LMFDB - Embedded newform 2940.1.o.d.2939.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,1,Mod(2939,2940)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2940.2939"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2940.o (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,8,0,0] 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:	$1.46725113714$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.2.2134623456000.5

Embedding invariants

Embedding label			2939.5
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	2940.2939
Dual form			2940.1.o.d.2939.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.382683 - 0.923880i) q^{2} +1.00000 q^{3} +(-0.707107 - 0.707107i) q^{4} +1.00000i q^{5} +(0.382683 - 0.923880i) q^{6} +(-0.923880 + 0.382683i) q^{8} +1.00000 q^{9} +(0.923880 + 0.382683i) q^{10} +(-0.707107 - 0.707107i) q^{12} +1.00000i q^{15} +1.00000i q^{16} +1.41421i q^{17} +(0.382683 - 0.923880i) q^{18} +1.84776 q^{19} +(0.707107 - 0.707107i) q^{20} -0.765367i q^{23} +(-0.923880 + 0.382683i) q^{24} -1.00000 q^{25} +1.00000 q^{27} +(0.923880 + 0.382683i) q^{30} -0.765367 q^{31} +(0.923880 + 0.382683i) q^{32} +(1.30656 + 0.541196i) q^{34} +(-0.707107 - 0.707107i) q^{36} +(0.707107 - 1.70711i) q^{38} +(-0.382683 - 0.923880i) q^{40} +1.00000i q^{45} +(-0.707107 - 0.292893i) q^{46} +1.41421 q^{47} +1.00000i q^{48} +(-0.382683 + 0.923880i) q^{50} +1.41421i q^{51} -1.84776 q^{53} +(0.382683 - 0.923880i) q^{54} +1.84776 q^{57} +(0.707107 - 0.707107i) q^{60} -1.84776i q^{61} +(-0.292893 + 0.707107i) q^{62} +(0.707107 - 0.707107i) q^{64} +(1.00000 - 1.00000i) q^{68} -0.765367i q^{69} +(-0.923880 + 0.382683i) q^{72} -1.00000 q^{75} +(-1.30656 - 1.30656i) q^{76} -1.41421i q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.41421 q^{83} -1.41421 q^{85} +(0.923880 + 0.382683i) q^{90} +(-0.541196 + 0.541196i) q^{92} -0.765367 q^{93} +(0.541196 - 1.30656i) q^{94} +1.84776i q^{95} +(0.923880 + 0.382683i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9} - 8 q^{25} + 8 q^{27} - 8 q^{62} + 8 q^{68} - 8 q^{75} - 8 q^{80} + 8 q^{81}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 - 8 * q^25 + 8 * q^27 - 8 * q^62 + 8 * q^68 - 8 * q^75 - 8 * q^80 + 8 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times$.

$n$	$1081$	$1177$	$1471$	$1961$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0.382683	−	0.923880i	0.382683	−	0.923880i
$2$	0.382683	−	0.923880i	0.382683	−	0.923880i
$3$	1.00000			1.00000
$3$	1.00000			1.00000
$4$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$5$			1.00000i			1.00000i
$5$			1.00000i			1.00000i
$6$	0.382683	−	0.923880i	0.382683	−	0.923880i
$7$		0			0
$7$		0			0
$8$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$9$	1.00000			1.00000
$10$	0.923880	+	0.382683i	0.923880	+	0.382683i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$			1.00000i			1.00000i
$16$			1.00000i			1.00000i
$17$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$17$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$18$	0.382683	−	0.923880i	0.382683	−	0.923880i
$19$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$19$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$20$	0.707107	−	0.707107i	0.707107	−	0.707107i
$21$		0			0
$22$		0			0
$23$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$23$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$24$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$25$	−1.00000			−1.00000
$26$		0			0
$27$	1.00000			1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	0.923880	+	0.382683i	0.923880	+	0.382683i
$31$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$31$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$32$	0.923880	+	0.382683i	0.923880	+	0.382683i
$33$		0			0
$34$	1.30656	+	0.541196i	1.30656	+	0.541196i
$35$		0			0
$36$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	0.707107	−	1.70711i	0.707107	−	1.70711i
$39$		0			0
$40$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$			1.00000i			1.00000i
$46$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$47$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$47$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$48$			1.00000i			1.00000i
$49$		0			0
$50$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$51$			1.41421i			1.41421i
$52$		0			0
$53$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$53$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$54$	0.382683	−	0.923880i	0.382683	−	0.923880i
$55$		0			0
$56$		0			0
$57$	1.84776			1.84776
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$	0.707107	−	0.707107i	0.707107	−	0.707107i
$61$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$61$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$62$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$63$		0			0
$64$	0.707107	−	0.707107i	0.707107	−	0.707107i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	1.00000	−	1.00000i	1.00000	−	1.00000i
$69$		−	0.765367i		−	0.765367i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	−1.00000			−1.00000
$76$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$77$		0			0
$78$		0			0
$79$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$79$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$80$	−1.00000			−1.00000
$81$	1.00000			1.00000
$82$		0			0
$83$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$83$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	−1.41421			−1.41421
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	0.923880	+	0.382683i	0.923880	+	0.382683i
$91$		0			0
$92$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$93$	−0.765367			−0.765367
$94$	0.541196	−	1.30656i	0.541196	−	1.30656i
$95$			1.84776i			1.84776i
$96$	0.923880	+	0.382683i	0.923880	+	0.382683i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$	0.707107	+	0.707107i	0.707107	+	0.707107i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	1.30656	+	0.541196i	1.30656	+	0.541196i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$107$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$107$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$108$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$113$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$114$	0.707107	−	1.70711i	0.707107	−	1.70711i
$115$	0.765367			0.765367
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$121$	−1.00000			−1.00000
$122$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$123$		0			0
$124$	0.541196	+	0.541196i	0.541196	+	0.541196i
$125$		−	1.00000i		−	1.00000i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$			1.00000i			1.00000i
$136$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$137$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$137$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$138$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$139$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$139$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$140$		0			0
$141$	1.41421			1.41421
$142$		0			0
$143$		0			0
$144$			1.00000i			1.00000i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$151$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$151$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$152$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$153$			1.41421i			1.41421i
$154$		0			0
$155$		−	0.765367i		−	0.765367i
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$159$	−1.84776			−1.84776
$160$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$161$		0			0
$162$	0.382683	−	0.923880i	0.382683	−	0.923880i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$171$	1.84776			1.84776
$172$		0			0
$173$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$173$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	0.707107	−	0.707107i	0.707107	−	0.707107i
$181$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$181$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$182$		0			0
$183$		−	1.84776i		−	1.84776i
$184$	0.292893	+	0.707107i	0.292893	+	0.707107i
$185$		0			0
$186$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$187$		0			0
$188$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$189$		0			0
$190$	1.70711	+	0.707107i	1.70711	+	0.707107i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	0.707107	−	0.707107i	0.707107	−	0.707107i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$197$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$198$		0			0
$199$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$199$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$200$	0.923880	−	0.382683i	0.923880	−	0.382683i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	1.00000	−	1.00000i	1.00000	−	1.00000i
$205$		0			0
$206$		0			0
$207$		−	0.765367i		−	0.765367i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$211$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$212$	1.30656	+	1.30656i	1.30656	+	1.30656i
$213$		0			0
$214$	1.70711	+	0.707107i	1.70711	+	0.707107i
$215$		0			0
$216$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−1.00000			−1.00000
$226$	0.292893	−	0.707107i	0.292893	−	0.707107i
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$229$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$229$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$230$	0.292893	−	0.707107i	0.292893	−	0.707107i
$231$		0			0
$232$		0			0
$233$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$233$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$234$		0			0
$235$			1.41421i			1.41421i
$236$		0			0
$237$		−	1.41421i		−	1.41421i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−1.00000			−1.00000
$241$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$241$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$242$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$243$	1.00000			1.00000
$244$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.707107	−	0.292893i	0.707107	−	0.292893i
$249$	−1.41421			−1.41421
$250$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−1.41421			−1.41421
$256$	−1.00000			−1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$263$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$264$		0			0
$265$		−	1.84776i		−	1.84776i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	0.923880	+	0.382683i	0.923880	+	0.382683i
$271$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$271$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$272$	−1.41421			−1.41421
$273$		0			0
$274$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$275$		0			0
$276$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	0.292893	−	0.707107i	0.292893	−	0.707107i
$279$	−0.765367			−0.765367
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	0.541196	−	1.30656i	0.541196	−	1.30656i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$			1.84776i			1.84776i
$286$		0			0
$287$		0			0
$288$	0.923880	+	0.382683i	0.923880	+	0.382683i
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.707107	+	0.707107i	0.707107	+	0.707107i
$301$		0			0
$302$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$303$		0			0
$304$			1.84776i			1.84776i
$305$	1.84776			1.84776
$306$	1.30656	+	0.541196i	1.30656	+	0.541196i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$317$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$317$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$318$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$319$		0			0
$320$	0.707107	+	0.707107i	0.707107	+	0.707107i
$321$			1.84776i			1.84776i
$322$		0			0
$323$			2.61313i			2.61313i
$324$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$331$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$332$	1.00000	+	1.00000i	1.00000	+	1.00000i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$339$	0.765367			0.765367
$340$	1.00000	+	1.00000i	1.00000	+	1.00000i
$341$		0			0
$342$	0.707107	−	1.70711i	0.707107	−	1.70711i
$343$		0			0
$344$		0			0
$345$	0.765367			0.765367
$346$	1.30656	+	0.541196i	1.30656	+	0.541196i
$347$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$347$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$348$		0			0
$349$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$349$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$361$	2.41421			2.41421
$362$	0.707107	+	0.292893i	0.707107	+	0.292893i
$363$	−1.00000			−1.00000
$364$		0			0
$365$		0			0
$366$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	0.765367			0.765367
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.541196	+	0.541196i	0.541196	+	0.541196i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		−	1.00000i		−	1.00000i
$376$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$377$		0			0
$378$		0			0
$379$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$379$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$380$	1.30656	−	1.30656i	1.30656	−	1.30656i
$381$		0			0
$382$		0			0
$383$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$383$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$384$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	1.08239			1.08239
$392$		0			0
$393$		0			0
$394$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$395$	1.41421			1.41421
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$399$		0			0
$400$		−	1.00000i		−	1.00000i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$			1.00000i			1.00000i
$406$		0			0
$407$		0			0
$408$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$409$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$409$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$410$		0			0
$411$	−0.765367			−0.765367
$412$		0			0
$413$		0			0
$414$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$415$		−	1.41421i		−	1.41421i
$416$		0			0
$417$	0.765367			0.765367
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$421$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$422$	−1.84776	−	0.765367i	−1.84776	−	0.765367i
$423$	1.41421			1.41421
$424$	1.70711	−	0.707107i	1.70711	−	0.707107i
$425$		−	1.41421i		−	1.41421i
$426$		0			0
$427$		0			0
$428$	1.30656	−	1.30656i	1.30656	−	1.30656i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$			1.00000i			1.00000i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.41421i		−	1.41421i
$438$		0			0
$439$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$439$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$443$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$451$		0			0
$452$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$453$		−	1.41421i		−	1.41421i
$454$		0			0
$455$		0			0
$456$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	1.70711	+	0.707107i	1.70711	+	0.707107i
$459$			1.41421i			1.41421i
$460$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		−	0.765367i		−	0.765367i
$466$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$	1.30656	+	0.541196i	1.30656	+	0.541196i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$475$	−1.84776			−1.84776
$476$		0			0
$477$	−1.84776			−1.84776
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$481$		0			0
$482$	0.707107	+	0.292893i	0.707107	+	0.292893i
$483$		0			0
$484$	0.707107	+	0.707107i	0.707107	+	0.707107i
$485$		0			0
$486$	0.382683	−	0.923880i	0.382683	−	0.923880i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	0.707107	+	1.70711i	0.707107	+	1.70711i
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	0.765367i		−	0.765367i
$497$		0			0
$498$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$499$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$499$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$500$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$501$		0			0
$502$		0			0
$503$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$503$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.00000			−1.00000
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$511$		0			0
$512$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$513$	1.84776			1.84776
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			1.41421i			1.41421i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$527$		−	1.08239i		−	1.08239i
$528$		0			0
$529$	0.414214			0.414214
$530$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.84776			−1.84776
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	0.707107	−	0.707107i	0.707107	−	0.707107i
$541$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$541$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$542$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$543$			0.765367i			0.765367i
$544$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	0.541196	+	0.541196i	0.541196	+	0.541196i
$549$		−	1.84776i		−	1.84776i
$550$		0			0
$551$		0			0
$552$	0.292893	+	0.707107i	0.292893	+	0.707107i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$557$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$557$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$558$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$563$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$564$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$565$			0.765367i			0.765367i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	1.70711	+	0.707107i	1.70711	+	0.707107i
$571$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$571$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			0.765367i			0.765367i
$576$	0.707107	−	0.707107i	0.707107	−	0.707107i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$587$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$	−1.41421			−1.41421
$590$		0			0
$591$	−0.765367			−0.765367
$592$		0			0
$593$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$593$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.84776			−1.84776
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	0.923880	−	0.382683i	0.923880	−	0.382683i
$601$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$601$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$602$		0			0
$603$		0			0
$604$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$605$		−	1.00000i		−	1.00000i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	1.70711	+	0.707107i	1.70711	+	0.707107i
$609$		0			0
$610$	0.707107	−	1.70711i	0.707107	−	1.70711i
$611$		0			0
$612$	1.00000	−	1.00000i	1.00000	−	1.00000i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$617$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$618$		0			0
$619$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$619$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$620$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$621$		−	0.765367i		−	0.765367i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$631$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$632$	0.541196	+	1.30656i	0.541196	+	1.30656i
$633$		−	2.00000i		−	2.00000i
$634$	0.707107	−	1.70711i	0.707107	−	1.70711i
$635$		0			0
$636$	1.30656	+	1.30656i	1.30656	+	1.30656i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.923880	−	0.382683i	0.923880	−	0.382683i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	1.70711	+	0.707107i	1.70711	+	0.707107i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$	2.41421	+	1.00000i	2.41421	+	1.00000i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$653$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$661$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$662$	−1.84776	−	0.765367i	−1.84776	−	0.765367i
$663$		0			0
$664$	1.30656	−	0.541196i	1.30656	−	0.541196i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−1.00000			−1.00000
$676$	0.707107	+	0.707107i	0.707107	+	0.707107i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	0.292893	−	0.707107i	0.292893	−	0.707107i
$679$		0			0
$680$	1.30656	−	0.541196i	1.30656	−	0.541196i
$681$		0			0
$682$		0			0
$683$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$683$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$684$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$685$		−	0.765367i		−	0.765367i
$686$		0			0
$687$			1.84776i			1.84776i
$688$		0			0
$689$		0			0
$690$	0.292893	−	0.707107i	0.292893	−	0.707107i
$691$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$691$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$692$	1.00000	−	1.00000i	1.00000	−	1.00000i
$693$		0			0
$694$	0.707107	+	0.292893i	0.707107	+	0.292893i
$695$			0.765367i			0.765367i
$696$		0			0
$697$		0			0
$698$	0.707107	+	0.292893i	0.707107	+	0.292893i
$699$	−1.84776			−1.84776
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$			1.41421i			1.41421i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$709$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$710$		0			0
$711$		−	1.41421i		−	1.41421i
$712$		0			0
$713$			0.585786i			0.585786i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	−1.00000			−1.00000
$721$		0			0
$722$	0.923880	−	2.23044i	0.923880	−	2.23044i
$723$			0.765367i			0.765367i
$724$	0.541196	−	0.541196i	0.541196	−	0.541196i
$725$		0			0
$726$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	0.292893	−	0.707107i	0.292893	−	0.707107i
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$743$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$744$	0.707107	−	0.292893i	0.707107	−	0.292893i
$745$		0			0
$746$		0			0
$747$	−1.41421			−1.41421
$748$		0			0
$749$		0			0
$750$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			1.41421i			1.41421i
$753$		0			0
$754$		0			0
$755$	1.41421			1.41421
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$759$		0			0
$760$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.41421			−1.41421
$766$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$767$		0			0
$768$	−1.00000			−1.00000
$769$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$769$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$773$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$774$		0			0
$775$	0.765367			0.765367
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.414214	−	1.00000i	0.414214	−	1.00000i
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	0.541196	+	0.541196i	0.541196	+	0.541196i
$789$		−	1.84776i		−	1.84776i
$790$	0.541196	−	1.30656i	0.541196	−	1.30656i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		−	1.84776i		−	1.84776i
$796$	1.30656	+	1.30656i	1.30656	+	1.30656i
$797$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$797$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$			2.00000i			2.00000i
$800$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	0.923880	+	0.382683i	0.923880	+	0.382683i
$811$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$811$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$812$		0			0
$813$	−1.84776			−1.84776
$814$		0			0
$815$		0			0
$816$	−1.41421			−1.41421
$817$		0			0
$818$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$827$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$828$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$829$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$829$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$830$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$831$		0			0
$832$		0			0
$833$		0			0
$834$	0.292893	−	0.707107i	0.292893	−	0.707107i
$835$		0			0
$836$		0			0
$837$	−0.765367			−0.765367
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$	0.541196	−	1.30656i	0.541196	−	1.30656i
$843$		0			0
$844$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$845$		−	1.00000i		−	1.00000i
$846$	0.541196	−	1.30656i	0.541196	−	1.30656i
$847$		0			0
$848$		−	1.84776i		−	1.84776i
$849$		0			0
$850$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$			1.84776i			1.84776i
$856$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$857$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$857$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$858$		0			0
$859$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$859$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$863$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$864$	0.923880	+	0.382683i	0.923880	+	0.382683i
$865$	−1.41421			−1.41421
$866$		0			0
$867$	−1.00000			−1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$	0.292893	−	0.707107i	0.292893	−	0.707107i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$887$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$887$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.61313			2.61313
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.707107	+	0.707107i	0.707107	+	0.707107i
$901$		−	2.61313i		−	2.61313i
$902$		0			0
$903$		0			0
$904$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$905$	−0.765367			−0.765367
$906$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$			1.84776i			1.84776i
$913$		0			0
$914$		0			0
$915$	1.84776			1.84776
$916$	1.30656	−	1.30656i	1.30656	−	1.30656i
$917$		0			0
$918$	1.30656	+	0.541196i	1.30656	+	0.541196i
$919$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$919$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$920$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$931$		0			0
$932$	1.30656	+	1.30656i	1.30656	+	1.30656i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	1.00000	−	1.00000i	1.00000	−	1.00000i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$947$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$948$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$949$		0			0
$950$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$951$	1.84776			1.84776
$952$		0			0
$953$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$953$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$954$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.707107	+	0.707107i	0.707107	+	0.707107i
$961$	−0.414214			−0.414214
$962$		0			0
$963$			1.84776i			1.84776i
$964$	0.541196	−	0.541196i	0.541196	−	0.541196i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	0.923880	−	0.382683i	0.923880	−	0.382683i
$969$			2.61313i			2.61313i
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.84776			1.84776
$977$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$977$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.00000			−2.00000			−1.00000			$\pi$
$983$	−2.00000			−2.00000			−1.00000			$\pi$
$984$		0			0
$985$		−	0.765367i		−	0.765367i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$993$		−	2.00000i		−	2.00000i
$994$		0			0
$995$		−	1.84776i		−	1.84776i
$996$	1.00000	+	1.00000i	1.00000	+	1.00000i
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	1.30656	+	0.541196i	1.30656	+	0.541196i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.1.o.d.2939.5	yes	8
3.2	odd	2		2940.1.o.c.2939.4	yes	8
4.3	odd	2		2940.1.o.c.2939.6	yes	8
5.4	even	2		2940.1.o.c.2939.4	yes	8
7.2	even	3		2940.1.be.e.2579.1		16
7.3	odd	6		2940.1.be.f.1979.6		16
7.4	even	3		2940.1.be.e.1979.6		16
7.5	odd	6		2940.1.be.f.2579.1		16
7.6	odd	2		2940.1.o.c.2939.5	yes	8
12.11	even	2	inner	2940.1.o.d.2939.3	yes	8
15.14	odd	2	CM	2940.1.o.d.2939.5	yes	8
20.19	odd	2	inner	2940.1.o.d.2939.3	yes	8
21.2	odd	6		2940.1.be.f.2579.8		16
21.5	even	6		2940.1.be.e.2579.8		16
21.11	odd	6		2940.1.be.f.1979.3		16
21.17	even	6		2940.1.be.e.1979.3		16
21.20	even	2	inner	2940.1.o.d.2939.4	yes	8
28.3	even	6		2940.1.be.e.1979.1		16
28.11	odd	6		2940.1.be.f.1979.1		16
28.19	even	6		2940.1.be.e.2579.6		16
28.23	odd	6		2940.1.be.f.2579.6		16
28.27	even	2	inner	2940.1.o.d.2939.6	yes	8
35.4	even	6		2940.1.be.f.1979.3		16
35.9	even	6		2940.1.be.f.2579.8		16
35.19	odd	6		2940.1.be.e.2579.8		16
35.24	odd	6		2940.1.be.e.1979.3		16
35.34	odd	2	inner	2940.1.o.d.2939.4	yes	8
60.59	even	2		2940.1.o.c.2939.6	yes	8
84.11	even	6		2940.1.be.e.1979.8		16
84.23	even	6		2940.1.be.e.2579.3		16
84.47	odd	6		2940.1.be.f.2579.3		16
84.59	odd	6		2940.1.be.f.1979.8		16
84.83	odd	2		2940.1.o.c.2939.3	✓	8
105.44	odd	6		2940.1.be.e.2579.1		16
105.59	even	6		2940.1.be.f.1979.6		16
105.74	odd	6		2940.1.be.e.1979.6		16
105.89	even	6		2940.1.be.f.2579.1		16
105.104	even	2		2940.1.o.c.2939.5	yes	8
140.19	even	6		2940.1.be.f.2579.3		16
140.39	odd	6		2940.1.be.e.1979.8		16
140.59	even	6		2940.1.be.f.1979.8		16
140.79	odd	6		2940.1.be.e.2579.3		16
140.139	even	2		2940.1.o.c.2939.3	✓	8
420.59	odd	6		2940.1.be.e.1979.1		16
420.179	even	6		2940.1.be.f.1979.1		16
420.299	odd	6		2940.1.be.e.2579.6		16
420.359	even	6		2940.1.be.f.2579.6		16
420.419	odd	2	inner	2940.1.o.d.2939.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2940.1.o.c.2939.3	✓	8	84.83	odd	2
2940.1.o.c.2939.3	✓	8	140.139	even	2
2940.1.o.c.2939.4	yes	8	3.2	odd	2
2940.1.o.c.2939.4	yes	8	5.4	even	2
2940.1.o.c.2939.5	yes	8	7.6	odd	2
2940.1.o.c.2939.5	yes	8	105.104	even	2
2940.1.o.c.2939.6	yes	8	4.3	odd	2
2940.1.o.c.2939.6	yes	8	60.59	even	2
2940.1.o.d.2939.3	yes	8	12.11	even	2	inner
2940.1.o.d.2939.3	yes	8	20.19	odd	2	inner
2940.1.o.d.2939.4	yes	8	21.20	even	2	inner
2940.1.o.d.2939.4	yes	8	35.34	odd	2	inner
2940.1.o.d.2939.5	yes	8	1.1	even	1	trivial
2940.1.o.d.2939.5	yes	8	15.14	odd	2	CM
2940.1.o.d.2939.6	yes	8	28.27	even	2	inner
2940.1.o.d.2939.6	yes	8	420.419	odd	2	inner
2940.1.be.e.1979.1		16	28.3	even	6
2940.1.be.e.1979.1		16	420.59	odd	6
2940.1.be.e.1979.3		16	21.17	even	6
2940.1.be.e.1979.3		16	35.24	odd	6
2940.1.be.e.1979.6		16	7.4	even	3
2940.1.be.e.1979.6		16	105.74	odd	6
2940.1.be.e.1979.8		16	84.11	even	6
2940.1.be.e.1979.8		16	140.39	odd	6
2940.1.be.e.2579.1		16	7.2	even	3
2940.1.be.e.2579.1		16	105.44	odd	6
2940.1.be.e.2579.3		16	84.23	even	6
2940.1.be.e.2579.3		16	140.79	odd	6
2940.1.be.e.2579.6		16	28.19	even	6
2940.1.be.e.2579.6		16	420.299	odd	6
2940.1.be.e.2579.8		16	21.5	even	6
2940.1.be.e.2579.8		16	35.19	odd	6
2940.1.be.f.1979.1		16	28.11	odd	6
2940.1.be.f.1979.1		16	420.179	even	6
2940.1.be.f.1979.3		16	21.11	odd	6
2940.1.be.f.1979.3		16	35.4	even	6
2940.1.be.f.1979.6		16	7.3	odd	6
2940.1.be.f.1979.6		16	105.59	even	6
2940.1.be.f.1979.8		16	84.59	odd	6
2940.1.be.f.1979.8		16	140.59	even	6
2940.1.be.f.2579.1		16	7.5	odd	6
2940.1.be.f.2579.1		16	105.89	even	6
2940.1.be.f.2579.3		16	84.47	odd	6
2940.1.be.f.2579.3		16	140.19	even	6
2940.1.be.f.2579.6		16	28.23	odd	6
2940.1.be.f.2579.6		16	420.359	even	6
2940.1.be.f.2579.8		16	21.2	odd	6
2940.1.be.f.2579.8		16	35.9	even	6

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:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2940.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.382683 - 0.923880i) q^{2} +1.00000 q^{3} +(-0.707107 - 0.707107i) q^{4} +1.00000i q^{5} +(0.382683 - 0.923880i) q^{6} +(-0.923880 + 0.382683i) q^{8} +1.00000 q^{9} +(0.923880 + 0.382683i) q^{10} +(-0.707107 - 0.707107i) q^{12} +1.00000i q^{15} +1.00000i q^{16} +1.41421i q^{17} +(0.382683 - 0.923880i) q^{18} +1.84776 q^{19} +(0.707107 - 0.707107i) q^{20} -0.765367i q^{23} +(-0.923880 + 0.382683i) q^{24} -1.00000 q^{25} +1.00000 q^{27} +(0.923880 + 0.382683i) q^{30} -0.765367 q^{31} +(0.923880 + 0.382683i) q^{32} +(1.30656 + 0.541196i) q^{34} +(-0.707107 - 0.707107i) q^{36} +(0.707107 - 1.70711i) q^{38} +(-0.382683 - 0.923880i) q^{40} +1.00000i q^{45} +(-0.707107 - 0.292893i) q^{46} +1.41421 q^{47} +1.00000i q^{48} +(-0.382683 + 0.923880i) q^{50} +1.41421i q^{51} -1.84776 q^{53} +(0.382683 - 0.923880i) q^{54} +1.84776 q^{57} +(0.707107 - 0.707107i) q^{60} -1.84776i q^{61} +(-0.292893 + 0.707107i) q^{62} +(0.707107 - 0.707107i) q^{64} +(1.00000 - 1.00000i) q^{68} -0.765367i q^{69} +(-0.923880 + 0.382683i) q^{72} -1.00000 q^{75} +(-1.30656 - 1.30656i) q^{76} -1.41421i q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.41421 q^{83} -1.41421 q^{85} +(0.923880 + 0.382683i) q^{90} +(-0.541196 + 0.541196i) q^{92} -0.765367 q^{93} +(0.541196 - 1.30656i) q^{94} +1.84776i q^{95} +(0.923880 + 0.382683i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9} - 8 q^{25} + 8 q^{27} - 8 q^{62} + 8 q^{68} - 8 q^{75} - 8 q^{80} + 8 q^{81}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 - 8 * q^25 + 8 * q^27 - 8 * q^62 + 8 * q^68 - 8 * q^75 - 8 * q^80 + 8 * q^81

Introduction

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