LMFDB - Embedded newform 3380.1.g.c.3379.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,1,Mod(3379,3380)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3380.3379");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3380.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.68683974270$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.38614472000.1

Embedding invariants

Embedding label			3379.6
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	3380.3379
Dual form			3380.1.g.c.3379.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.24698 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.24698i q^{6} +1.80194i q^{7} -1.00000i q^{8} +0.554958 q^{9} -1.00000 q^{10} -1.24698 q^{12} -1.80194 q^{14} +1.24698i q^{15} +1.00000 q^{16} +0.554958i q^{18} -1.00000i q^{20} +2.24698i q^{21} +0.445042 q^{23} -1.24698i q^{24} -1.00000 q^{25} -0.554958 q^{27} -1.80194i q^{28} +1.24698 q^{29} -1.24698 q^{30} +1.00000i q^{32} -1.80194 q^{35} -0.554958 q^{36} +1.00000 q^{40} -1.80194i q^{41} -2.24698 q^{42} +0.445042 q^{43} +0.554958i q^{45} +0.445042i q^{46} +0.445042i q^{47} +1.24698 q^{48} -2.24698 q^{49} -1.00000i q^{50} -0.554958i q^{54} +1.80194 q^{56} +1.24698i q^{58} -1.24698i q^{60} -1.80194 q^{61} +1.00000i q^{63} -1.00000 q^{64} -0.445042i q^{67} +0.554958 q^{69} -1.80194i q^{70} -0.554958i q^{72} -1.24698 q^{75} +1.00000i q^{80} -1.24698 q^{81} +1.80194 q^{82} +1.24698i q^{83} -2.24698i q^{84} +0.445042i q^{86} +1.55496 q^{87} +0.445042i q^{89} -0.554958 q^{90} -0.445042 q^{92} -0.445042 q^{94} +1.24698i q^{96} -2.24698i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{4} + 4 q^{9} - 6 q^{10} + 2 q^{12} - 2 q^{14} + 6 q^{16} + 2 q^{23} - 6 q^{25} - 4 q^{27} - 2 q^{29} + 2 q^{30} - 2 q^{35} - 4 q^{36} + 6 q^{40} - 4 q^{42} + 2 q^{43} - 2 q^{48} - 4 q^{49}+ \cdots - 2 q^{94}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^4 + 4 * q^9 - 6 * q^10 + 2 * q^12 - 2 * q^14 + 6 * q^16 + 2 * q^23 - 6 * q^25 - 4 * q^27 - 2 * q^29 + 2 * q^30 - 2 * q^35 - 4 * q^36 + 6 * q^40 - 4 * q^42 + 2 * q^43 - 2 * q^48 - 4 * q^49 + 2 * q^56 - 2 * q^61 - 6 * q^64 + 4 * q^69 + 2 * q^75 + 2 * q^81 + 2 * q^82 + 10 * q^87 - 4 * q^90 - 2 * q^92 - 2 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.00000i	1.00000i
$2$	1.00000i	1.00000i
$3$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$3$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$4$	−1.00000	−1.00000
$5$	1.00000i	1.00000i
$5$	1.00000i	1.00000i
$6$	1.24698i	1.24698i
$7$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$7$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$8$	− 1.00000i	− 1.00000i
$9$	0.554958	0.554958
$10$	−1.00000	−1.00000
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.24698	−1.24698
$13$	0	0
$13$	0	0
$14$	−1.80194	−1.80194
$15$	1.24698i	1.24698i
$16$	1.00000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.554958i	0.554958i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	− 1.00000i	− 1.00000i
$21$	2.24698i	2.24698i
$22$	0	0
$23$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$23$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$24$	− 1.24698i	− 1.24698i
$25$	−1.00000	−1.00000
$26$	0	0
$27$	−0.554958	−0.554958
$28$	− 1.80194i	− 1.80194i
$29$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$29$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$30$	−1.24698	−1.24698
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000i	1.00000i
$33$	0	0
$34$	0	0
$35$	−1.80194	−1.80194
$36$	−0.554958	−0.554958
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	1.00000	1.00000
$41$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$41$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$42$	−2.24698	−2.24698
$43$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$43$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$44$	0	0
$45$	0.554958i	0.554958i
$46$	0.445042i	0.445042i
$47$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$47$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$48$	1.24698	1.24698
$49$	−2.24698	−2.24698
$50$	− 1.00000i	− 1.00000i
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	− 0.554958i	− 0.554958i
$55$	0	0
$56$	1.80194	1.80194
$57$	0	0
$58$	1.24698i	1.24698i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	− 1.24698i	− 1.24698i
$61$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$61$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$62$	0	0
$63$	1.00000i	1.00000i
$64$	−1.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$67$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$68$	0	0
$69$	0.554958	0.554958
$70$	− 1.80194i	− 1.80194i
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 0.554958i	− 0.554958i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	−1.24698	−1.24698
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.00000i	1.00000i
$81$	−1.24698	−1.24698
$82$	1.80194	1.80194
$83$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$83$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$84$	− 2.24698i	− 2.24698i
$85$	0	0
$86$	0.445042i	0.445042i
$87$	1.55496	1.55496
$88$	0	0
$89$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$89$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$90$	−0.554958	−0.554958
$91$	0	0
$92$	−0.445042	−0.445042
$93$	0	0
$94$	−0.445042	−0.445042
$95$	0	0
$96$	1.24698i	1.24698i
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	− 2.24698i	− 2.24698i
$99$	0	0
$100$	1.00000	1.00000
$101$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$101$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$102$	0	0
$103$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$103$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$104$	0	0
$105$	−2.24698	−2.24698
$106$	0	0
$107$	2.00000	2.00000	1.00000			$0$
$107$	2.00000	2.00000	1.00000			$0$
$108$	0.554958	0.554958
$109$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$109$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$110$	0	0
$111$	0	0
$112$	1.80194i	1.80194i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0.445042i	0.445042i
$116$	−1.24698	−1.24698
$117$	0	0
$118$	0	0
$119$	0	0
$120$	1.24698	1.24698
$121$	−1.00000	−1.00000
$122$	− 1.80194i	− 1.80194i
$123$	− 2.24698i	− 2.24698i
$124$	0	0
$125$	− 1.00000i	− 1.00000i
$126$	−1.00000	−1.00000
$127$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$127$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$128$	− 1.00000i	− 1.00000i
$129$	0.554958	0.554958
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0.445042	0.445042
$135$	− 0.554958i	− 0.554958i
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0.554958i	0.554958i
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	1.80194	1.80194
$141$	0.554958i	0.554958i
$142$	0	0
$143$	0	0
$144$	0.554958	0.554958
$145$	1.24698i	1.24698i
$146$	0	0
$147$	−2.80194	−2.80194
$148$	0	0
$149$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$149$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$150$	− 1.24698i	− 1.24698i
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−1.00000	−1.00000
$161$	0.801938i	0.801938i
$162$	− 1.24698i	− 1.24698i
$163$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$163$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$164$	1.80194i	1.80194i
$165$	0	0
$166$	−1.24698	−1.24698
$167$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$167$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$168$	2.24698	2.24698
$169$	0	0
$170$	0	0
$171$	0	0
$172$	−0.445042	−0.445042
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	1.55496i	1.55496i
$175$	− 1.80194i	− 1.80194i
$176$	0	0
$177$	0	0
$178$	−0.445042	−0.445042
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	− 0.554958i	− 0.554958i
$181$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$181$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$182$	0	0
$183$	−2.24698	−2.24698
$184$	− 0.445042i	− 0.445042i
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 0.445042i	− 0.445042i
$189$	− 1.00000i	− 1.00000i
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1.24698	−1.24698
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	2.24698	2.24698
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.00000i	1.00000i
$201$	− 0.554958i	− 0.554958i
$202$	0.445042i	0.445042i
$203$	2.24698i	2.24698i
$204$	0	0
$205$	1.80194	1.80194
$206$	1.80194i	1.80194i
$207$	0.246980	0.246980
$208$	0	0
$209$	0	0
$210$	− 2.24698i	− 2.24698i
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	2.00000i	2.00000i
$215$	0.445042i	0.445042i
$216$	0.554958i	0.554958i
$217$	0	0
$218$	1.80194	1.80194
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$223$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$224$	−1.80194	−1.80194
$225$	−0.554958	−0.554958
$226$	0	0
$227$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$227$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$228$	0	0
$229$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$229$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$230$	−0.445042	−0.445042
$231$	0	0
$232$	− 1.24698i	− 1.24698i
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	−0.445042	−0.445042
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	1.24698i	1.24698i
$241$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$241$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$242$	− 1.00000i	− 1.00000i
$243$	−1.00000	−1.00000
$244$	1.80194	1.80194
$245$	− 2.24698i	− 2.24698i
$246$	2.24698	2.24698
$247$	0	0
$248$	0	0
$249$	1.55496i	1.55496i
$250$	1.00000	1.00000
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	− 1.00000i	− 1.00000i
$253$	0	0
$254$	1.80194i	1.80194i
$255$	0	0
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0.554958i	0.554958i
$259$	0	0
$260$	0	0
$261$	0.692021	0.692021
$262$	0	0
$263$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$263$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.554958i	0.554958i
$268$	0.445042i	0.445042i
$269$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$269$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$270$	0.554958	0.554958
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−0.554958	−0.554958
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	1.80194i	1.80194i
$281$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$281$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$282$	−0.554958	−0.554958
$283$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$283$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.24698	3.24698
$288$	0.554958i	0.554958i
$289$	1.00000	1.00000
$290$	−1.24698	−1.24698
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	− 2.80194i	− 2.80194i
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−1.24698	−1.24698
$299$	0	0
$300$	1.24698	1.24698
$301$	0.801938i	0.801938i
$302$	0	0
$303$	0.554958	0.554958
$304$	0	0
$305$	− 1.80194i	− 1.80194i
$306$	0	0
$307$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$307$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$308$	0	0
$309$	2.24698	2.24698
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	−1.00000	−1.00000
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	− 1.00000i	− 1.00000i
$321$	2.49396	2.49396
$322$	−0.801938	−0.801938
$323$	0	0
$324$	1.24698	1.24698
$325$	0	0
$326$	1.24698	1.24698
$327$	− 2.24698i	− 2.24698i
$328$	−1.80194	−1.80194
$329$	−0.801938	−0.801938
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	− 1.24698i	− 1.24698i
$333$	0	0
$334$	−1.80194	−1.80194
$335$	0.445042	0.445042
$336$	2.24698i	2.24698i
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 2.24698i	− 2.24698i
$344$	− 0.445042i	− 0.445042i
$345$	0.554958i	0.554958i
$346$	0	0
$347$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$347$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$348$	−1.55496	−1.55496
$349$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$349$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$350$	1.80194	1.80194
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	− 0.445042i	− 0.445042i
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0.554958	0.554958
$361$	−1.00000	−1.00000
$362$	0.445042i	0.445042i
$363$	−1.24698	−1.24698
$364$	0	0
$365$	0	0
$366$	− 2.24698i	− 2.24698i
$367$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$367$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$368$	0.445042	0.445042
$369$	− 1.00000i	− 1.00000i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	− 1.24698i	− 1.24698i
$376$	0.445042	0.445042
$377$	0	0
$378$	1.00000	1.00000
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	2.24698	2.24698
$382$	0	0
$383$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$383$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$384$	− 1.24698i	− 1.24698i
$385$	0	0
$386$	0	0
$387$	0.246980	0.246980
$388$	0	0
$389$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$389$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$390$	0	0
$391$	0	0
$392$	2.24698i	2.24698i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	−1.00000	−1.00000
$401$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$401$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$402$	0.554958	0.554958
$403$	0	0
$404$	−0.445042	−0.445042
$405$	− 1.24698i	− 1.24698i
$406$	−2.24698	−2.24698
$407$	0	0
$408$	0	0
$409$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$409$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$410$	1.80194i	1.80194i
$411$	0	0
$412$	−1.80194	−1.80194
$413$	0	0
$414$	0.246980i	0.246980i
$415$	−1.24698	−1.24698
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	2.24698	2.24698
$421$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$421$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$422$	0	0
$423$	0.246980i	0.246980i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.24698i	− 3.24698i
$428$	−2.00000	−2.00000
$429$	0	0
$430$	−0.445042	−0.445042
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−0.554958	−0.554958
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	1.55496i	1.55496i
$436$	1.80194i	1.80194i
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−1.24698	−1.24698
$442$	0	0
$443$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$443$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$444$	0	0
$445$	−0.445042	−0.445042
$446$	−1.24698	−1.24698
$447$	1.55496i	1.55496i
$448$	− 1.80194i	− 1.80194i
$449$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$449$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$450$	− 0.554958i	− 0.554958i
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−1.24698	−1.24698
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	1.24698	1.24698
$459$	0	0
$460$	− 0.445042i	− 0.445042i
$461$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$461$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$462$	0	0
$463$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$463$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$464$	1.24698	1.24698
$465$	0	0
$466$	0	0
$467$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$467$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$468$	0	0
$469$	0.801938	0.801938
$470$	− 0.445042i	− 0.445042i
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−1.24698	−1.24698
$481$	0	0
$482$	−0.445042	−0.445042
$483$	1.00000i	1.00000i
$484$	1.00000	1.00000
$485$	0	0
$486$	− 1.00000i	− 1.00000i
$487$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$487$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$488$	1.80194i	1.80194i
$489$	− 1.55496i	− 1.55496i
$490$	2.24698	2.24698
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	2.24698i	2.24698i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−1.55496	−1.55496
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	1.00000i	1.00000i
$501$	2.24698i	2.24698i
$502$	0	0
$503$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$503$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$504$	1.00000	1.00000
$505$	0.445042i	0.445042i
$506$	0	0
$507$	0	0
$508$	−1.80194	−1.80194
$509$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$509$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	1.00000i
$513$	0	0
$514$	0	0
$515$	1.80194i	1.80194i
$516$	−0.554958	−0.554958
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$521$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$522$	0.692021i	0.692021i
$523$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$523$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$524$	0	0
$525$	− 2.24698i	− 2.24698i
$526$	− 1.80194i	− 1.80194i
$527$	0	0
$528$	0	0
$529$	−0.801938	−0.801938
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−0.554958	−0.554958
$535$	2.00000i	2.00000i
$536$	−0.445042	−0.445042
$537$	0	0
$538$	1.24698i	1.24698i
$539$	0	0
$540$	0.554958i	0.554958i
$541$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$541$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$542$	0	0
$543$	0.554958	0.554958
$544$	0	0
$545$	1.80194	1.80194
$546$	0	0
$547$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$547$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$548$	0	0
$549$	−1.00000	−1.00000
$550$	0	0
$551$	0	0
$552$	− 0.554958i	− 0.554958i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	−1.80194	−1.80194
$561$	0	0
$562$	1.24698	1.24698
$563$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$563$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$564$	− 0.554958i	− 0.554958i
$565$	0	0
$566$	− 1.24698i	− 1.24698i
$567$	− 2.24698i	− 2.24698i
$568$	0	0
$569$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$569$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	3.24698i	3.24698i
$575$	−0.445042	−0.445042
$576$	−0.554958	−0.554958
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	1.00000i	1.00000i
$579$	0	0
$580$	− 1.24698i	− 1.24698i
$581$	−2.24698	−2.24698
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$587$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$588$	2.80194	2.80194
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	− 1.24698i	− 1.24698i
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	1.24698i	1.24698i
$601$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$601$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$602$	−0.801938	−0.801938
$603$	− 0.246980i	− 0.246980i
$604$	0	0
$605$	− 1.00000i	− 1.00000i
$606$	0.554958i	0.554958i
$607$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$607$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$608$	0	0
$609$	2.80194i	2.80194i
$610$	1.80194	1.80194
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	1.24698	1.24698
$615$	2.24698	2.24698
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	2.24698i	2.24698i
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−0.246980	−0.246980
$622$	0	0
$623$	−0.801938	−0.801938
$624$	0	0
$625$	1.00000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	− 1.00000i	− 1.00000i
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.80194i	1.80194i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.00000	1.00000
$641$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$641$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$642$	2.49396i	2.49396i
$643$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$643$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$644$	− 0.801938i	− 0.801938i
$645$	0.554958i	0.554958i
$646$	0	0
$647$	−2.00000	−2.00000	−1.00000			$\pi$
$647$	−2.00000	−2.00000	−1.00000			$\pi$
$648$	1.24698i	1.24698i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	1.24698i	1.24698i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	2.24698	2.24698
$655$	0	0
$656$	− 1.80194i	− 1.80194i
$657$	0	0
$658$	− 0.801938i	− 0.801938i
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$661$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$662$	0	0
$663$	0	0
$664$	1.24698	1.24698
$665$	0	0
$666$	0	0
$667$	0.554958	0.554958
$668$	− 1.80194i	− 1.80194i
$669$	1.55496i	1.55496i
$670$	0.445042i	0.445042i
$671$	0	0
$672$	−2.24698	−2.24698
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0.554958	0.554958
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.55496i	1.55496i
$682$	0	0
$683$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$683$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$684$	0	0
$685$	0	0
$686$	2.24698	2.24698
$687$	− 1.55496i	− 1.55496i
$688$	0.445042	0.445042
$689$	0	0
$690$	−0.554958	−0.554958
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	1.24698i	1.24698i
$695$	0	0
$696$	− 1.55496i	− 1.55496i
$697$	0	0
$698$	2.00000	2.00000
$699$	0	0
$700$	1.80194i	1.80194i
$701$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$701$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−0.554958	−0.554958
$706$	0	0
$707$	0.801938i	0.801938i
$708$	0	0
$709$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$709$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$710$	0	0
$711$	0	0
$712$	0.445042	0.445042
$713$	0	0
$714$	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			$\pi$
$720$	0.554958i	0.554958i
$721$	3.24698i	3.24698i
$722$	− 1.00000i	− 1.00000i
$723$	0.554958i	0.554958i
$724$	−0.445042	−0.445042
$725$	−1.24698	−1.24698
$726$	− 1.24698i	− 1.24698i
$727$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$727$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	2.24698	2.24698
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	1.24698i	1.24698i
$735$	− 2.80194i	− 2.80194i
$736$	0.445042i	0.445042i
$737$	0	0
$738$	1.00000	1.00000
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$743$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$744$	0	0
$745$	−1.24698	−1.24698
$746$	0	0
$747$	0.692021i	0.692021i
$748$	0	0
$749$	3.60388i	3.60388i
$750$	1.24698	1.24698
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0.445042i	0.445042i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	1.00000i	1.00000i
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$761$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$762$	2.24698i	2.24698i
$763$	3.24698	3.24698
$764$	0	0
$765$	0	0
$766$	−1.24698	−1.24698
$767$	0	0
$768$	1.24698	1.24698
$769$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$769$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0.246980i	0.246980i
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0.445042i	0.445042i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.692021	−0.692021
$784$	−2.24698	−2.24698
$785$	0	0
$786$	0	0
$787$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$787$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$788$	0	0
$789$	−2.24698	−2.24698
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	− 1.00000i	− 1.00000i
$801$	0.246980i	0.246980i
$802$	−1.80194	−1.80194
$803$	0	0
$804$	0.554958i	0.554958i
$805$	−0.801938	−0.801938
$806$	0	0
$807$	1.55496	1.55496
$808$	− 0.445042i	− 0.445042i
$809$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$809$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$810$	1.24698	1.24698
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	− 2.24698i	− 2.24698i
$813$	0	0
$814$	0	0
$815$	1.24698	1.24698
$816$	0	0
$817$	0	0
$818$	1.80194	1.80194
$819$	0	0
$820$	−1.80194	−1.80194
$821$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$821$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$822$	0	0
$823$	−2.00000	−2.00000	−1.00000			$\pi$
$823$	−2.00000	−2.00000	−1.00000			$\pi$
$824$	− 1.80194i	− 1.80194i
$825$	0	0
$826$	0	0
$827$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$827$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$828$	−0.246980	−0.246980
$829$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$829$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$830$	− 1.24698i	− 1.24698i
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.80194	−1.80194
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	2.24698i	2.24698i
$841$	0.554958	0.554958
$842$	−1.24698	−1.24698
$843$	− 1.55496i	− 1.55496i
$844$	0	0
$845$	0	0
$846$	−0.246980	−0.246980
$847$	− 1.80194i	− 1.80194i
$848$	0	0
$849$	−1.55496	−1.55496
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	3.24698	3.24698
$855$	0	0
$856$	− 2.00000i	− 2.00000i
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	− 0.445042i	− 0.445042i
$861$	4.04892	4.04892
$862$	0	0
$863$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$863$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$864$	− 0.554958i	− 0.554958i
$865$	0	0
$866$	0	0
$867$	1.24698	1.24698
$868$	0	0
$869$	0	0
$870$	−1.55496	−1.55496
$871$	0	0
$872$	−1.80194	−1.80194
$873$	0	0
$874$	0	0
$875$	1.80194	1.80194
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.00000	−2.00000	−1.00000			$\pi$
$881$	−2.00000	−2.00000	−1.00000			$\pi$
$882$	− 1.24698i	− 1.24698i
$883$	−2.00000	−2.00000	−1.00000			$\pi$
$883$	−2.00000	−2.00000	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	1.24698i	1.24698i
$887$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$887$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$888$	0	0
$889$	3.24698i	3.24698i
$890$	− 0.445042i	− 0.445042i
$891$	0	0
$892$	− 1.24698i	− 1.24698i
$893$	0	0
$894$	−1.55496	−1.55496
$895$	0	0
$896$	1.80194	1.80194
$897$	0	0
$898$	1.24698	1.24698
$899$	0	0
$900$	0.554958	0.554958
$901$	0	0
$902$	0	0
$903$	1.00000i	1.00000i
$904$	0	0
$905$	0.445042i	0.445042i
$906$	0	0
$907$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$907$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$908$	− 1.24698i	− 1.24698i
$909$	0.246980	0.246980
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 2.24698i	− 2.24698i
$916$	1.24698i	1.24698i
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0.445042	0.445042
$921$	− 1.55496i	− 1.55496i
$922$	0.445042	0.445042
$923$	0	0
$924$	0	0
$925$	0	0
$926$	2.00000	2.00000
$927$	1.00000	1.00000
$928$	1.24698i	1.24698i
$929$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$929$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0.445042i	0.445042i
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0.801938i	0.801938i
$939$	0	0
$940$	0.445042	0.445042
$941$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$941$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$942$	0	0
$943$	− 0.801938i	− 0.801938i
$944$	0	0
$945$	1.00000	1.00000
$946$	0	0
$947$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$947$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	− 1.24698i	− 1.24698i
$961$	−1.00000	−1.00000
$962$	0	0
$963$	1.10992	1.10992
$964$	− 0.445042i	− 0.445042i
$965$	0	0
$966$	−1.00000	−1.00000
$967$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$967$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$968$	1.00000i	1.00000i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	1.00000	1.00000
$973$	0	0
$974$	0.445042	0.445042
$975$	0	0
$976$	−1.80194	−1.80194
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	1.55496	1.55496
$979$	0	0
$980$	2.24698i	2.24698i
$981$	− 1.00000i	− 1.00000i
$982$	0	0
$983$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$983$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$984$	−2.24698	−2.24698
$985$	0	0
$986$	0	0
$987$	−1.00000	−1.00000
$988$	0	0
$989$	0.198062	0.198062
$990$	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
$996$	− 1.55496i	− 1.55496i
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.1.g.c.3379.6		6
4.3	odd	2		3380.1.g.d.3379.1		6
5.4	even	2		3380.1.g.d.3379.1		6
13.2	odd	12		3380.1.v.e.2219.1		6
13.3	even	3		3380.1.w.f.1499.1		12
13.4	even	6		3380.1.w.f.699.1		12
13.5	odd	4		3380.1.h.d.339.3	yes	3
13.6	odd	12		3380.1.v.e.3019.1		6
13.7	odd	12		3380.1.v.g.3019.1		6
13.8	odd	4		3380.1.h.b.339.3	✓	3
13.9	even	3		3380.1.w.f.699.4		12
13.10	even	6		3380.1.w.f.1499.4		12
13.11	odd	12		3380.1.v.g.2219.1		6
13.12	even	2	inner	3380.1.g.c.3379.3		6
20.19	odd	2	CM	3380.1.g.c.3379.6		6
52.3	odd	6		3380.1.w.e.1499.6		12
52.7	even	12		3380.1.v.d.3019.3		6
52.11	even	12		3380.1.v.d.2219.3		6
52.15	even	12		3380.1.v.f.2219.3		6
52.19	even	12		3380.1.v.f.3019.3		6
52.23	odd	6		3380.1.w.e.1499.3		12
52.31	even	4		3380.1.h.c.339.1	yes	3
52.35	odd	6		3380.1.w.e.699.3		12
52.43	odd	6		3380.1.w.e.699.6		12
52.47	even	4		3380.1.h.e.339.1	yes	3
52.51	odd	2		3380.1.g.d.3379.4		6
65.4	even	6		3380.1.w.e.699.6		12
65.9	even	6		3380.1.w.e.699.3		12
65.19	odd	12		3380.1.v.f.3019.3		6
65.24	odd	12		3380.1.v.d.2219.3		6
65.29	even	6		3380.1.w.e.1499.6		12
65.34	odd	4		3380.1.h.e.339.1	yes	3
65.44	odd	4		3380.1.h.c.339.1	yes	3
65.49	even	6		3380.1.w.e.1499.3		12
65.54	odd	12		3380.1.v.f.2219.3		6
65.59	odd	12		3380.1.v.d.3019.3		6
65.64	even	2		3380.1.g.d.3379.4		6
260.19	even	12		3380.1.v.e.3019.1		6
260.59	even	12		3380.1.v.g.3019.1		6
260.99	even	4		3380.1.h.b.339.3	✓	3
260.119	even	12		3380.1.v.e.2219.1		6
260.139	odd	6		3380.1.w.f.699.4		12
260.159	odd	6		3380.1.w.f.1499.1		12
260.179	odd	6		3380.1.w.f.1499.4		12
260.199	odd	6		3380.1.w.f.699.1		12
260.219	even	12		3380.1.v.g.2219.1		6
260.239	even	4		3380.1.h.d.339.3	yes	3
260.259	odd	2	inner	3380.1.g.c.3379.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.1.g.c.3379.3		6	13.12	even	2	inner
3380.1.g.c.3379.3		6	260.259	odd	2	inner
3380.1.g.c.3379.6		6	1.1	even	1	trivial
3380.1.g.c.3379.6		6	20.19	odd	2	CM
3380.1.g.d.3379.1		6	4.3	odd	2
3380.1.g.d.3379.1		6	5.4	even	2
3380.1.g.d.3379.4		6	52.51	odd	2
3380.1.g.d.3379.4		6	65.64	even	2
3380.1.h.b.339.3	✓	3	13.8	odd	4
3380.1.h.b.339.3	✓	3	260.99	even	4
3380.1.h.c.339.1	yes	3	52.31	even	4
3380.1.h.c.339.1	yes	3	65.44	odd	4
3380.1.h.d.339.3	yes	3	13.5	odd	4
3380.1.h.d.339.3	yes	3	260.239	even	4
3380.1.h.e.339.1	yes	3	52.47	even	4
3380.1.h.e.339.1	yes	3	65.34	odd	4
3380.1.v.d.2219.3		6	52.11	even	12
3380.1.v.d.2219.3		6	65.24	odd	12
3380.1.v.d.3019.3		6	52.7	even	12
3380.1.v.d.3019.3		6	65.59	odd	12
3380.1.v.e.2219.1		6	13.2	odd	12
3380.1.v.e.2219.1		6	260.119	even	12
3380.1.v.e.3019.1		6	13.6	odd	12
3380.1.v.e.3019.1		6	260.19	even	12
3380.1.v.f.2219.3		6	52.15	even	12
3380.1.v.f.2219.3		6	65.54	odd	12
3380.1.v.f.3019.3		6	52.19	even	12
3380.1.v.f.3019.3		6	65.19	odd	12
3380.1.v.g.2219.1		6	13.11	odd	12
3380.1.v.g.2219.1		6	260.219	even	12
3380.1.v.g.3019.1		6	13.7	odd	12
3380.1.v.g.3019.1		6	260.59	even	12
3380.1.w.e.699.3		12	52.35	odd	6
3380.1.w.e.699.3		12	65.9	even	6
3380.1.w.e.699.6		12	52.43	odd	6
3380.1.w.e.699.6		12	65.4	even	6
3380.1.w.e.1499.3		12	52.23	odd	6
3380.1.w.e.1499.3		12	65.49	even	6
3380.1.w.e.1499.6		12	52.3	odd	6
3380.1.w.e.1499.6		12	65.29	even	6
3380.1.w.f.699.1		12	13.4	even	6
3380.1.w.f.699.1		12	260.199	odd	6
3380.1.w.f.699.4		12	13.9	even	3
3380.1.w.f.699.4		12	260.139	odd	6
3380.1.w.f.1499.1		12	13.3	even	3
3380.1.w.f.1499.1		12	260.159	odd	6
3380.1.w.f.1499.4		12	13.10	even	6
3380.1.w.f.1499.4		12	260.179	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3380.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.24698 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.24698i q^{6} +1.80194i q^{7} -1.00000i q^{8} +0.554958 q^{9} -1.00000 q^{10} -1.24698 q^{12} -1.80194 q^{14} +1.24698i q^{15} +1.00000 q^{16} +0.554958i q^{18} -1.00000i q^{20} +2.24698i q^{21} +0.445042 q^{23} -1.24698i q^{24} -1.00000 q^{25} -0.554958 q^{27} -1.80194i q^{28} +1.24698 q^{29} -1.24698 q^{30} +1.00000i q^{32} -1.80194 q^{35} -0.554958 q^{36} +1.00000 q^{40} -1.80194i q^{41} -2.24698 q^{42} +0.445042 q^{43} +0.554958i q^{45} +0.445042i q^{46} +0.445042i q^{47} +1.24698 q^{48} -2.24698 q^{49} -1.00000i q^{50} -0.554958i q^{54} +1.80194 q^{56} +1.24698i q^{58} -1.24698i q^{60} -1.80194 q^{61} +1.00000i q^{63} -1.00000 q^{64} -0.445042i q^{67} +0.554958 q^{69} -1.80194i q^{70} -0.554958i q^{72} -1.24698 q^{75} +1.00000i q^{80} -1.24698 q^{81} +1.80194 q^{82} +1.24698i q^{83} -2.24698i q^{84} +0.445042i q^{86} +1.55496 q^{87} +0.445042i q^{89} -0.554958 q^{90} -0.445042 q^{92} -0.445042 q^{94} +1.24698i q^{96} -2.24698i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{4} + 4 q^{9} - 6 q^{10} + 2 q^{12} - 2 q^{14} + 6 q^{16} + 2 q^{23} - 6 q^{25} - 4 q^{27} - 2 q^{29} + 2 q^{30} - 2 q^{35} - 4 q^{36} + 6 q^{40} - 4 q^{42} + 2 q^{43} - 2 q^{48} - 4 q^{49}+ \cdots - 2 q^{94}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^4 + 4 * q^9 - 6 * q^10 + 2 * q^12 - 2 * q^14 + 6 * q^16 + 2 * q^23 - 6 * q^25 - 4 * q^27 - 2 * q^29 + 2 * q^30 - 2 * q^35 - 4 * q^36 + 6 * q^40 - 4 * q^42 + 2 * q^43 - 2 * q^48 - 4 * q^49 + 2 * q^56 - 2 * q^61 - 6 * q^64 + 4 * q^69 + 2 * q^75 + 2 * q^81 + 2 * q^82 + 10 * q^87 - 4 * q^90 - 2 * q^92 - 2 * q^94

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