0000			−0.846154
$677$	−27.0000	−	27.0000i	−1.03769	−	1.03769i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	−27.0000	−	27.0000i	−1.03769	−	1.03769i	−0.0384331	−	0.999261i	$-0.512237\pi$
$678$		0			0
$679$		0			0
$680$	−24.0000	+	12.0000i	−0.920358	+	0.460179i
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	21.0000	+	7.00000i	0.802369	+	0.267456i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			18.0000i			0.685745i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−22.0000	−	22.0000i	−0.836315	−	0.836315i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−24.0000	−	24.0000i	−0.909065	−	0.909065i
$698$	−36.0000	+	36.0000i	−1.36262	+	1.36262i
$699$		0			0
$700$		0			0
$701$	52.0000			1.96401			0.982006	−	0.188847i	$-0.0604752\pi$
$701$	52.0000			1.96401			0.982006	+	0.188847i	$0.0604752\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−18.0000			−0.677439
$707$		0			0
$708$		0			0
$709$			44.0000i			1.65245i	0.563337	+	0.826227i	$0.309517\pi$
$709$			44.0000i			1.65245i	−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$	−32.0000	−	32.0000i	−1.19925	−	1.19925i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−12.0000	−	24.0000i	−0.447214	−	0.894427i
$721$		0			0
$722$	19.0000	+	19.0000i	0.707107	+	0.707107i
$723$		0			0
$724$		−	36.0000i		−	1.33793i
$725$	16.0000	+	12.0000i	0.594225	+	0.445669i
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$	−44.0000	+	22.0000i	−1.62851	+	0.814257i
$731$		0			0
$732$		0			0
$733$	29.0000	−	29.0000i	1.07114	−	1.07114i	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	29.0000	−	29.0000i	1.07114	−	1.07114i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	24.0000	−	24.0000i	0.883452	−	0.883452i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−14.0000	+	42.0000i	−0.514650	+	1.54395i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	−14.0000	−	28.0000i	−0.512920	−	1.02584i
$746$	22.0000			0.805477
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$			8.00000i			0.291343i
$755$		0			0
$756$		0			0
$757$	−17.0000	−	17.0000i	−0.617876	−	0.617876i	0.327111	−	0.944986i	$-0.393925\pi$
$757$	−17.0000	−	17.0000i	−0.617876	−	0.617876i	−0.944986	+	0.327111i	$0.893925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−38.0000			−1.37750			−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000			−1.37750			−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−9.00000	+	27.0000i	−0.325396	+	0.976187i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$	38.0000	+	38.0000i	1.36765	+	1.36765i
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.791285	−	0.611448i	$-0.209412\pi$
$774$		0			0
$775$		0			0
$776$	52.0000			1.86669
$777$		0			0
$778$	34.0000	−	34.0000i	1.21896	−	1.21896i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	28.0000i		−	1.00000i
$785$	51.0000	+	17.0000i	1.82027	+	0.606756i
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	−26.0000	+	26.0000i	−0.926212	+	0.926212i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−12.0000	+	12.0000i	−0.426132	+	0.426132i
$794$		−	26.0000i		−	0.922705i
$795$		0			0
$796$		0			0
$797$	−37.0000	−	37.0000i	−1.31061	−	1.31061i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−37.0000	−	37.0000i	−1.31061	−	1.31061i	−0.389640	−	0.920967i	$-0.627401\pi$
$798$		0			0
$799$		0			0
$800$	28.0000	−	4.00000i	0.989949	−	0.141421i
$801$	−48.0000			−1.69600
$802$	−2.00000	−	2.00000i	−0.0706225	−	0.0706225i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	4.00000	−	4.00000i	0.140720	−	0.140720i
$809$		−	56.0000i		−	1.96886i	−0.175791	−	0.984428i	$-0.556248\pi$
$809$		−	56.0000i		−	1.96886i	0.175791	−	0.984428i	$-0.443752\pi$
$810$	−27.0000	−	9.00000i	−0.948683	−	0.316228i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−6.00000	+	6.00000i	−0.209785	+	0.209785i
$819$		0			0
$820$	16.0000	+	32.0000i	0.558744	+	1.11749i
$821$	−28.0000			−0.977207			−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000			−0.977207			−0.488603	+	0.872506i	$0.662493\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$			54.0000i			1.87550i	0.347314	+	0.937749i	$0.387094\pi$
$829$			54.0000i			1.87550i	−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$		0			0
$832$	8.00000	+	8.00000i	0.277350	+	0.277350i
$833$	−21.0000	+	21.0000i	−0.727607	+	0.727607i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$	28.0000	+	28.0000i	0.964944	+	0.964944i
$843$		0			0
$844$		0			0
$845$	−11.0000	−	22.0000i	−0.378412	−	0.756823i
$846$		0			0
$847$		0			0
$848$	−36.0000	+	36.0000i	−1.23625	+	1.23625i
$849$		0			0
$850$	−24.0000	−	18.0000i	−0.823193	−	0.617395i
$851$		0			0
$852$		0			0
$853$	−41.0000	+	41.0000i	−1.40381	+	1.40381i	−0.616308	+	0.787505i	$0.711372\pi$
$853$	−41.0000	+	41.0000i	−1.40381	+	1.40381i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	33.0000	+	33.0000i	1.12726	+	1.12726i	0.990621	+	0.136637i	$0.0436295\pi$
$857$	33.0000	+	33.0000i	1.12726	+	1.12726i	0.136637	+	0.990621i	$0.456370\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$	11.0000	−	33.0000i	0.374011	−	1.12203i
$866$	−58.0000			−1.97092
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−12.0000	−	12.0000i	−0.406371	−	0.406371i
$873$	39.0000	−	39.0000i	1.31995	−	1.31995i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.0000	+	23.0000i	0.776655	+	0.776655i	0.979260	−	0.202606i	$-0.0649409\pi$
$877$	23.0000	+	23.0000i	0.776655	+	0.776655i	−0.202606	+	0.979260i	$0.564941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	32.0000			1.07811			0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000			1.07811			0.539054	+	0.842271i	$0.318782\pi$
$882$	−21.0000	−	21.0000i	−0.707107	−	0.707107i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		−	12.0000i		−	0.403604i
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$	16.0000	−	48.0000i	0.536321	−	1.60896i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	14.0000	−	14.0000i	0.467186	−	0.467186i
$899$		0			0
$900$	18.0000	−	24.0000i	0.600000	−	0.800000i
$901$	54.0000			1.79900
$902$		0			0
$903$		0			0
$904$			4.00000i			0.133038i
$905$	36.0000	−	18.0000i	1.19668	−	0.598340i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		−	6.00000i		−	0.199007i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$			34.0000i			1.12462i
$915$		0			0
$916$	−8.00000			−0.264327
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	38.0000	+	38.0000i	1.25146	+	1.25146i
$923$		0			0
$924$		0			0
$925$	−49.0000	+	7.00000i	−1.61111	+	0.230159i
$926$		0			0
$927$		0			0
$928$	−16.0000	+	16.0000i	−0.525226	+	0.525226i
$929$		−	46.0000i		−	1.50921i	−0.656179	−	0.754606i	$-0.727828\pi$
$929$		−	46.0000i		−	1.50921i	0.656179	−	0.754606i	$-0.272172\pi$
$930$		0			0
$931$		0			0
$932$	−42.0000	−	42.0000i	−1.37576	−	1.37576i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	12.0000			0.392232
$937$	43.0000	+	43.0000i	1.40475	+	1.40475i	0.784046	+	0.620703i	$0.213153\pi$
$937$	43.0000	+	43.0000i	1.40475	+	1.40475i	0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−58.0000			−1.89075			−0.945373	−	0.325991i	$-0.894302\pi$
$941$	−58.0000			−1.89075			−0.945373	+	0.325991i	$0.894302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$		−	22.0000i		−	0.714150i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−41.0000	+	41.0000i	−1.32812	+	1.32812i	−0.421111	+	0.907009i	$0.638360\pi$
$953$	−41.0000	+	41.0000i	−1.32812	+	1.32812i	−0.907009	+	0.421111i	$0.861640\pi$
$954$			54.0000i			1.74831i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	−14.0000	−	14.0000i	−0.451378	−	0.451378i
$963$		0			0
$964$		−	16.0000i		−	0.515325i
$965$	−19.0000	+	57.0000i	−0.611632	+	1.83489i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	22.0000	−	22.0000i	0.707107	−	0.707107i
$969$		0			0
$970$	26.0000	+	52.0000i	0.834810	+	1.66962i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−48.0000			−1.53644
$977$	−27.0000	−	27.0000i	−0.863807	−	0.863807i	0.127971	−	0.991778i	$-0.459153\pi$
$977$	−27.0000	−	27.0000i	−0.863807	−	0.863807i	−0.991778	+	0.127971i	$0.959153\pi$
$978$		0			0
$979$		0			0
$980$	28.0000	−	14.0000i	0.894427	−	0.447214i
$981$	−18.0000			−0.574696
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	−39.0000	−	13.0000i	−1.24264	−	0.414214i
$986$	24.0000			0.764316
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−37.0000	−	37.0000i	−1.17180	−	1.17180i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−37.0000	−	37.0000i	−1.17180	−	1.17180i	−0.190022	−	0.981780i	$-0.560856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.2.e.a.7.1	yes	2
3.2	odd	2		180.2.k.c.127.1		2
4.3	odd	2	CM	20.2.e.a.7.1	yes	2
5.2	odd	4		100.2.e.b.43.1		2
5.3	odd	4	inner	20.2.e.a.3.1	✓	2
5.4	even	2		100.2.e.b.7.1		2
7.2	even	3		980.2.x.d.67.1		4
7.3	odd	6		980.2.x.c.667.1		4
7.4	even	3		980.2.x.d.667.1		4
7.5	odd	6		980.2.x.c.67.1		4
7.6	odd	2		980.2.k.a.687.1		2
8.3	odd	2		320.2.n.e.127.1		2
8.5	even	2		320.2.n.e.127.1		2
12.11	even	2		180.2.k.c.127.1		2
15.2	even	4		900.2.k.c.343.1		2
15.8	even	4		180.2.k.c.163.1		2
15.14	odd	2		900.2.k.c.307.1		2
16.3	odd	4		1280.2.o.j.127.1		2
16.5	even	4		1280.2.o.g.127.1		2
16.11	odd	4		1280.2.o.g.127.1		2
16.13	even	4		1280.2.o.j.127.1		2
20.3	even	4	inner	20.2.e.a.3.1	✓	2
20.7	even	4		100.2.e.b.43.1		2
20.19	odd	2		100.2.e.b.7.1		2
28.3	even	6		980.2.x.c.667.1		4
28.11	odd	6		980.2.x.d.667.1		4
28.19	even	6		980.2.x.c.67.1		4
28.23	odd	6		980.2.x.d.67.1		4
28.27	even	2		980.2.k.a.687.1		2
35.3	even	12		980.2.x.c.863.1		4
35.13	even	4		980.2.k.a.883.1		2
35.18	odd	12		980.2.x.d.863.1		4
35.23	odd	12		980.2.x.d.263.1		4
35.33	even	12		980.2.x.c.263.1		4
40.3	even	4		320.2.n.e.63.1		2
40.13	odd	4		320.2.n.e.63.1		2
40.19	odd	2		1600.2.n.h.1407.1		2
40.27	even	4		1600.2.n.h.1343.1		2
40.29	even	2		1600.2.n.h.1407.1		2
40.37	odd	4		1600.2.n.h.1343.1		2
60.23	odd	4		180.2.k.c.163.1		2
60.47	odd	4		900.2.k.c.343.1		2
60.59	even	2		900.2.k.c.307.1		2
80.3	even	4		1280.2.o.g.383.1		2
80.13	odd	4		1280.2.o.g.383.1		2
80.43	even	4		1280.2.o.j.383.1		2
80.53	odd	4		1280.2.o.j.383.1		2
140.3	odd	12		980.2.x.c.863.1		4
140.23	even	12		980.2.x.d.263.1		4
140.83	odd	4		980.2.k.a.883.1		2
140.103	odd	12		980.2.x.c.263.1		4
140.123	even	12		980.2.x.d.863.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.2.e.a.3.1	✓	2	5.3	odd	4	inner
20.2.e.a.3.1	✓	2	20.3	even	4	inner
20.2.e.a.7.1	yes	2	1.1	even	1	trivial
20.2.e.a.7.1	yes	2	4.3	odd	2	CM
100.2.e.b.7.1		2	5.4	even	2
100.2.e.b.7.1		2	20.19	odd	2
100.2.e.b.43.1		2	5.2	odd	4
100.2.e.b.43.1		2	20.7	even	4
180.2.k.c.127.1		2	3.2	odd	2
180.2.k.c.127.1		2	12.11	even	2
180.2.k.c.163.1		2	15.8	even	4
180.2.k.c.163.1		2	60.23	odd	4
320.2.n.e.63.1		2	40.3	even	4
320.2.n.e.63.1		2	40.13	odd	4
320.2.n.e.127.1		2	8.3	odd	2
320.2.n.e.127.1		2	8.5	even	2
900.2.k.c.307.1		2	15.14	odd	2
900.2.k.c.307.1		2	60.59	even	2
900.2.k.c.343.1		2	15.2	even	4
900.2.k.c.343.1		2	60.47	odd	4
980.2.k.a.687.1		2	7.6	odd	2
980.2.k.a.687.1		2	28.27	even	2
980.2.k.a.883.1		2	35.13	even	4
980.2.k.a.883.1		2	140.83	odd	4
980.2.x.c.67.1		4	7.5	odd	6
980.2.x.c.67.1		4	28.19	even	6
980.2.x.c.263.1		4	35.33	even	12
980.2.x.c.263.1		4	140.103	odd	12
980.2.x.c.667.1		4	7.3	odd	6
980.2.x.c.667.1		4	28.3	even	6
980.2.x.c.863.1		4	35.3	even	12
980.2.x.c.863.1		4	140.3	odd	12
980.2.x.d.67.1		4	7.2	even	3
980.2.x.d.67.1		4	28.23	odd	6
980.2.x.d.263.1		4	35.23	odd	12
980.2.x.d.263.1		4	140.23	even	12
980.2.x.d.667.1		4	7.4	even	3
980.2.x.d.667.1		4	28.11	odd	6
980.2.x.d.863.1		4	35.18	odd	12
980.2.x.d.863.1		4	140.123	even	12
1280.2.o.g.127.1		2	16.5	even	4
1280.2.o.g.127.1		2	16.11	odd	4
1280.2.o.g.383.1		2	80.3	even	4
1280.2.o.g.383.1		2	80.13	odd	4
1280.2.o.j.127.1		2	16.3	odd	4
1280.2.o.j.127.1		2	16.13	even	4
1280.2.o.j.383.1		2	80.43	even	4
1280.2.o.j.383.1		2	80.53	odd	4
1600.2.n.h.1343.1		2	40.27	even	4
1600.2.n.h.1343.1		2	40.37	odd	4
1600.2.n.h.1407.1		2	40.19	odd	2
1600.2.n.h.1407.1		2	40.29	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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(-2.00000 + 1.00000i) q^{5} +(2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +(3.00000 + 1.00000i) q^{10} +(-1.00000 + 1.00000i) q^{13} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{17} +(-3.00000 + 3.00000i) q^{18} +(-2.00000 - 4.00000i) q^{20} +(3.00000 - 4.00000i) q^{25} +2.00000 q^{26} +4.00000i q^{29} +(4.00000 + 4.00000i) q^{32} -6.00000i q^{34} +6.00000 q^{36} +(-7.00000 - 7.00000i) q^{37} +(-2.00000 + 6.00000i) q^{40} -8.00000 q^{41} +(3.00000 + 6.00000i) q^{45} +7.00000i q^{49} +(-7.00000 + 1.00000i) q^{50} +(-2.00000 - 2.00000i) q^{52} +(9.00000 - 9.00000i) q^{53} +(4.00000 - 4.00000i) q^{58} +12.0000 q^{61} -8.00000i q^{64} +(1.00000 - 3.00000i) q^{65} +(-6.00000 + 6.00000i) q^{68} +(-6.00000 - 6.00000i) q^{72} +(-11.0000 + 11.0000i) q^{73} +14.0000i q^{74} +(8.00000 - 4.00000i) q^{80} -9.00000 q^{81} +(8.00000 + 8.00000i) q^{82} +(-9.00000 - 3.00000i) q^{85} -16.0000i q^{89} +(3.00000 - 9.00000i) q^{90} +(13.0000 + 13.0000i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{5} + 4 q^{8} + 6 q^{10} - 2 q^{13} - 8 q^{16} + 6 q^{17} - 6 q^{18} - 4 q^{20} + 6 q^{25} + 4 q^{26} + 8 q^{32} + 12 q^{36} - 14 q^{37} - 4 q^{40} - 16 q^{41} + 6 q^{45} - 14 q^{50}+ \cdots + 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 + 6 * q^10 - 2 * q^13 - 8 * q^16 + 6 * q^17 - 6 * q^18 - 4 * q^20 + 6 * q^25 + 4 * q^26 + 8 * q^32 + 12 * q^36 - 14 * q^37 - 4 * q^40 - 16 * q^41 + 6 * q^45 - 14 * q^50 - 4 * q^52 + 18 * q^53 + 8 * q^58 + 24 * q^61 + 2 * q^65 - 12 * q^68 - 12 * q^72 - 22 * q^73 + 16 * q^80 - 18 * q^81 + 16 * q^82 - 18 * q^85 + 6 * q^90 + 26 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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