LMFDB - Embedded newform 1352.1.h.a.675.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,1,Mod(675,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, 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("1352.675");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1352.h (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:	$0.674735897080$
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 7.1.2471326208.1

Embedding invariants

Embedding label			675.3
Root			$1.24698i$ of defining polynomial
Character	$\chi$	$=$	1352.675
Dual form			1352.1.h.a.675.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.24698 q^{3} -1.00000 q^{4} -1.24698i q^{6} +1.00000i q^{8} +0.554958 q^{9} +O(q^{10})$	$q-1.00000i q^{2} +1.24698 q^{3} -1.00000 q^{4} -1.24698i q^{6} +1.00000i q^{8} +0.554958 q^{9} -1.80194i q^{11} -1.24698 q^{12} +1.00000 q^{16} +1.80194 q^{17} -0.554958i q^{18} +0.445042i q^{19} -1.80194 q^{22} +1.24698i q^{24} -1.00000 q^{25} -0.554958 q^{27} -1.00000i q^{32} -2.24698i q^{33} -1.80194i q^{34} -0.554958 q^{36} +0.445042 q^{38} +0.445042i q^{41} +0.445042 q^{43} +1.80194i q^{44} +1.24698 q^{48} -1.00000 q^{49} +1.00000i q^{50} +2.24698 q^{51} +0.554958i q^{54} +0.554958i q^{57} +1.24698i q^{59} -1.00000 q^{64} -2.24698 q^{66} -1.24698i q^{67} -1.80194 q^{68} +0.554958i q^{72} +1.24698i q^{73} -1.24698 q^{75} -0.445042i q^{76} -1.24698 q^{81} +0.445042 q^{82} +0.445042i q^{83} -0.445042i q^{86} +1.80194 q^{88} +1.24698i q^{89} -1.24698i q^{96} +1.80194i q^{97} +1.00000i q^{98} -1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{4} + 4 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 6 * q^4 + 4 * q^9	$ 6 q - 2 q^{3} - 6 q^{4} + 4 q^{9} + 2 q^{12} + 6 q^{16} + 2 q^{17} - 2 q^{22} - 6 q^{25} - 4 q^{27} - 4 q^{36} + 2 q^{38} + 2 q^{43} - 2 q^{48} - 6 q^{49} + 4 q^{51} - 6 q^{64} - 4 q^{66} - 2 q^{68} + 2 q^{75} + 2 q^{81} + 2 q^{82} + 2 q^{88}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^4 + 4 * q^9 + 2 * q^12 + 6 * q^16 + 2 * q^17 - 2 * q^22 - 6 * q^25 - 4 * q^27 - 4 * q^36 + 2 * q^38 + 2 * q^43 - 2 * q^48 - 6 * q^49 + 4 * q^51 - 6 * q^64 - 4 * q^66 - 2 * q^68 + 2 * q^75 + 2 * q^81 + 2 * q^82 + 2 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1185$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	− 1.24698i	− 1.24698i
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000i	1.00000i
$9$	0.554958	0.554958
$10$	0	0
$11$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$11$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$12$	−1.24698	−1.24698
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	1.00000	1.00000
$17$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$17$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$18$	− 0.554958i	− 0.554958i
$19$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$19$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$20$	0	0
$21$	0	0
$22$	−1.80194	−1.80194
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.24698i	1.24698i
$25$	−1.00000	−1.00000
$26$	0	0
$27$	−0.554958	−0.554958
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 1.00000i
$33$	− 2.24698i	− 2.24698i
$34$	− 1.80194i	− 1.80194i
$35$	0	0
$36$	−0.554958	−0.554958
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0.445042	0.445042
$39$	0	0
$40$	0	0
$41$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$41$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$42$	0	0
$43$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$43$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$44$	1.80194i	1.80194i
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.24698	1.24698
$49$	−1.00000	−1.00000
$50$	1.00000i	1.00000i
$51$	2.24698	2.24698
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0.554958i	0.554958i
$55$	0	0
$56$	0	0
$57$	0.554958i	0.554958i
$58$	0	0
$59$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$59$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−1.00000
$65$	0	0
$66$	−2.24698	−2.24698
$67$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$67$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$68$	−1.80194	−1.80194
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0.554958i	0.554958i
$73$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$73$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$74$	0	0
$75$	−1.24698	−1.24698
$76$	− 0.445042i	− 0.445042i
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−1.24698	−1.24698
$82$	0.445042	0.445042
$83$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$83$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$84$	0	0
$85$	0	0
$86$	− 0.445042i	− 0.445042i
$87$	0	0
$88$	1.80194	1.80194
$89$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$89$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	− 1.24698i	− 1.24698i
$97$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$97$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$98$	1.00000i	1.00000i
$99$	− 1.00000i	− 1.00000i
$100$	1.00000	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	− 2.24698i	− 2.24698i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$107$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$108$	0.554958	0.554958
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$113$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$114$	0.554958	0.554958
$115$	0	0
$116$	0	0
$117$	0	0
$118$	1.24698	1.24698
$119$	0	0
$120$	0	0
$121$	−2.24698	−2.24698
$122$	0	0
$123$	0.554958i	0.554958i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000i	1.00000i
$129$	0.554958	0.554958
$130$	0	0
$131$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$131$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$132$	2.24698i	2.24698i
$133$	0	0
$134$	−1.24698	−1.24698
$135$	0	0
$136$	1.80194i	1.80194i
$137$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$137$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$138$	0	0
$139$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$139$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0.554958	0.554958
$145$	0	0
$146$	1.24698	1.24698
$147$	−1.24698	−1.24698
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	1.24698i	1.24698i
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−0.445042	−0.445042
$153$	1.00000	1.00000
$154$	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
$160$	0	0
$161$	0	0
$162$	1.24698i	1.24698i
$163$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$163$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$164$	− 0.445042i	− 0.445042i
$165$	0	0
$166$	0.445042	0.445042
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.246980i	0.246980i
$172$	−0.445042	−0.445042
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	− 1.80194i	− 1.80194i
$177$	1.55496i	1.55496i
$178$	1.24698	1.24698
$179$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$179$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.24698i	− 3.24698i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1.24698	−1.24698
$193$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$193$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$194$	1.80194	1.80194
$195$	0	0
$196$	1.00000	1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−1.00000	−1.00000
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 1.00000i	− 1.00000i
$201$	− 1.55496i	− 1.55496i
$202$	0	0
$203$	0	0
$204$	−2.24698	−2.24698
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.801938	0.801938
$210$	0	0
$211$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$211$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$212$	0	0
$213$	0	0
$214$	0.445042i	0.445042i
$215$	0	0
$216$	− 0.554958i	− 0.554958i
$217$	0	0
$218$	0	0
$219$	1.55496i	1.55496i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−0.554958	−0.554958
$226$	− 1.24698i	− 1.24698i
$227$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$227$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$228$	− 0.554958i	− 0.554958i
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$233$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$234$	0	0
$235$	0	0
$236$	− 1.24698i	− 1.24698i
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$241$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$242$	2.24698i	2.24698i
$243$	−1.00000	−1.00000
$244$	0	0
$245$	0	0
$246$	0.554958	0.554958
$247$	0	0
$248$	0	0
$249$	0.554958i	0.554958i
$250$	0	0
$251$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$251$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$257$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$258$	− 0.554958i	− 0.554958i
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0.445042i	0.445042i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	2.24698	2.24698
$265$	0	0
$266$	0	0
$267$	1.55496i	1.55496i
$268$	1.24698i	1.24698i
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	1.80194	1.80194
$273$	0	0
$274$	1.24698	1.24698
$275$	1.80194i	1.80194i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.80194i	1.80194i
$279$	0	0
$280$	0	0
$281$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$281$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$282$	0	0
$283$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$283$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 0.554958i	− 0.554958i
$289$	2.24698	2.24698
$290$	0	0
$291$	2.24698i	2.24698i
$292$	− 1.24698i	− 1.24698i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	1.24698i	1.24698i
$295$	0	0
$296$	0	0
$297$	1.00000i	1.00000i
$298$	0	0
$299$	0	0
$300$	1.24698	1.24698
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0.445042i	0.445042i
$305$	0	0
$306$	− 1.00000i	− 1.00000i
$307$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$307$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$313$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−0.554958	−0.554958
$322$	0	0
$323$	0.801938i	0.801938i
$324$	1.24698	1.24698
$325$	0	0
$326$	−1.80194	−1.80194
$327$	0	0
$328$	−0.445042	−0.445042
$329$	0	0
$330$	0	0
$331$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$331$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$332$	− 0.445042i	− 0.445042i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$337$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$338$	0	0
$339$	1.55496	1.55496
$340$	0	0
$341$	0	0
$342$	0.246980	0.246980
$343$	0	0
$344$	0.445042i	0.445042i
$345$	0	0
$346$	0	0
$347$	2.00000	2.00000	1.00000			$0$
$347$	2.00000	2.00000	1.00000			$0$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−1.80194	−1.80194
$353$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$353$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$354$	1.55496	1.55496
$355$	0	0
$356$	− 1.24698i	− 1.24698i
$357$	0	0
$358$	− 1.80194i	− 1.80194i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	0.801938	0.801938
$362$	0	0
$363$	−2.80194	−2.80194
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0.246980i	0.246980i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−3.24698	−3.24698
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$379$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.24698i	1.24698i
$385$	0	0
$386$	−0.445042	−0.445042
$387$	0.246980	0.246980
$388$	− 1.80194i	− 1.80194i
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	− 1.00000i	− 1.00000i
$393$	−0.554958	−0.554958
$394$	0	0
$395$	0	0
$396$	1.00000i	1.00000i
$397$	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	$-0.642857\pi$
$401$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$402$	−1.55496	−1.55496
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	2.24698i	2.24698i
$409$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$409$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$410$	0	0
$411$	1.55496i	1.55496i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.24698	−2.24698
$418$	− 0.801938i	− 0.801938i
$419$	2.00000	2.00000	1.00000			$0$
$419$	2.00000	2.00000	1.00000			$0$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	1.80194i	1.80194i
$423$	0	0
$424$	0	0
$425$	−1.80194	−1.80194
$426$	0	0
$427$	0	0
$428$	0.445042	0.445042
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−0.554958	−0.554958
$433$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$433$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.55496	1.55496
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−0.554958	−0.554958
$442$	0	0
$443$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$443$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$449$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$450$	0.554958i	0.554958i
$451$	0.801938	0.801938
$452$	−1.24698	−1.24698
$453$	0	0
$454$	−1.24698	−1.24698
$455$	0	0
$456$	−0.554958	−0.554958
$457$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$457$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$458$	0	0
$459$	−1.00000	−1.00000
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	1.24698i	1.24698i
$467$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$467$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.24698	−1.24698
$473$	− 0.801938i	− 0.801938i
$474$	0	0
$475$	− 0.445042i	− 0.445042i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	2.00000	2.00000
$483$	0	0
$484$	2.24698	2.24698
$485$	0	0
$486$	1.00000i	1.00000i
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	− 2.24698i	− 2.24698i
$490$	0	0
$491$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$491$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$492$	− 0.554958i	− 0.554958i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0.554958	0.554958
$499$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$499$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$500$	0	0
$501$	0	0
$502$	− 1.80194i	− 1.80194i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 1.00000i
$513$	− 0.246980i	− 0.246980i
$514$	− 0.445042i	− 0.445042i
$515$	0	0
$516$	−0.554958	−0.554958
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$521$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$522$	0	0
$523$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$523$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$524$	0.445042	0.445042
$525$	0	0
$526$	0	0
$527$	0	0
$528$	− 2.24698i	− 2.24698i
$529$	1.00000	1.00000
$530$	0	0
$531$	0.692021i	0.692021i
$532$	0	0
$533$	0	0
$534$	1.55496	1.55496
$535$	0	0
$536$	1.24698	1.24698
$537$	2.24698	2.24698
$538$	0	0
$539$	1.80194i	1.80194i
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	− 1.80194i	− 1.80194i
$545$	0	0
$546$	0	0
$547$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$547$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$548$	− 1.24698i	− 1.24698i
$549$	0	0
$550$	1.80194	1.80194
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	1.80194	1.80194
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 4.04892i	− 4.04892i
$562$	−1.80194	−1.80194
$563$	−2.00000	−2.00000	−1.00000			$\pi$
$563$	−2.00000	−2.00000	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	− 0.445042i	− 0.445042i
$567$	0	0
$568$	0	0
$569$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$569$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$570$	0	0
$571$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$571$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−0.554958	−0.554958
$577$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$577$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$578$	− 2.24698i	− 2.24698i
$579$	− 0.554958i	− 0.554958i
$580$	0	0
$581$	0	0
$582$	2.24698	2.24698
$583$	0	0
$584$	−1.24698	−1.24698
$585$	0	0
$586$	0	0
$587$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$587$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$588$	1.24698	1.24698
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$593$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$594$	1.00000	1.00000
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	− 1.24698i	− 1.24698i
$601$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$601$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$602$	0	0
$603$	− 0.692021i	− 0.692021i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0.445042	0.445042
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−1.00000	−1.00000
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−1.80194	−1.80194
$615$	0	0
$616$	0	0
$617$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$617$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$618$	0	0
$619$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$619$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	1.80194i	1.80194i
$627$	1.00000	1.00000
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−2.24698	−2.24698
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$641$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$642$	0.554958i	0.554958i
$643$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$643$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$644$	0	0
$645$	0	0
$646$	0.801938	0.801938
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	− 1.24698i	− 1.24698i
$649$	2.24698	2.24698
$650$	0	0
$651$	0	0
$652$	1.80194i	1.80194i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0.445042i	0.445042i
$657$	0.692021i	0.692021i
$658$	0	0
$659$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$659$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	−2.00000	−2.00000
$663$	0	0
$664$	−0.445042	−0.445042
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$673$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$674$	− 0.445042i	− 0.445042i
$675$	0.554958	0.554958
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	− 1.55496i	− 1.55496i
$679$	0	0
$680$	0	0
$681$	− 1.55496i	− 1.55496i
$682$	0	0
$683$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$683$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$684$	− 0.246980i	− 0.246980i
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0.445042	0.445042
$689$	0	0
$690$	0	0
$691$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$691$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$692$	0	0
$693$	0	0
$694$	− 2.00000i	− 2.00000i
$695$	0	0
$696$	0	0
$697$	0.801938i	0.801938i
$698$	0	0
$699$	−1.55496	−1.55496
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	1.80194i	1.80194i
$705$	0	0
$706$	1.80194	1.80194
$707$	0	0
$708$	− 1.55496i	− 1.55496i
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	0	0
$712$	−1.24698	−1.24698
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−1.80194	−1.80194
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	− 0.801938i	− 0.801938i
$723$	2.49396i	2.49396i
$724$	0	0
$725$	0	0
$726$	2.80194i	2.80194i
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.801938	0.801938
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.24698	−2.24698
$738$	0.246980	0.246980
$739$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$739$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.246980i	0.246980i
$748$	3.24698i	3.24698i
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	2.24698	2.24698
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−1.24698	−1.24698
$759$	0	0
$760$	0	0
$761$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$761$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.24698	1.24698
$769$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$769$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$770$	0	0
$771$	0.554958	0.554958
$772$	0.445042i	0.445042i
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	− 0.246980i	− 0.246980i
$775$	0	0
$776$	−1.80194	−1.80194
$777$	0	0
$778$	0	0
$779$	−0.198062	−0.198062
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.00000	−1.00000
$785$	0	0
$786$	0.554958i	0.554958i
$787$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$787$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	1.00000	1.00000
$793$	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
$801$	0.692021i	0.692021i
$802$	−1.80194	−1.80194
$803$	2.24698	2.24698
$804$	1.55496i	1.55496i
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$809$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$810$	0	0
$811$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$811$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	2.24698	2.24698
$817$	0.198062i	0.198062i
$818$	−1.24698	−1.24698
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	1.55496	1.55496
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	2.24698i	2.24698i
$826$	0	0
$827$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$827$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.80194	−1.80194
$834$	2.24698i	2.24698i
$835$	0	0
$836$	−0.801938	−0.801938
$837$	0	0
$838$	− 2.00000i	− 2.00000i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	− 2.24698i	− 2.24698i
$844$	1.80194	1.80194
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.554958	0.554958
$850$	1.80194i	1.80194i
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	− 0.445042i	− 0.445042i
$857$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$857$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$858$	0	0
$859$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$859$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0.554958i	0.554958i
$865$	0	0
$866$	1.24698i	1.24698i
$867$	2.80194	2.80194
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.00000i	1.00000i
$874$	0	0
$875$	0	0
$876$	− 1.55496i	− 1.55496i
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$881$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$882$	0.554958i	0.554958i
$883$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$883$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$884$	0	0
$885$	0	0
$886$	1.80194i	1.80194i
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.24698i	2.24698i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−0.445042	−0.445042
$899$	0	0
$900$	0.554958	0.554958
$901$	0	0
$902$	− 0.801938i	− 0.801938i
$903$	0	0
$904$	1.24698i	1.24698i
$905$	0	0
$906$	0	0
$907$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$907$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$908$	1.24698i	1.24698i
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.554958i	0.554958i
$913$	0.801938	0.801938
$914$	−2.00000	−2.00000
$915$	0	0
$916$	0	0
$917$	0	0
$918$	1.00000i	1.00000i
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	− 2.24698i	− 2.24698i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$929$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$930$	0	0
$931$	− 0.445042i	− 0.445042i
$932$	1.24698	1.24698
$933$	0	0
$934$	1.24698i	1.24698i
$935$	0	0
$936$	0	0
$937$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$937$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$938$	0	0
$939$	−2.24698	−2.24698
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	1.24698i	1.24698i
$945$	0	0
$946$	−0.801938	−0.801938
$947$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$947$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$948$	0	0
$949$	0	0
$950$	−0.445042	−0.445042
$951$	0	0
$952$	0	0
$953$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$953$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.00000	−1.00000
$962$	0	0
$963$	−0.246980	−0.246980
$964$	− 2.00000i	− 2.00000i
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	− 2.24698i	− 2.24698i
$969$	1.00000i	1.00000i
$970$	0	0
$971$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$971$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$972$	1.00000	1.00000
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$977$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$978$	−2.24698	−2.24698
$979$	2.24698	2.24698
$980$	0	0
$981$	0	0
$982$	1.24698i	1.24698i
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−0.554958	−0.554958
$985$	0	0
$986$	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	0
$993$	− 2.49396i	− 2.49396i
$994$	0	0
$995$	0	0
$996$	− 0.554958i	− 0.554958i
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.80194	1.80194
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.1.h.a.675.3		6
8.3	odd	2	CM	1352.1.h.a.675.3		6
13.2	odd	12		1352.1.n.c.867.1		6
13.3	even	3		1352.1.p.c.147.4		12
13.4	even	6		1352.1.p.c.699.4		12
13.5	odd	4		1352.1.g.b.339.3	✓	3
13.6	odd	12		1352.1.n.c.315.1		6
13.7	odd	12		1352.1.n.b.315.1		6
13.8	odd	4		1352.1.g.c.339.3	yes	3
13.9	even	3		1352.1.p.c.699.1		12
13.10	even	6		1352.1.p.c.147.1		12
13.11	odd	12		1352.1.n.b.867.1		6
13.12	even	2	inner	1352.1.h.a.675.6		6
104.3	odd	6		1352.1.p.c.147.4		12
104.11	even	12		1352.1.n.b.867.1		6
104.19	even	12		1352.1.n.c.315.1		6
104.35	odd	6		1352.1.p.c.699.1		12
104.43	odd	6		1352.1.p.c.699.4		12
104.51	odd	2	inner	1352.1.h.a.675.6		6
104.59	even	12		1352.1.n.b.315.1		6
104.67	even	12		1352.1.n.c.867.1		6
104.75	odd	6		1352.1.p.c.147.1		12
104.83	even	4		1352.1.g.b.339.3	✓	3
104.99	even	4		1352.1.g.c.339.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1352.1.g.b.339.3	✓	3	13.5	odd	4
1352.1.g.b.339.3	✓	3	104.83	even	4
1352.1.g.c.339.3	yes	3	13.8	odd	4
1352.1.g.c.339.3	yes	3	104.99	even	4
1352.1.h.a.675.3		6	1.1	even	1	trivial
1352.1.h.a.675.3		6	8.3	odd	2	CM
1352.1.h.a.675.6		6	13.12	even	2	inner
1352.1.h.a.675.6		6	104.51	odd	2	inner
1352.1.n.b.315.1		6	13.7	odd	12
1352.1.n.b.315.1		6	104.59	even	12
1352.1.n.b.867.1		6	13.11	odd	12
1352.1.n.b.867.1		6	104.11	even	12
1352.1.n.c.315.1		6	13.6	odd	12
1352.1.n.c.315.1		6	104.19	even	12
1352.1.n.c.867.1		6	13.2	odd	12
1352.1.n.c.867.1		6	104.67	even	12
1352.1.p.c.147.1		12	13.10	even	6
1352.1.p.c.147.1		12	104.75	odd	6
1352.1.p.c.147.4		12	13.3	even	3
1352.1.p.c.147.4		12	104.3	odd	6
1352.1.p.c.699.1		12	13.9	even	3
1352.1.p.c.699.1		12	104.35	odd	6
1352.1.p.c.699.4		12	13.4	even	6
1352.1.p.c.699.4		12	104.43	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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1352.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.24698 q^{3} -1.00000 q^{4} -1.24698i q^{6} +1.00000i q^{8} +0.554958 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.24698 q^{3} -1.00000 q^{4} -1.24698i q^{6} +1.00000i q^{8} +0.554958 q^{9} -1.80194i q^{11} -1.24698 q^{12} +1.00000 q^{16} +1.80194 q^{17} -0.554958i q^{18} +0.445042i q^{19} -1.80194 q^{22} +1.24698i q^{24} -1.00000 q^{25} -0.554958 q^{27} -1.00000i q^{32} -2.24698i q^{33} -1.80194i q^{34} -0.554958 q^{36} +0.445042 q^{38} +0.445042i q^{41} +0.445042 q^{43} +1.80194i q^{44} +1.24698 q^{48} -1.00000 q^{49} +1.00000i q^{50} +2.24698 q^{51} +0.554958i q^{54} +0.554958i q^{57} +1.24698i q^{59} -1.00000 q^{64} -2.24698 q^{66} -1.24698i q^{67} -1.80194 q^{68} +0.554958i q^{72} +1.24698i q^{73} -1.24698 q^{75} -0.445042i q^{76} -1.24698 q^{81} +0.445042 q^{82} +0.445042i q^{83} -0.445042i q^{86} +1.80194 q^{88} +1.24698i q^{89} -1.24698i q^{96} +1.80194i q^{97} +1.00000i q^{98} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{4} + 4 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 6 * q^4 + 4 * q^9	\( 6 q - 2 q^{3} - 6 q^{4} + 4 q^{9} + 2 q^{12} + 6 q^{16} + 2 q^{17} - 2 q^{22} - 6 q^{25} - 4 q^{27} - 4 q^{36} + 2 q^{38} + 2 q^{43} - 2 q^{48} - 6 q^{49} + 4 q^{51} - 6 q^{64} - 4 q^{66} - 2 q^{68} + 2 q^{75} + 2 q^{81} + 2 q^{82} + 2 q^{88}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^4 + 4 * q^9 + 2 * q^12 + 6 * q^16 + 2 * q^17 - 2 * q^22 - 6 * q^25 - 4 * q^27 - 4 * q^36 + 2 * q^38 + 2 * q^43 - 2 * q^48 - 6 * q^49 + 4 * q^51 - 6 * q^64 - 4 * q^66 - 2 * q^68 + 2 * q^75 + 2 * q^81 + 2 * q^82 + 2 * q^88

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