LMFDB - Embedded newform 1740.1.v.a.347.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1740,1,Mod(347,1740)] 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("1740.347"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1740, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1740 = 2^{2} \cdot 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1740.v (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$0.868373121981$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.3784500.2

Embedding invariants

Embedding label			347.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1740.347
Dual form			1740.1.v.a.1043.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.707107 - 0.707107i) q^{2} +(-0.707107 + 0.707107i) q^{3} -1.00000i q^{4} -1.00000 q^{5} +1.00000i q^{6} +(-0.707107 - 0.707107i) q^{8} -1.00000i q^{9} +(-0.707107 + 0.707107i) q^{10} +1.41421i q^{11} +(0.707107 + 0.707107i) q^{12} +(-1.00000 + 1.00000i) q^{13} +(0.707107 - 0.707107i) q^{15} -1.00000 q^{16} +(-0.707107 - 0.707107i) q^{18} +1.41421i q^{19} +1.00000i q^{20} +(1.00000 + 1.00000i) q^{22} +1.00000 q^{24} +1.00000 q^{25} +1.41421i q^{26} +(0.707107 + 0.707107i) q^{27} -1.00000 q^{29} -1.00000i q^{30} +1.41421 q^{31} +(-0.707107 + 0.707107i) q^{32} +(-1.00000 - 1.00000i) q^{33} -1.00000 q^{36} +(1.00000 + 1.00000i) q^{38} -1.41421i q^{39} +(0.707107 + 0.707107i) q^{40} +1.41421 q^{44} +1.00000i q^{45} +(-1.41421 + 1.41421i) q^{47} +(0.707107 - 0.707107i) q^{48} +1.00000i q^{49} +(0.707107 - 0.707107i) q^{50} +(1.00000 + 1.00000i) q^{52} +(-1.00000 - 1.00000i) q^{53} +1.00000 q^{54} -1.41421i q^{55} +(-1.00000 - 1.00000i) q^{57} +(-0.707107 + 0.707107i) q^{58} +(-0.707107 - 0.707107i) q^{60} +(1.00000 - 1.00000i) q^{62} +1.00000i q^{64} +(1.00000 - 1.00000i) q^{65} -1.41421 q^{66} +(-0.707107 + 0.707107i) q^{72} +(-0.707107 + 0.707107i) q^{75} +1.41421 q^{76} +(-1.00000 - 1.00000i) q^{78} +1.41421i q^{79} +1.00000 q^{80} -1.00000 q^{81} +(0.707107 - 0.707107i) q^{87} +(1.00000 - 1.00000i) q^{88} +(0.707107 + 0.707107i) q^{90} +(-1.00000 + 1.00000i) q^{93} +2.00000i q^{94} -1.41421i q^{95} -1.00000i q^{96} +(0.707107 + 0.707107i) q^{98} +1.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{13} - 4 q^{16} + 4 q^{22} + 4 q^{24} + 4 q^{25} - 4 q^{29} - 4 q^{33} - 4 q^{36} + 4 q^{38} + 4 q^{52} - 4 q^{53} + 4 q^{54} - 4 q^{57} + 4 q^{62} + 4 q^{65} - 4 q^{78} + 4 q^{80} - 4 q^{81}+ \cdots - 4 q^{93}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^13 - 4 * q^16 + 4 * q^22 + 4 * q^24 + 4 * q^25 - 4 * q^29 - 4 * q^33 - 4 * q^36 + 4 * q^38 + 4 * q^52 - 4 * q^53 + 4 * q^54 - 4 * q^57 + 4 * q^62 + 4 * q^65 - 4 * q^78 + 4 * q^80 - 4 * q^81 + 4 * q^88 - 4 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1740\mathbb{Z}\right)^\times$.

$n$	$581$	$697$	$871$	$901$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0.707107	−	0.707107i	0.707107	−	0.707107i
$2$	0.707107	−	0.707107i	0.707107	−	0.707107i
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$4$		−	1.00000i		−	1.00000i
$5$	−1.00000			−1.00000
$5$	−1.00000			−1.00000
$6$			1.00000i			1.00000i
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$9$		−	1.00000i		−	1.00000i
$10$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$11$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$11$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$12$	0.707107	+	0.707107i	0.707107	+	0.707107i
$13$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$13$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$14$		0			0
$15$	0.707107	−	0.707107i	0.707107	−	0.707107i
$16$	−1.00000			−1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$19$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$19$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$20$			1.00000i			1.00000i
$21$		0			0
$22$	1.00000	+	1.00000i	1.00000	+	1.00000i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$	1.00000			1.00000
$25$	1.00000			1.00000
$26$			1.41421i			1.41421i
$27$	0.707107	+	0.707107i	0.707107	+	0.707107i
$28$		0			0
$29$	−1.00000			−1.00000
$29$	−1.00000			−1.00000
$30$		−	1.00000i		−	1.00000i
$31$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$31$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$32$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$33$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$34$		0			0
$35$		0			0
$36$	−1.00000			−1.00000
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	1.00000	+	1.00000i	1.00000	+	1.00000i
$39$		−	1.41421i		−	1.41421i
$40$	0.707107	+	0.707107i	0.707107	+	0.707107i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$	1.41421			1.41421
$45$			1.00000i			1.00000i
$46$		0			0
$47$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$47$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$48$	0.707107	−	0.707107i	0.707107	−	0.707107i
$49$			1.00000i			1.00000i
$50$	0.707107	−	0.707107i	0.707107	−	0.707107i
$51$		0			0
$52$	1.00000	+	1.00000i	1.00000	+	1.00000i
$53$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$53$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$54$	1.00000			1.00000
$55$		−	1.41421i		−	1.41421i
$56$		0			0
$57$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$58$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	1.00000	−	1.00000i	1.00000	−	1.00000i
$63$		0			0
$64$			1.00000i			1.00000i
$65$	1.00000	−	1.00000i	1.00000	−	1.00000i
$66$	−1.41421			−1.41421
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$76$	1.41421			1.41421
$77$		0			0
$78$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$79$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$79$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$80$	1.00000			1.00000
$81$	−1.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.707107	−	0.707107i	0.707107	−	0.707107i
$88$	1.00000	−	1.00000i	1.00000	−	1.00000i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	0.707107	+	0.707107i	0.707107	+	0.707107i
$91$		0			0
$92$		0			0
$93$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$94$			2.00000i			2.00000i
$95$		−	1.41421i		−	1.41421i
$96$		−	1.00000i		−	1.00000i
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	0.707107	+	0.707107i	0.707107	+	0.707107i
$99$	1.41421			1.41421
$100$		−	1.00000i		−	1.00000i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$	1.41421			1.41421
$105$		0			0
$106$	−1.41421			−1.41421
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	0.707107	−	0.707107i	0.707107	−	0.707107i
$109$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$109$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$110$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$	−1.41421			−1.41421
$115$		0			0
$116$			1.00000i			1.00000i
$117$	1.00000	+	1.00000i	1.00000	+	1.00000i
$118$		0			0
$119$		0			0
$120$	−1.00000			−1.00000
$121$	−1.00000			−1.00000
$122$		0			0
$123$		0			0
$124$		−	1.41421i		−	1.41421i
$125$	−1.00000			−1.00000
$126$		0			0
$127$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$127$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$128$	0.707107	+	0.707107i	0.707107	+	0.707107i
$129$		0			0
$130$		−	1.41421i		−	1.41421i
$131$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$131$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$132$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$133$		0			0
$134$		0			0
$135$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		−	2.00000i		−	2.00000i
$142$		0			0
$143$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$144$			1.00000i			1.00000i
$145$	1.00000			1.00000
$146$		0			0
$147$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$148$		0			0
$149$	2.00000			2.00000			1.00000			$0$
$149$	2.00000			2.00000			1.00000			$0$
$150$			1.00000i			1.00000i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	1.00000	−	1.00000i	1.00000	−	1.00000i
$153$		0			0
$154$		0			0
$155$	−1.41421			−1.41421
$156$	−1.41421			−1.41421
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$	1.00000	+	1.00000i	1.00000	+	1.00000i
$159$	1.41421			1.41421
$160$	0.707107	−	0.707107i	0.707107	−	0.707107i
$161$		0			0
$162$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$	1.00000	+	1.00000i	1.00000	+	1.00000i
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$		−	1.00000i		−	1.00000i
$170$		0			0
$171$	1.41421			1.41421
$172$		0			0
$173$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$173$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$174$		−	1.00000i		−	1.00000i
$175$		0			0
$176$		−	1.41421i		−	1.41421i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	1.00000			1.00000
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$			1.41421i			1.41421i
$187$		0			0
$188$	1.41421	+	1.41421i	1.41421	+	1.41421i
$189$		0			0
$190$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$191$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$191$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$192$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$			1.41421i			1.41421i
$196$	1.00000			1.00000
$197$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$197$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$198$	1.00000	−	1.00000i	1.00000	−	1.00000i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.00000	−	1.00000i	1.00000	−	1.00000i
$209$	−2.00000			−2.00000
$210$		0			0
$211$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$211$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$212$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		−	1.00000i		−	1.00000i
$217$		0			0
$218$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$219$		0			0
$220$	−1.41421			−1.41421
$221$		0			0
$222$		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
$225$		−	1.00000i		−	1.00000i
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	0.707107	+	0.707107i	0.707107	+	0.707107i
$233$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$233$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$234$	1.41421			1.41421
$235$	1.41421	−	1.41421i	1.41421	−	1.41421i
$236$		0			0
$237$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$241$	2.00000			2.00000			1.00000			$0$
$241$	2.00000			2.00000			1.00000			$0$
$242$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$243$	0.707107	−	0.707107i	0.707107	−	0.707107i
$244$		0			0
$245$		−	1.00000i		−	1.00000i
$246$		0			0
$247$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$248$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$249$		0			0
$250$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$251$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$251$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$252$		0			0
$253$		0			0
$254$	−2.00000			−2.00000
$255$		0			0
$256$	1.00000			1.00000
$257$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$257$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$261$			1.00000i			1.00000i
$262$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$			1.41421i			1.41421i
$265$	1.00000	+	1.00000i	1.00000	+	1.00000i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	−1.00000			−1.00000
$271$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$271$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.41421i			1.41421i
$276$		0			0
$277$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$277$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$278$		0			0
$279$		−	1.41421i		−	1.41421i
$280$		0			0
$281$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$281$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$282$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$	1.00000	+	1.00000i	1.00000	+	1.00000i
$286$	−2.00000			−2.00000
$287$		0			0
$288$	0.707107	+	0.707107i	0.707107	+	0.707107i
$289$		−	1.00000i		−	1.00000i
$290$	0.707107	−	0.707107i	0.707107	−	0.707107i
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$	−1.00000			−1.00000
$295$		0			0
$296$		0			0
$297$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$298$	1.41421	−	1.41421i	1.41421	−	1.41421i
$299$		0			0
$300$	0.707107	+	0.707107i	0.707107	+	0.707107i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	1.41421i		−	1.41421i
$305$		0			0
$306$		0			0
$307$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$307$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$311$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$311$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$312$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$313$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$313$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$	1.41421			1.41421
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$	1.00000	−	1.00000i	1.00000	−	1.00000i
$319$		−	1.41421i		−	1.41421i
$320$		−	1.00000i		−	1.00000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			1.00000i			1.00000i
$325$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$326$		0			0
$327$	1.41421	+	1.41421i	1.41421	+	1.41421i
$328$		0			0
$329$		0			0
$330$	1.41421			1.41421
$331$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$331$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$339$		0			0
$340$		0			0
$341$			2.00000i			2.00000i
$342$	1.00000	−	1.00000i	1.00000	−	1.00000i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.41421			1.41421
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	−1.41421			−1.41421
$352$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$353$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$353$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$359$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$360$	0.707107	−	0.707107i	0.707107	−	0.707107i
$361$	−1.00000			−1.00000
$362$		0			0
$363$	0.707107	−	0.707107i	0.707107	−	0.707107i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.00000	+	1.00000i	1.00000	+	1.00000i
$373$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$373$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$374$		0			0
$375$	0.707107	−	0.707107i	0.707107	−	0.707107i
$376$	2.00000			2.00000
$377$	1.00000	−	1.00000i	1.00000	−	1.00000i
$378$		0			0
$379$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$379$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$380$	−1.41421			−1.41421
$381$	2.00000			2.00000
$382$	1.00000	+	1.00000i	1.00000	+	1.00000i
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$	−1.00000			−1.00000
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	1.00000	+	1.00000i	1.00000	+	1.00000i
$391$		0			0
$392$	0.707107	−	0.707107i	0.707107	−	0.707107i
$393$	1.00000	+	1.00000i	1.00000	+	1.00000i
$394$		−	1.41421i		−	1.41421i
$395$		−	1.41421i		−	1.41421i
$396$		−	1.41421i		−	1.41421i
$397$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$397$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−1.00000			−1.00000
$401$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$401$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$402$		0			0
$403$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$404$		0			0
$405$	1.00000			1.00000
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		−	1.41421i		−	1.41421i
$417$		0			0
$418$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$423$	1.41421	+	1.41421i	1.41421	+	1.41421i
$424$			1.41421i			1.41421i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	2.00000			2.00000
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$436$	−2.00000			−2.00000
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$			1.41421i			1.41421i
$457$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$457$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	1.00000			1.00000
$465$	1.00000	−	1.00000i	1.00000	−	1.00000i
$466$	1.41421			1.41421
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$	1.00000	−	1.00000i	1.00000	−	1.00000i
$469$		0			0
$470$		−	2.00000i		−	2.00000i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−1.41421			−1.41421
$475$			1.41421i			1.41421i
$476$		0			0
$477$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$478$		0			0
$479$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$479$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$480$			1.00000i			1.00000i
$481$		0			0
$482$	1.41421	−	1.41421i	1.41421	−	1.41421i
$483$		0			0
$484$			1.00000i			1.00000i
$485$		0			0
$486$		−	1.00000i		−	1.00000i
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$		0			0
$490$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$491$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$491$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$		0			0
$494$	−2.00000			−2.00000
$495$	−1.41421			−1.41421
$496$	−1.41421			−1.41421
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$			1.00000i			1.00000i
$501$		0			0
$502$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.707107	+	0.707107i	0.707107	+	0.707107i
$508$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	0.707107	−	0.707107i	0.707107	−	0.707107i
$513$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$514$			1.41421i			1.41421i
$515$		0			0
$516$		0			0
$517$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$518$		0			0
$519$	−1.41421			−1.41421
$520$	−1.41421			−1.41421
$521$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$521$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$522$	0.707107	+	0.707107i	0.707107	+	0.707107i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$	−1.41421			−1.41421
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.00000	+	1.00000i	1.00000	+	1.00000i
$529$			1.00000i			1.00000i
$530$	1.41421			1.41421
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.41421			−1.41421
$540$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$543$		0			0
$544$		0			0
$545$			2.00000i			2.00000i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		0			0
$550$	1.00000	+	1.00000i	1.00000	+	1.00000i
$551$		−	1.41421i		−	1.41421i
$552$		0			0
$553$		0			0
$554$	1.41421			1.41421
$555$		0			0
$556$		0			0
$557$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$557$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$558$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.41421	+	1.41421i	1.41421	+	1.41421i
$563$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$563$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$564$	−2.00000			−2.00000
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	1.41421			1.41421
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$573$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$574$		0			0
$575$		0			0
$576$	1.00000			1.00000
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$579$		0			0
$580$		−	1.00000i		−	1.00000i
$581$		0			0
$582$		0			0
$583$	1.41421	−	1.41421i	1.41421	−	1.41421i
$584$		0			0
$585$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$589$			2.00000i			2.00000i
$590$		0			0
$591$			1.41421i			1.41421i
$592$		0			0
$593$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$593$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$594$			1.41421i			1.41421i
$595$		0			0
$596$		−	2.00000i		−	2.00000i
$597$		0			0
$598$		0			0
$599$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$599$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$600$	1.00000			1.00000
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.00000			1.00000
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$609$		0			0
$610$		0			0
$611$		−	2.82843i		−	2.82843i
$612$		0			0
$613$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$613$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$614$	2.00000			2.00000
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$619$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$620$			1.41421i			1.41421i
$621$		0			0
$622$	1.00000	+	1.00000i	1.00000	+	1.00000i
$623$		0			0
$624$			1.41421i			1.41421i
$625$	1.00000			1.00000
$626$			1.41421i			1.41421i
$627$	1.41421	−	1.41421i	1.41421	−	1.41421i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	1.00000	−	1.00000i	1.00000	−	1.00000i
$633$	1.00000	−	1.00000i	1.00000	−	1.00000i
$634$		0			0
$635$	1.41421	+	1.41421i	1.41421	+	1.41421i
$636$		−	1.41421i		−	1.41421i
$637$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$638$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$639$		0			0
$640$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		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	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	0.707107	+	0.707107i	0.707107	+	0.707107i
$649$		0			0
$650$			1.41421i			1.41421i
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$	2.00000			2.00000
$655$			1.41421i			1.41421i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$659$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$660$	1.00000	−	1.00000i	1.00000	−	1.00000i
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	1.00000	−	1.00000i	1.00000	−	1.00000i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$673$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$674$		0			0
$675$	0.707107	+	0.707107i	0.707107	+	0.707107i
$676$	−1.00000			−1.00000
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	1.41421	+	1.41421i	1.41421	+	1.41421i
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		−	1.41421i		−	1.41421i
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.00000			2.00000
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	1.00000	−	1.00000i	1.00000	−	1.00000i
$693$		0			0
$694$		0			0
$695$		0			0
$696$	−1.00000			−1.00000
$697$		0			0
$698$		0			0
$699$	−1.41421			−1.41421
$700$		0			0
$701$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$701$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$702$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$703$		0			0
$704$	−1.41421			−1.41421
$705$			2.00000i			2.00000i
$706$	1.41421			1.41421
$707$		0			0
$708$		0			0
$709$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$709$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$710$		0			0
$711$	1.41421			1.41421
$712$		0			0
$713$		0			0
$714$		0			0
$715$	1.41421	+	1.41421i	1.41421	+	1.41421i
$716$		0			0
$717$		0			0
$718$	1.00000	−	1.00000i	1.00000	−	1.00000i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		−	1.00000i		−	1.00000i
$721$		0			0
$722$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$723$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$724$		0			0
$725$	−1.00000			−1.00000
$726$		−	1.00000i		−	1.00000i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$	0.707107	+	0.707107i	0.707107	+	0.707107i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$739$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$740$		0			0
$741$	2.00000			2.00000
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$	1.41421			1.41421
$745$	−2.00000			−2.00000
$746$			1.41421i			1.41421i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		−	1.00000i		−	1.00000i
$751$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$751$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$752$	1.41421	−	1.41421i	1.41421	−	1.41421i
$753$	1.00000	+	1.00000i	1.00000	+	1.00000i
$754$		−	1.41421i		−	1.41421i
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	1.00000	+	1.00000i	1.00000	+	1.00000i
$759$		0			0
$760$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	1.41421	−	1.41421i	1.41421	−	1.41421i
$763$		0			0
$764$	1.41421			1.41421
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		−	1.41421i		−	1.41421i
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$	1.41421			1.41421
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	1.41421			1.41421
$781$		0			0
$782$		0			0
$783$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$784$		−	1.00000i		−	1.00000i
$785$		0			0
$786$	1.41421			1.41421
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$789$		0			0
$790$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$791$		0			0
$792$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$793$		0			0
$794$	−1.41421			−1.41421
$795$	−1.41421			−1.41421
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$801$		0			0
$802$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$803$		0			0
$804$		0			0
$805$		0			0
$806$			2.00000i			2.00000i
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	0.707107	−	0.707107i	0.707107	−	0.707107i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	1.00000	−	1.00000i	1.00000	−	1.00000i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$823$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$824$		0			0
$825$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$826$		0			0
$827$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$827$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−1.41421			−1.41421
$832$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$833$		0			0
$834$		0			0
$835$		0			0
$836$			2.00000i			2.00000i
$837$	1.00000	+	1.00000i	1.00000	+	1.00000i
$838$		0			0
$839$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$839$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$844$			1.41421i			1.41421i
$845$			1.00000i			1.00000i
$846$	2.00000			2.00000
$847$		0			0
$848$	1.00000	+	1.00000i	1.00000	+	1.00000i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$	−1.41421			−1.41421
$856$		0			0
$857$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$857$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$858$	1.41421	−	1.41421i	1.41421	−	1.41421i
$859$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$859$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$	−1.00000			−1.00000
$865$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$866$		0			0
$867$	0.707107	+	0.707107i	0.707107	+	0.707107i
$868$		0			0
$869$	−2.00000			−2.00000
$870$			1.00000i			1.00000i
$871$		0			0
$872$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$877$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$			1.41421i			1.41421i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	0.707107	−	0.707107i	0.707107	−	0.707107i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		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
$890$		0			0
$891$		−	1.41421i		−	1.41421i
$892$		0			0
$893$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$894$			2.00000i			2.00000i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.41421			−1.41421
$900$	−1.00000			−1.00000
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$911$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$912$	1.00000	+	1.00000i	1.00000	+	1.00000i
$913$		0			0
$914$	−1.41421			−1.41421
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−2.00000			−2.00000
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	0.707107	−	0.707107i	0.707107	−	0.707107i
$929$	−2.00000			−2.00000			−1.00000			$\pi$
$929$	−2.00000			−2.00000			−1.00000			$\pi$
$930$		−	1.41421i		−	1.41421i
$931$	−1.41421			−1.41421
$932$	1.00000	−	1.00000i	1.00000	−	1.00000i
$933$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$934$		0			0
$935$		0			0
$936$		−	1.41421i		−	1.41421i
$937$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$937$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$938$		0			0
$939$		−	1.41421i		−	1.41421i
$940$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$949$		0			0
$950$	1.00000	+	1.00000i	1.00000	+	1.00000i
$951$		0			0
$952$		0			0
$953$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$953$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$954$			1.41421i			1.41421i
$955$		−	1.41421i		−	1.41421i
$956$		0			0
$957$	1.00000	+	1.00000i	1.00000	+	1.00000i
$958$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$959$		0			0
$960$	0.707107	+	0.707107i	0.707107	+	0.707107i
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$		−	2.00000i		−	2.00000i
$965$		0			0
$966$		0			0
$967$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$967$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$968$	0.707107	+	0.707107i	0.707107	+	0.707107i
$969$		0			0
$970$		0			0
$971$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$971$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$972$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$973$		0			0
$974$		0			0
$975$		−	1.41421i		−	1.41421i
$976$		0			0
$977$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$977$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$978$		0			0
$979$		0			0
$980$	−1.00000			−1.00000
$981$	−2.00000			−2.00000
$982$	1.00000	+	1.00000i	1.00000	+	1.00000i
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$986$		0			0
$987$		0			0
$988$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$989$		0			0
$990$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$993$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1740.1.v.a.347.2	yes	4
3.2	odd	2		1740.1.v.b.347.1	yes	4
4.3	odd	2	inner	1740.1.v.a.347.1	✓	4
5.3	odd	4		1740.1.v.b.1043.1	yes	4
12.11	even	2		1740.1.v.b.347.2	yes	4
15.8	even	4	inner	1740.1.v.a.1043.2	yes	4
20.3	even	4		1740.1.v.b.1043.2	yes	4
29.28	even	2	inner	1740.1.v.a.347.1	✓	4
60.23	odd	4	inner	1740.1.v.a.1043.1	yes	4
87.86	odd	2		1740.1.v.b.347.2	yes	4
116.115	odd	2	CM	1740.1.v.a.347.2	yes	4
145.28	odd	4		1740.1.v.b.1043.2	yes	4
348.347	even	2		1740.1.v.b.347.1	yes	4
435.173	even	4	inner	1740.1.v.a.1043.1	yes	4
580.463	even	4		1740.1.v.b.1043.1	yes	4
1740.1043	odd	4	inner	1740.1.v.a.1043.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1740.1.v.a.347.1	✓	4	4.3	odd	2	inner
1740.1.v.a.347.1	✓	4	29.28	even	2	inner
1740.1.v.a.347.2	yes	4	1.1	even	1	trivial
1740.1.v.a.347.2	yes	4	116.115	odd	2	CM
1740.1.v.a.1043.1	yes	4	60.23	odd	4	inner
1740.1.v.a.1043.1	yes	4	435.173	even	4	inner
1740.1.v.a.1043.2	yes	4	15.8	even	4	inner
1740.1.v.a.1043.2	yes	4	1740.1043	odd	4	inner
1740.1.v.b.347.1	yes	4	3.2	odd	2
1740.1.v.b.347.1	yes	4	348.347	even	2
1740.1.v.b.347.2	yes	4	12.11	even	2
1740.1.v.b.347.2	yes	4	87.86	odd	2
1740.1.v.b.1043.1	yes	4	5.3	odd	4
1740.1.v.b.1043.1	yes	4	580.463	even	4
1740.1.v.b.1043.2	yes	4	20.3	even	4
1740.1.v.b.1043.2	yes	4	145.28	odd	4

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:	\( N \)	\(=\)	\( 1740 = 2^{2} \cdot 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1740.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{2} +(-0.707107 + 0.707107i) q^{3} -1.00000i q^{4} -1.00000 q^{5} +1.00000i q^{6} +(-0.707107 - 0.707107i) q^{8} -1.00000i q^{9} +(-0.707107 + 0.707107i) q^{10} +1.41421i q^{11} +(0.707107 + 0.707107i) q^{12} +(-1.00000 + 1.00000i) q^{13} +(0.707107 - 0.707107i) q^{15} -1.00000 q^{16} +(-0.707107 - 0.707107i) q^{18} +1.41421i q^{19} +1.00000i q^{20} +(1.00000 + 1.00000i) q^{22} +1.00000 q^{24} +1.00000 q^{25} +1.41421i q^{26} +(0.707107 + 0.707107i) q^{27} -1.00000 q^{29} -1.00000i q^{30} +1.41421 q^{31} +(-0.707107 + 0.707107i) q^{32} +(-1.00000 - 1.00000i) q^{33} -1.00000 q^{36} +(1.00000 + 1.00000i) q^{38} -1.41421i q^{39} +(0.707107 + 0.707107i) q^{40} +1.41421 q^{44} +1.00000i q^{45} +(-1.41421 + 1.41421i) q^{47} +(0.707107 - 0.707107i) q^{48} +1.00000i q^{49} +(0.707107 - 0.707107i) q^{50} +(1.00000 + 1.00000i) q^{52} +(-1.00000 - 1.00000i) q^{53} +1.00000 q^{54} -1.41421i q^{55} +(-1.00000 - 1.00000i) q^{57} +(-0.707107 + 0.707107i) q^{58} +(-0.707107 - 0.707107i) q^{60} +(1.00000 - 1.00000i) q^{62} +1.00000i q^{64} +(1.00000 - 1.00000i) q^{65} -1.41421 q^{66} +(-0.707107 + 0.707107i) q^{72} +(-0.707107 + 0.707107i) q^{75} +1.41421 q^{76} +(-1.00000 - 1.00000i) q^{78} +1.41421i q^{79} +1.00000 q^{80} -1.00000 q^{81} +(0.707107 - 0.707107i) q^{87} +(1.00000 - 1.00000i) q^{88} +(0.707107 + 0.707107i) q^{90} +(-1.00000 + 1.00000i) q^{93} +2.00000i q^{94} -1.41421i q^{95} -1.00000i q^{96} +(0.707107 + 0.707107i) q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{13} - 4 q^{16} + 4 q^{22} + 4 q^{24} + 4 q^{25} - 4 q^{29} - 4 q^{33} - 4 q^{36} + 4 q^{38} + 4 q^{52} - 4 q^{53} + 4 q^{54} - 4 q^{57} + 4 q^{62} + 4 q^{65} - 4 q^{78} + 4 q^{80} - 4 q^{81}+ \cdots - 4 q^{93}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^13 - 4 * q^16 + 4 * q^22 + 4 * q^24 + 4 * q^25 - 4 * q^29 - 4 * q^33 - 4 * q^36 + 4 * q^38 + 4 * q^52 - 4 * q^53 + 4 * q^54 - 4 * q^57 + 4 * q^62 + 4 * q^65 - 4 * q^78 + 4 * q^80 - 4 * q^81 + 4 * q^88 - 4 * q^93

Introduction

L-functions

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