LMFDB - Embedded newform 2535.1.f.b.1184.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2535,1,Mod(1184,2535)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2535, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("2535.1184");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2535 = 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2535.f (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$1.26512980702$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.16290480375.1
Artin image:	$D_7$
Artin field:	Galois closure of 7.1.16290480375.1

Embedding invariants

Embedding label			1184.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	2535.1184

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.445042 q^{2} +1.00000 q^{3} -0.801938 q^{4} +1.00000 q^{5} -0.445042 q^{6} +0.801938 q^{8} +1.00000 q^{9} +O(q^{10})$	$q-0.445042 q^{2} +1.00000 q^{3} -0.801938 q^{4} +1.00000 q^{5} -0.445042 q^{6} +0.801938 q^{8} +1.00000 q^{9} -0.445042 q^{10} -0.801938 q^{12} +1.00000 q^{15} +0.445042 q^{16} +1.24698 q^{17} -0.445042 q^{18} -1.80194 q^{19} -0.801938 q^{20} -0.445042 q^{23} +0.801938 q^{24} +1.00000 q^{25} +1.00000 q^{27} -0.445042 q^{30} +1.24698 q^{31} -1.00000 q^{32} -0.554958 q^{34} -0.801938 q^{36} +0.801938 q^{38} +0.801938 q^{40} +1.00000 q^{45} +0.198062 q^{46} -1.80194 q^{47} +0.445042 q^{48} +1.00000 q^{49} -0.445042 q^{50} +1.24698 q^{51} -1.80194 q^{53} -0.445042 q^{54} -1.80194 q^{57} -0.801938 q^{60} +1.24698 q^{61} -0.554958 q^{62} -1.00000 q^{68} -0.445042 q^{69} +0.801938 q^{72} +1.00000 q^{75} +1.44504 q^{76} -0.445042 q^{79} +0.445042 q^{80} +1.00000 q^{81} +1.24698 q^{83} +1.24698 q^{85} -0.445042 q^{90} +0.356896 q^{92} +1.24698 q^{93} +0.801938 q^{94} -1.80194 q^{95} -1.00000 q^{96} -0.445042 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 + 3 * q^3 + 2 * q^4 + 3 * q^5 - q^6 - 2 * q^8 + 3 * q^9	$ 3 q - q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 2 q^{8} + 3 q^{9} - q^{10} + 2 q^{12} + 3 q^{15} + q^{16} - q^{17} - q^{18} - q^{19} + 2 q^{20} - q^{23} - 2 q^{24} + 3 q^{25} + 3 q^{27} - q^{30} - q^{31} - 3 q^{32} - 2 q^{34} + 2 q^{36} - 2 q^{38} - 2 q^{40} + 3 q^{45} + 5 q^{46} - q^{47} + q^{48} + 3 q^{49} - q^{50} - q^{51} - q^{53} - q^{54} - q^{57} + 2 q^{60} - q^{61} - 2 q^{62} - 3 q^{68} - q^{69} - 2 q^{72} + 3 q^{75} + 4 q^{76} - q^{79} + q^{80} + 3 q^{81} - q^{83} - q^{85} - q^{90} - 3 q^{92} - q^{93} - 2 q^{94} - q^{95} - 3 q^{96} - q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + 2 * q^4 + 3 * q^5 - q^6 - 2 * q^8 + 3 * q^9 - q^10 + 2 * q^12 + 3 * q^15 + q^16 - q^17 - q^18 - q^19 + 2 * q^20 - q^23 - 2 * q^24 + 3 * q^25 + 3 * q^27 - q^30 - q^31 - 3 * q^32 - 2 * q^34 + 2 * q^36 - 2 * q^38 - 2 * q^40 + 3 * q^45 + 5 * q^46 - q^47 + q^48 + 3 * q^49 - q^50 - q^51 - q^53 - q^54 - q^57 + 2 * q^60 - q^61 - 2 * q^62 - 3 * q^68 - q^69 - 2 * q^72 + 3 * q^75 + 4 * q^76 - q^79 + q^80 + 3 * q^81 - q^83 - q^85 - q^90 - 3 * q^92 - q^93 - 2 * q^94 - q^95 - 3 * q^96 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2535\mathbb{Z}\right)^\times$.

$n$	$1522$	$1691$	$1861$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$2$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$3$	1.00000	1.00000
$3$	1.00000	1.00000
$4$	−0.801938	−0.801938
$5$	1.00000	1.00000
$5$	1.00000	1.00000
$6$	−0.445042	−0.445042
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0.801938	0.801938
$9$	1.00000	1.00000
$10$	−0.445042	−0.445042
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−0.801938	−0.801938
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.00000	1.00000
$16$	0.445042	0.445042
$17$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$17$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$18$	−0.445042	−0.445042
$19$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$19$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$20$	−0.801938	−0.801938
$21$	0	0
$22$	0	0
$23$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$23$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$24$	0.801938	0.801938
$25$	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			$\pi$
$30$	−0.445042	−0.445042
$31$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$31$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$32$	−1.00000	−1.00000
$33$	0	0
$34$	−0.554958	−0.554958
$35$	0	0
$36$	−0.801938	−0.801938
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0.801938	0.801938
$39$	0	0
$40$	0.801938	0.801938
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	1.00000	1.00000
$46$	0.198062	0.198062
$47$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$47$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$48$	0.445042	0.445042
$49$	1.00000	1.00000
$50$	−0.445042	−0.445042
$51$	1.24698	1.24698
$52$	0	0
$53$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$53$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$54$	−0.445042	−0.445042
$55$	0	0
$56$	0	0
$57$	−1.80194	−1.80194
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−0.801938	−0.801938
$61$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$61$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$62$	−0.554958	−0.554958
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.00000	−1.00000
$69$	−0.445042	−0.445042
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.801938	0.801938
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	1.00000	1.00000
$76$	1.44504	1.44504
$77$	0	0
$78$	0	0
$79$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$79$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$80$	0.445042	0.445042
$81$	1.00000	1.00000
$82$	0	0
$83$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$83$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$84$	0	0
$85$	1.24698	1.24698
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−0.445042	−0.445042
$91$	0	0
$92$	0.356896	0.356896
$93$	1.24698	1.24698
$94$	0.801938	0.801938
$95$	−1.80194	−1.80194
$96$	−1.00000	−1.00000
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−0.445042	−0.445042
$99$	0	0
$100$	−0.801938	−0.801938
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−0.554958	−0.554958
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0.801938	0.801938
$107$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$107$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$108$	−0.801938	−0.801938
$109$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$109$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$113$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$114$	0.801938	0.801938
$115$	−0.445042	−0.445042
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0.801938	0.801938
$121$	1.00000	1.00000
$122$	−0.554958	−0.554958
$123$	0	0
$124$	−1.00000	−1.00000
$125$	1.00000	1.00000
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	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
$135$	1.00000	1.00000
$136$	1.00000	1.00000
$137$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$137$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$138$	0.198062	0.198062
$139$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$139$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$140$	0	0
$141$	−1.80194	−1.80194
$142$	0	0
$143$	0	0
$144$	0.445042	0.445042
$145$	0	0
$146$	0	0
$147$	1.00000	1.00000
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−0.445042	−0.445042
$151$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$151$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$152$	−1.44504	−1.44504
$153$	1.24698	1.24698
$154$	0	0
$155$	1.24698	1.24698
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0.198062	0.198062
$159$	−1.80194	−1.80194
$160$	−1.00000	−1.00000
$161$	0	0
$162$	−0.445042	−0.445042
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−0.554958	−0.554958
$167$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$167$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$168$	0	0
$169$	0	0
$170$	−0.554958	−0.554958
$171$	−1.80194	−1.80194
$172$	0	0
$173$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$173$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.801938	−0.801938
$181$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$181$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$182$	0	0
$183$	1.24698	1.24698
$184$	−0.356896	−0.356896
$185$	0	0
$186$	−0.554958	−0.554958
$187$	0	0
$188$	1.44504	1.44504
$189$	0	0
$190$	0.801938	0.801938
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−0.801938	−0.801938
$197$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$197$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$198$	0	0
$199$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$199$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$200$	0.801938	0.801938
$201$	0	0
$202$	0	0
$203$	0	0
$204$	−1.00000	−1.00000
$205$	0	0
$206$	0	0
$207$	−0.445042	−0.445042
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$211$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$212$	1.44504	1.44504
$213$	0	0
$214$	0.801938	0.801938
$215$	0	0
$216$	0.801938	0.801938
$217$	0	0
$218$	0.198062	0.198062
$219$	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
$226$	0.198062	0.198062
$227$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$227$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$228$	1.44504	1.44504
$229$	2.00000	2.00000	1.00000			$0$
$229$	2.00000	2.00000	1.00000			$0$
$230$	0.198062	0.198062
$231$	0	0
$232$	0	0
$233$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$233$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$234$	0	0
$235$	−1.80194	−1.80194
$236$	0	0
$237$	−0.445042	−0.445042
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0.445042	0.445042
$241$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$241$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$242$	−0.445042	−0.445042
$243$	1.00000	1.00000
$244$	−1.00000	−1.00000
$245$	1.00000	1.00000
$246$	0	0
$247$	0	0
$248$	1.00000	1.00000
$249$	1.24698	1.24698
$250$	−0.445042	−0.445042
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.24698	1.24698
$256$	−0.445042	−0.445042
$257$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$257$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$263$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$264$	0	0
$265$	−1.80194	−1.80194
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.445042	−0.445042
$271$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$271$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$272$	0.554958	0.554958
$273$	0	0
$274$	−0.554958	−0.554958
$275$	0	0
$276$	0.356896	0.356896
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0.198062	0.198062
$279$	1.24698	1.24698
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0.801938	0.801938
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	−1.80194	−1.80194
$286$	0	0
$287$	0	0
$288$	−1.00000	−1.00000
$289$	0.554958	0.554958
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$293$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$294$	−0.445042	−0.445042
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−0.801938	−0.801938
$301$	0	0
$302$	0.198062	0.198062
$303$	0	0
$304$	−0.801938	−0.801938
$305$	1.24698	1.24698
$306$	−0.554958	−0.554958
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	−0.554958	−0.554958
$311$	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
$315$	0	0
$316$	0.356896	0.356896
$317$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$317$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$318$	0.801938	0.801938
$319$	0	0
$320$	0	0
$321$	−1.80194	−1.80194
$322$	0	0
$323$	−2.24698	−2.24698
$324$	−0.801938	−0.801938
$325$	0	0
$326$	0	0
$327$	−0.445042	−0.445042
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$331$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$332$	−1.00000	−1.00000
$333$	0	0
$334$	−0.554958	−0.554958
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	−0.445042	−0.445042
$340$	−1.00000	−1.00000
$341$	0	0
$342$	0.801938	0.801938
$343$	0	0
$344$	0	0
$345$	−0.445042	−0.445042
$346$	0.198062	0.198062
$347$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$347$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$348$	0	0
$349$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$349$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$353$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0.801938	0.801938
$361$	2.24698	2.24698
$362$	0.801938	0.801938
$363$	1.00000	1.00000
$364$	0	0
$365$	0	0
$366$	−0.554958	−0.554958
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−0.198062	−0.198062
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−1.00000	−1.00000
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	1.00000	1.00000
$376$	−1.44504	−1.44504
$377$	0	0
$378$	0	0
$379$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$379$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$380$	1.44504	1.44504
$381$	0	0
$382$	0	0
$383$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$383$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$384$	1.00000	1.00000
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	−0.554958	−0.554958
$392$	0.801938	0.801938
$393$	0	0
$394$	0.801938	0.801938
$395$	−0.445042	−0.445042
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−0.554958	−0.554958
$399$	0	0
$400$	0.445042	0.445042
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	1.00000
$406$	0	0
$407$	0	0
$408$	1.00000	1.00000
$409$	2.00000	2.00000	1.00000			$0$
$409$	2.00000	2.00000	1.00000			$0$
$410$	0	0
$411$	1.24698	1.24698
$412$	0	0
$413$	0	0
$414$	0.198062	0.198062
$415$	1.24698	1.24698
$416$	0	0
$417$	−0.445042	−0.445042
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$421$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$422$	0.801938	0.801938
$423$	−1.80194	−1.80194
$424$	−1.44504	−1.44504
$425$	1.24698	1.24698
$426$	0	0
$427$	0	0
$428$	1.44504	1.44504
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.445042	0.445042
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0.356896	0.356896
$437$	0.801938	0.801938
$438$	0	0
$439$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$439$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	0	0
$443$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$443$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$444$	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			$\pi$
$450$	−0.445042	−0.445042
$451$	0	0
$452$	0.356896	0.356896
$453$	−0.445042	−0.445042
$454$	0.801938	0.801938
$455$	0	0
$456$	−1.44504	−1.44504
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−0.890084	−0.890084
$459$	1.24698	1.24698
$460$	0.356896	0.356896
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	1.24698	1.24698
$466$	0.801938	0.801938
$467$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$467$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$468$	0	0
$469$	0	0
$470$	0.801938	0.801938
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0.198062	0.198062
$475$	−1.80194	−1.80194
$476$	0	0
$477$	−1.80194	−1.80194
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−1.00000	−1.00000
$481$	0	0
$482$	0.198062	0.198062
$483$	0	0
$484$	−0.801938	−0.801938
$485$	0	0
$486$	−0.445042	−0.445042
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	1.00000	1.00000
$489$	0	0
$490$	−0.445042	−0.445042
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0.554958	0.554958
$497$	0	0
$498$	−0.554958	−0.554958
$499$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$499$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$500$	−0.801938	−0.801938
$501$	1.24698	1.24698
$502$	0	0
$503$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$503$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−0.554958	−0.554958
$511$	0	0
$512$	−0.801938	−0.801938
$513$	−1.80194	−1.80194
$514$	0.801938	0.801938
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−0.445042	−0.445042
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	−0.554958	−0.554958
$527$	1.55496	1.55496
$528$	0	0
$529$	−0.801938	−0.801938
$530$	0.801938	0.801938
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1.80194	−1.80194
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	−0.801938	−0.801938
$541$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$541$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$542$	0.198062	0.198062
$543$	−1.80194	−1.80194
$544$	−1.24698	−1.24698
$545$	−0.445042	−0.445042
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−1.00000	−1.00000
$549$	1.24698	1.24698
$550$	0	0
$551$	0	0
$552$	−0.356896	−0.356896
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0.356896	0.356896
$557$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$557$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$558$	−0.554958	−0.554958
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$563$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$564$	1.44504	1.44504
$565$	−0.445042	−0.445042
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0.801938	0.801938
$571$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$571$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.445042	−0.445042
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−0.246980	−0.246980
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0.801938	0.801938
$587$	2.00000	2.00000	1.00000			$0$
$587$	2.00000	2.00000	1.00000			$0$
$588$	−0.801938	−0.801938
$589$	−2.24698	−2.24698
$590$	0	0
$591$	−1.80194	−1.80194
$592$	0	0
$593$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$593$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.24698	1.24698
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0.801938	0.801938
$601$	−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	0
$604$	0.356896	0.356896
$605$	1.00000	1.00000
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.80194	1.80194
$609$	0	0
$610$	−0.554958	−0.554958
$611$	0	0
$612$	−1.00000	−1.00000
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$617$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$618$	0	0
$619$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$619$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$620$	−1.00000	−1.00000
$621$	−0.445042	−0.445042
$622$	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
$631$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$631$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$632$	−0.356896	−0.356896
$633$	−1.80194	−1.80194
$634$	0.198062	0.198062
$635$	0	0
$636$	1.44504	1.44504
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.00000	1.00000
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0.801938	0.801938
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	1.00000	1.00000
$647$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$647$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$648$	0.801938	0.801938
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$653$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$654$	0.198062	0.198062
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$661$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$662$	−0.554958	−0.554958
$663$	0	0
$664$	1.00000	1.00000
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−1.00000	−1.00000
$669$	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	0
$677$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$677$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$678$	0.198062	0.198062
$679$	0	0
$680$	1.00000	1.00000
$681$	−1.80194	−1.80194
$682$	0	0
$683$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$683$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$684$	1.44504	1.44504
$685$	1.24698	1.24698
$686$	0	0
$687$	2.00000	2.00000
$688$	0	0
$689$	0	0
$690$	0.198062	0.198062
$691$	2.00000	2.00000	1.00000			$0$
$691$	2.00000	2.00000	1.00000			$0$
$692$	0.356896	0.356896
$693$	0	0
$694$	−0.554958	−0.554958
$695$	−0.445042	−0.445042
$696$	0	0
$697$	0	0
$698$	0.801938	0.801938
$699$	−1.80194	−1.80194
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−1.80194	−1.80194
$706$	0.198062	0.198062
$707$	0	0
$708$	0	0
$709$	2.00000	2.00000	1.00000			$0$
$709$	2.00000	2.00000	1.00000			$0$
$710$	0	0
$711$	−0.445042	−0.445042
$712$	0	0
$713$	−0.554958	−0.554958
$714$	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			$\pi$
$720$	0.445042	0.445042
$721$	0	0
$722$	−1.00000	−1.00000
$723$	−0.445042	−0.445042
$724$	1.44504	1.44504
$725$	0	0
$726$	−0.445042	−0.445042
$727$	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
$732$	−1.00000	−1.00000
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	1.00000	1.00000
$736$	0.445042	0.445042
$737$	0	0
$738$	0	0
$739$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$739$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$743$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$744$	1.00000	1.00000
$745$	0	0
$746$	0	0
$747$	1.24698	1.24698
$748$	0	0
$749$	0	0
$750$	−0.445042	−0.445042
$751$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$751$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$752$	−0.801938	−0.801938
$753$	0	0
$754$	0	0
$755$	−0.445042	−0.445042
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0.198062	0.198062
$759$	0	0
$760$	−1.44504	−1.44504
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.24698	1.24698
$766$	0.198062	0.198062
$767$	0	0
$768$	−0.445042	−0.445042
$769$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$769$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$770$	0	0
$771$	−1.80194	−1.80194
$772$	0	0
$773$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$773$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$774$	0	0
$775$	1.24698	1.24698
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0.246980	0.246980
$783$	0	0
$784$	0.445042	0.445042
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	1.44504	1.44504
$789$	1.24698	1.24698
$790$	0.198062	0.198062
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1.80194	−1.80194
$796$	−1.00000	−1.00000
$797$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$797$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$798$	0	0
$799$	−2.24698	−2.24698
$800$	−1.00000	−1.00000
$801$	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			$\pi$
$810$	−0.445042	−0.445042
$811$	2.00000	2.00000	1.00000			$0$
$811$	2.00000	2.00000	1.00000			$0$
$812$	0	0
$813$	−0.445042	−0.445042
$814$	0	0
$815$	0	0
$816$	0.554958	0.554958
$817$	0	0
$818$	−0.890084	−0.890084
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−0.554958	−0.554958
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.00000	2.00000	1.00000			$0$
$827$	2.00000	2.00000	1.00000			$0$
$828$	0.356896	0.356896
$829$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$829$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$830$	−0.554958	−0.554958
$831$	0	0
$832$	0	0
$833$	1.24698	1.24698
$834$	0.198062	0.198062
$835$	1.24698	1.24698
$836$	0	0
$837$	1.24698	1.24698
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0.198062	0.198062
$843$	0	0
$844$	1.44504	1.44504
$845$	0	0
$846$	0.801938	0.801938
$847$	0	0
$848$	−0.801938	−0.801938
$849$	0	0
$850$	−0.554958	−0.554958
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	−1.80194	−1.80194
$856$	−1.44504	−1.44504
$857$	2.00000	2.00000	1.00000			$0$
$857$	2.00000	2.00000	1.00000			$0$
$858$	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
$863$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$863$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$864$	−1.00000	−1.00000
$865$	−0.445042	−0.445042
$866$	0	0
$867$	0.554958	0.554958
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−0.356896	−0.356896
$873$	0	0
$874$	−0.356896	−0.356896
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0.801938	0.801938
$879$	−1.80194	−1.80194
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−0.445042	−0.445042
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0.198062	0.198062
$887$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$887$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.24698	3.24698
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−0.801938	−0.801938
$901$	−2.24698	−2.24698
$902$	0	0
$903$	0	0
$904$	−0.356896	−0.356896
$905$	−1.80194	−1.80194
$906$	0.198062	0.198062
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	1.44504	1.44504
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−0.801938	−0.801938
$913$	0	0
$914$	0	0
$915$	1.24698	1.24698
$916$	−1.60388	−1.60388
$917$	0	0
$918$	−0.554958	−0.554958
$919$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$919$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$920$	−0.356896	−0.356896
$921$	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	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	−0.554958	−0.554958
$931$	−1.80194	−1.80194
$932$	1.44504	1.44504
$933$	0	0
$934$	0.801938	0.801938
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	1.44504	1.44504
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$947$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$948$	0.356896	0.356896
$949$	0	0
$950$	0.801938	0.801938
$951$	−0.445042	−0.445042
$952$	0	0
$953$	2.00000	2.00000	1.00000			$0$
$953$	2.00000	2.00000	1.00000			$0$
$954$	0.801938	0.801938
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	0.554958	0.554958
$962$	0	0
$963$	−1.80194	−1.80194
$964$	0.356896	0.356896
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.801938	0.801938
$969$	−2.24698	−2.24698
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.801938	−0.801938
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0.554958	0.554958
$977$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$977$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$978$	0	0
$979$	0	0
$980$	−0.801938	−0.801938
$981$	−0.445042	−0.445042
$982$	0	0
$983$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$983$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$984$	0	0
$985$	−1.80194	−1.80194
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$991$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$992$	−1.24698	−1.24698
$993$	1.24698	1.24698
$994$	0	0
$995$	1.24698	1.24698
$996$	−1.00000	−1.00000
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−0.554958	−0.554958
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2535.1.f.b.1184.2	yes	3
3.2	odd	2		2535.1.f.c.1184.2	yes	3
5.4	even	2		2535.1.f.c.1184.2	yes	3
13.2	odd	12		2535.1.y.b.1499.3		12
13.3	even	3		2535.1.x.c.2219.2		6
13.4	even	6		2535.1.x.a.2174.2		6
13.5	odd	4		2535.1.e.b.2534.4		6
13.6	odd	12		2535.1.y.b.1544.4		12
13.7	odd	12		2535.1.y.b.1544.3		12
13.8	odd	4		2535.1.e.b.2534.3		6
13.9	even	3		2535.1.x.c.2174.2		6
13.10	even	6		2535.1.x.a.2219.2		6
13.11	odd	12		2535.1.y.b.1499.4		12
13.12	even	2		2535.1.f.d.1184.2	yes	3
15.14	odd	2	CM	2535.1.f.b.1184.2	yes	3
39.2	even	12		2535.1.y.c.1499.4		12
39.5	even	4		2535.1.e.a.2534.3		6
39.8	even	4		2535.1.e.a.2534.4		6
39.11	even	12		2535.1.y.c.1499.3		12
39.17	odd	6		2535.1.x.d.2174.2		6
39.20	even	12		2535.1.y.c.1544.4		12
39.23	odd	6		2535.1.x.d.2219.2		6
39.29	odd	6		2535.1.x.b.2219.2		6
39.32	even	12		2535.1.y.c.1544.3		12
39.35	odd	6		2535.1.x.b.2174.2		6
39.38	odd	2		2535.1.f.a.1184.2	✓	3
65.4	even	6		2535.1.x.d.2174.2		6
65.9	even	6		2535.1.x.b.2174.2		6
65.19	odd	12		2535.1.y.c.1544.3		12
65.24	odd	12		2535.1.y.c.1499.3		12
65.29	even	6		2535.1.x.b.2219.2		6
65.34	odd	4		2535.1.e.a.2534.4		6
65.44	odd	4		2535.1.e.a.2534.3		6
65.49	even	6		2535.1.x.d.2219.2		6
65.54	odd	12		2535.1.y.c.1499.4		12
65.59	odd	12		2535.1.y.c.1544.4		12
65.64	even	2		2535.1.f.a.1184.2	✓	3
195.29	odd	6		2535.1.x.c.2219.2		6
195.44	even	4		2535.1.e.b.2534.4		6
195.59	even	12		2535.1.y.b.1544.3		12
195.74	odd	6		2535.1.x.c.2174.2		6
195.89	even	12		2535.1.y.b.1499.4		12
195.119	even	12		2535.1.y.b.1499.3		12
195.134	odd	6		2535.1.x.a.2174.2		6
195.149	even	12		2535.1.y.b.1544.4		12
195.164	even	4		2535.1.e.b.2534.3		6
195.179	odd	6		2535.1.x.a.2219.2		6
195.194	odd	2		2535.1.f.d.1184.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2535.1.e.a.2534.3		6	39.5	even	4
2535.1.e.a.2534.3		6	65.44	odd	4
2535.1.e.a.2534.4		6	39.8	even	4
2535.1.e.a.2534.4		6	65.34	odd	4
2535.1.e.b.2534.3		6	13.8	odd	4
2535.1.e.b.2534.3		6	195.164	even	4
2535.1.e.b.2534.4		6	13.5	odd	4
2535.1.e.b.2534.4		6	195.44	even	4
2535.1.f.a.1184.2	✓	3	39.38	odd	2
2535.1.f.a.1184.2	✓	3	65.64	even	2
2535.1.f.b.1184.2	yes	3	1.1	even	1	trivial
2535.1.f.b.1184.2	yes	3	15.14	odd	2	CM
2535.1.f.c.1184.2	yes	3	3.2	odd	2
2535.1.f.c.1184.2	yes	3	5.4	even	2
2535.1.f.d.1184.2	yes	3	13.12	even	2
2535.1.f.d.1184.2	yes	3	195.194	odd	2
2535.1.x.a.2174.2		6	13.4	even	6
2535.1.x.a.2174.2		6	195.134	odd	6
2535.1.x.a.2219.2		6	13.10	even	6
2535.1.x.a.2219.2		6	195.179	odd	6
2535.1.x.b.2174.2		6	39.35	odd	6
2535.1.x.b.2174.2		6	65.9	even	6
2535.1.x.b.2219.2		6	39.29	odd	6
2535.1.x.b.2219.2		6	65.29	even	6
2535.1.x.c.2174.2		6	13.9	even	3
2535.1.x.c.2174.2		6	195.74	odd	6
2535.1.x.c.2219.2		6	13.3	even	3
2535.1.x.c.2219.2		6	195.29	odd	6
2535.1.x.d.2174.2		6	39.17	odd	6
2535.1.x.d.2174.2		6	65.4	even	6
2535.1.x.d.2219.2		6	39.23	odd	6
2535.1.x.d.2219.2		6	65.49	even	6
2535.1.y.b.1499.3		12	13.2	odd	12
2535.1.y.b.1499.3		12	195.119	even	12
2535.1.y.b.1499.4		12	13.11	odd	12
2535.1.y.b.1499.4		12	195.89	even	12
2535.1.y.b.1544.3		12	13.7	odd	12
2535.1.y.b.1544.3		12	195.59	even	12
2535.1.y.b.1544.4		12	13.6	odd	12
2535.1.y.b.1544.4		12	195.149	even	12
2535.1.y.c.1499.3		12	39.11	even	12
2535.1.y.c.1499.3		12	65.24	odd	12
2535.1.y.c.1499.4		12	39.2	even	12
2535.1.y.c.1499.4		12	65.54	odd	12
2535.1.y.c.1544.3		12	39.32	even	12
2535.1.y.c.1544.3		12	65.19	odd	12
2535.1.y.c.1544.4		12	39.20	even	12
2535.1.y.c.1544.4		12	65.59	odd	12

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:	\( N \)	\(=\)	\( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2535.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.445042 q^{2} +1.00000 q^{3} -0.801938 q^{4} +1.00000 q^{5} -0.445042 q^{6} +0.801938 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.445042 q^{2} +1.00000 q^{3} -0.801938 q^{4} +1.00000 q^{5} -0.445042 q^{6} +0.801938 q^{8} +1.00000 q^{9} -0.445042 q^{10} -0.801938 q^{12} +1.00000 q^{15} +0.445042 q^{16} +1.24698 q^{17} -0.445042 q^{18} -1.80194 q^{19} -0.801938 q^{20} -0.445042 q^{23} +0.801938 q^{24} +1.00000 q^{25} +1.00000 q^{27} -0.445042 q^{30} +1.24698 q^{31} -1.00000 q^{32} -0.554958 q^{34} -0.801938 q^{36} +0.801938 q^{38} +0.801938 q^{40} +1.00000 q^{45} +0.198062 q^{46} -1.80194 q^{47} +0.445042 q^{48} +1.00000 q^{49} -0.445042 q^{50} +1.24698 q^{51} -1.80194 q^{53} -0.445042 q^{54} -1.80194 q^{57} -0.801938 q^{60} +1.24698 q^{61} -0.554958 q^{62} -1.00000 q^{68} -0.445042 q^{69} +0.801938 q^{72} +1.00000 q^{75} +1.44504 q^{76} -0.445042 q^{79} +0.445042 q^{80} +1.00000 q^{81} +1.24698 q^{83} +1.24698 q^{85} -0.445042 q^{90} +0.356896 q^{92} +1.24698 q^{93} +0.801938 q^{94} -1.80194 q^{95} -1.00000 q^{96} -0.445042 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 + 3 * q^3 + 2 * q^4 + 3 * q^5 - q^6 - 2 * q^8 + 3 * q^9	\( 3 q - q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 2 q^{8} + 3 q^{9} - q^{10} + 2 q^{12} + 3 q^{15} + q^{16} - q^{17} - q^{18} - q^{19} + 2 q^{20} - q^{23} - 2 q^{24} + 3 q^{25} + 3 q^{27} - q^{30} - q^{31} - 3 q^{32} - 2 q^{34} + 2 q^{36} - 2 q^{38} - 2 q^{40} + 3 q^{45} + 5 q^{46} - q^{47} + q^{48} + 3 q^{49} - q^{50} - q^{51} - q^{53} - q^{54} - q^{57} + 2 q^{60} - q^{61} - 2 q^{62} - 3 q^{68} - q^{69} - 2 q^{72} + 3 q^{75} + 4 q^{76} - q^{79} + q^{80} + 3 q^{81} - q^{83} - q^{85} - q^{90} - 3 q^{92} - q^{93} - 2 q^{94} - q^{95} - 3 q^{96} - q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 2 * q^4 + 3 * q^5 - q^6 - 2 * q^8 + 3 * q^9 - q^10 + 2 * q^12 + 3 * q^15 + q^16 - q^17 - q^18 - q^19 + 2 * q^20 - q^23 - 2 * q^24 + 3 * q^25 + 3 * q^27 - q^30 - q^31 - 3 * q^32 - 2 * q^34 + 2 * q^36 - 2 * q^38 - 2 * q^40 + 3 * q^45 + 5 * q^46 - q^47 + q^48 + 3 * q^49 - q^50 - q^51 - q^53 - q^54 - q^57 + 2 * q^60 - q^61 - 2 * q^62 - 3 * q^68 - q^69 - 2 * q^72 + 3 * q^75 + 4 * q^76 - q^79 + q^80 + 3 * q^81 - q^83 - q^85 - q^90 - 3 * q^92 - q^93 - 2 * q^94 - q^95 - 3 * q^96 - q^98

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