LMFDB - Embedded newform 2535.1.f.a.1184.3

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

Embedding invariants

Embedding label			1184.3
Root			$-1.24698$ 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)

$f(q)$	$=$	$q+1.24698 q^{2} -1.00000 q^{3} +0.554958 q^{4} +1.00000 q^{5} -1.24698 q^{6} -0.554958 q^{8} +1.00000 q^{9} +O(q^{10})$	$q+1.24698 q^{2} -1.00000 q^{3} +0.554958 q^{4} +1.00000 q^{5} -1.24698 q^{6} -0.554958 q^{8} +1.00000 q^{9} +1.24698 q^{10} -0.554958 q^{12} -1.00000 q^{15} -1.24698 q^{16} +1.80194 q^{17} +1.24698 q^{18} +0.445042 q^{19} +0.554958 q^{20} -1.24698 q^{23} +0.554958 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.24698 q^{30} +1.80194 q^{31} -1.00000 q^{32} +2.24698 q^{34} +0.554958 q^{36} +0.554958 q^{38} -0.554958 q^{40} +1.00000 q^{45} -1.55496 q^{46} -0.445042 q^{47} +1.24698 q^{48} +1.00000 q^{49} +1.24698 q^{50} -1.80194 q^{51} +0.445042 q^{53} -1.24698 q^{54} -0.445042 q^{57} -0.554958 q^{60} -1.80194 q^{61} +2.24698 q^{62} +1.00000 q^{68} +1.24698 q^{69} -0.554958 q^{72} -1.00000 q^{75} +0.246980 q^{76} +1.24698 q^{79} -1.24698 q^{80} +1.00000 q^{81} -1.80194 q^{83} +1.80194 q^{85} +1.24698 q^{90} -0.692021 q^{92} -1.80194 q^{93} -0.554958 q^{94} +0.445042 q^{95} +1.00000 q^{96} +1.24698 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$2$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$3$	−1.00000	−1.00000
$3$	−1.00000	−1.00000
$4$	0.554958	0.554958
$5$	1.00000	1.00000
$5$	1.00000	1.00000
$6$	−1.24698	−1.24698
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−0.554958	−0.554958
$9$	1.00000	1.00000
$10$	1.24698	1.24698
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−0.554958	−0.554958
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.00000	−1.00000
$16$	−1.24698	−1.24698
$17$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$17$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$18$	1.24698	1.24698
$19$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$19$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$20$	0.554958	0.554958
$21$	0	0
$22$	0	0
$23$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$23$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$24$	0.554958	0.554958
$25$	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			$\pi$
$30$	−1.24698	−1.24698
$31$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$31$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$32$	−1.00000	−1.00000
$33$	0	0
$34$	2.24698	2.24698
$35$	0	0
$36$	0.554958	0.554958
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0.554958	0.554958
$39$	0	0
$40$	−0.554958	−0.554958
$41$	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
$46$	−1.55496	−1.55496
$47$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$47$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$48$	1.24698	1.24698
$49$	1.00000	1.00000
$50$	1.24698	1.24698
$51$	−1.80194	−1.80194
$52$	0	0
$53$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$53$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$54$	−1.24698	−1.24698
$55$	0	0
$56$	0	0
$57$	−0.445042	−0.445042
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−0.554958	−0.554958
$61$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$61$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$62$	2.24698	2.24698
$63$	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
$69$	1.24698	1.24698
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.554958	−0.554958
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−1.00000	−1.00000
$76$	0.246980	0.246980
$77$	0	0
$78$	0	0
$79$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$79$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$80$	−1.24698	−1.24698
$81$	1.00000	1.00000
$82$	0	0
$83$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$83$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$84$	0	0
$85$	1.80194	1.80194
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	1.24698	1.24698
$91$	0	0
$92$	−0.692021	−0.692021
$93$	−1.80194	−1.80194
$94$	−0.554958	−0.554958
$95$	0.445042	0.445042
$96$	1.00000	1.00000
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.24698	1.24698
$99$	0	0
$100$	0.554958	0.554958
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−2.24698	−2.24698
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0.554958	0.554958
$107$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$107$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$108$	−0.554958	−0.554958
$109$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$109$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$113$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$114$	−0.554958	−0.554958
$115$	−1.24698	−1.24698
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0.554958	0.554958
$121$	1.00000	1.00000
$122$	−2.24698	−2.24698
$123$	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
$137$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$137$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$138$	1.55496	1.55496
$139$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$139$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$140$	0	0
$141$	0.445042	0.445042
$142$	0	0
$143$	0	0
$144$	−1.24698	−1.24698
$145$	0	0
$146$	0	0
$147$	−1.00000	−1.00000
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−1.24698	−1.24698
$151$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$151$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$152$	−0.246980	−0.246980
$153$	1.80194	1.80194
$154$	0	0
$155$	1.80194	1.80194
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	1.55496	1.55496
$159$	−0.445042	−0.445042
$160$	−1.00000	−1.00000
$161$	0	0
$162$	1.24698	1.24698
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−2.24698	−2.24698
$167$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$167$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$168$	0	0
$169$	0	0
$170$	2.24698	2.24698
$171$	0.445042	0.445042
$172$	0	0
$173$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$173$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$174$	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			$\pi$
$180$	0.554958	0.554958
$181$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$181$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$182$	0	0
$183$	1.80194	1.80194
$184$	0.692021	0.692021
$185$	0	0
$186$	−2.24698	−2.24698
$187$	0	0
$188$	−0.246980	−0.246980
$189$	0	0
$190$	0.554958	0.554958
$191$	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
$196$	0.554958	0.554958
$197$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$197$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$198$	0	0
$199$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$199$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$200$	−0.554958	−0.554958
$201$	0	0
$202$	0	0
$203$	0	0
$204$	−1.00000	−1.00000
$205$	0	0
$206$	0	0
$207$	−1.24698	−1.24698
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$211$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$212$	0.246980	0.246980
$213$	0	0
$214$	0.554958	0.554958
$215$	0	0
$216$	0.554958	0.554958
$217$	0	0
$218$	−1.55496	−1.55496
$219$	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
$226$	−1.55496	−1.55496
$227$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$227$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$228$	−0.246980	−0.246980
$229$	−2.00000	−2.00000	−1.00000			$\pi$
$229$	−2.00000	−2.00000	−1.00000			$\pi$
$230$	−1.55496	−1.55496
$231$	0	0
$232$	0	0
$233$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$233$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$234$	0	0
$235$	−0.445042	−0.445042
$236$	0	0
$237$	−1.24698	−1.24698
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	1.24698	1.24698
$241$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$241$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$242$	1.24698	1.24698
$243$	−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
$249$	1.80194	1.80194
$250$	1.24698	1.24698
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−1.80194	−1.80194
$256$	1.24698	1.24698
$257$	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	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$263$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$264$	0	0
$265$	0.445042	0.445042
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−1.24698	−1.24698
$271$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$271$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$272$	−2.24698	−2.24698
$273$	0	0
$274$	−2.24698	−2.24698
$275$	0	0
$276$	0.692021	0.692021
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.55496	1.55496
$279$	1.80194	1.80194
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0.554958	0.554958
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	−0.445042	−0.445042
$286$	0	0
$287$	0	0
$288$	−1.00000	−1.00000
$289$	2.24698	2.24698
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$293$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$294$	−1.24698	−1.24698
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−0.554958	−0.554958
$301$	0	0
$302$	−1.55496	−1.55496
$303$	0	0
$304$	−0.554958	−0.554958
$305$	−1.80194	−1.80194
$306$	2.24698	2.24698
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	2.24698	2.24698
$311$	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
$316$	0.692021	0.692021
$317$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$317$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$318$	−0.554958	−0.554958
$319$	0	0
$320$	0	0
$321$	−0.445042	−0.445042
$322$	0	0
$323$	0.801938	0.801938
$324$	0.554958	0.554958
$325$	0	0
$326$	0	0
$327$	1.24698	1.24698
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$331$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$332$	−1.00000	−1.00000
$333$	0	0
$334$	−2.24698	−2.24698
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	1.24698	1.24698
$340$	1.00000	1.00000
$341$	0	0
$342$	0.554958	0.554958
$343$	0	0
$344$	0	0
$345$	1.24698	1.24698
$346$	−1.55496	−1.55496
$347$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$347$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$348$	0	0
$349$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$349$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$353$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$354$	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			$\pi$
$360$	−0.554958	−0.554958
$361$	−0.801938	−0.801938
$362$	−0.554958	−0.554958
$363$	−1.00000	−1.00000
$364$	0	0
$365$	0	0
$366$	2.24698	2.24698
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	1.55496	1.55496
$369$	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
$376$	0.246980	0.246980
$377$	0	0
$378$	0	0
$379$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$379$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$380$	0.246980	0.246980
$381$	0	0
$382$	0	0
$383$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$383$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$384$	−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
$391$	−2.24698	−2.24698
$392$	−0.554958	−0.554958
$393$	0	0
$394$	−0.554958	−0.554958
$395$	1.24698	1.24698
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−2.24698	−2.24698
$399$	0	0
$400$	−1.24698	−1.24698
$401$	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			$\pi$
$409$	−2.00000	−2.00000	−1.00000			$\pi$
$410$	0	0
$411$	1.80194	1.80194
$412$	0	0
$413$	0	0
$414$	−1.55496	−1.55496
$415$	−1.80194	−1.80194
$416$	0	0
$417$	−1.24698	−1.24698
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$421$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$422$	−0.554958	−0.554958
$423$	−0.445042	−0.445042
$424$	−0.246980	−0.246980
$425$	1.80194	1.80194
$426$	0	0
$427$	0	0
$428$	0.246980	0.246980
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.24698	1.24698
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	−0.692021	−0.692021
$437$	−0.554958	−0.554958
$438$	0	0
$439$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$439$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	0	0
$443$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$443$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$444$	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			$\pi$
$450$	1.24698	1.24698
$451$	0	0
$452$	−0.692021	−0.692021
$453$	1.24698	1.24698
$454$	−0.554958	−0.554958
$455$	0	0
$456$	0.246980	0.246980
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−2.49396	−2.49396
$459$	−1.80194	−1.80194
$460$	−0.692021	−0.692021
$461$	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
$465$	−1.80194	−1.80194
$466$	0.554958	0.554958
$467$	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	0
$470$	−0.554958	−0.554958
$471$	0	0
$472$	0	0
$473$	0	0
$474$	−1.55496	−1.55496
$475$	0.445042	0.445042
$476$	0	0
$477$	0.445042	0.445042
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	1.00000	1.00000
$481$	0	0
$482$	−1.55496	−1.55496
$483$	0	0
$484$	0.554958	0.554958
$485$	0	0
$486$	−1.24698	−1.24698
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	1.00000	1.00000
$489$	0	0
$490$	1.24698	1.24698
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−2.24698	−2.24698
$497$	0	0
$498$	2.24698	2.24698
$499$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$499$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$500$	0.554958	0.554958
$501$	1.80194	1.80194
$502$	0	0
$503$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$503$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$504$	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			$\pi$
$510$	−2.24698	−2.24698
$511$	0	0
$512$	0.554958	0.554958
$513$	−0.445042	−0.445042
$514$	0.554958	0.554958
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1.24698	1.24698
$520$	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
$526$	2.24698	2.24698
$527$	3.24698	3.24698
$528$	0	0
$529$	0.554958	0.554958
$530$	0.554958	0.554958
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0.445042	0.445042
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	−0.554958	−0.554958
$541$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$541$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$542$	−1.55496	−1.55496
$543$	0.445042	0.445042
$544$	−1.80194	−1.80194
$545$	−1.24698	−1.24698
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−1.00000	−1.00000
$549$	−1.80194	−1.80194
$550$	0	0
$551$	0	0
$552$	−0.692021	−0.692021
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0.692021	0.692021
$557$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$557$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$558$	2.24698	2.24698
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$563$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$564$	0.246980	0.246980
$565$	−1.24698	−1.24698
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	−0.554958	−0.554958
$571$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$571$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.24698	−1.24698
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	2.80194	2.80194
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−0.554958	−0.554958
$587$	2.00000	2.00000	1.00000			$0$
$587$	2.00000	2.00000	1.00000			$0$
$588$	−0.554958	−0.554958
$589$	0.801938	0.801938
$590$	0	0
$591$	0.445042	0.445042
$592$	0	0
$593$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$593$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.80194	1.80194
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0.554958	0.554958
$601$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$601$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$602$	0	0
$603$	0	0
$604$	−0.692021	−0.692021
$605$	1.00000	1.00000
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−0.445042	−0.445042
$609$	0	0
$610$	−2.24698	−2.24698
$611$	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
$617$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$617$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$618$	0	0
$619$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$619$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$620$	1.00000	1.00000
$621$	1.24698	1.24698
$622$	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
$631$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$631$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$632$	−0.692021	−0.692021
$633$	0.445042	0.445042
$634$	1.55496	1.55496
$635$	0	0
$636$	−0.246980	−0.246980
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.00000	1.00000
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−0.554958	−0.554958
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	1.00000	1.00000
$647$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$647$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$648$	−0.554958	−0.554958
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$653$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$654$	1.55496	1.55496
$655$	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
$661$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$661$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$662$	2.24698	2.24698
$663$	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
$677$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$677$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$678$	1.55496	1.55496
$679$	0	0
$680$	−1.00000	−1.00000
$681$	0.445042	0.445042
$682$	0	0
$683$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$683$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$684$	0.246980	0.246980
$685$	−1.80194	−1.80194
$686$	0	0
$687$	2.00000	2.00000
$688$	0	0
$689$	0	0
$690$	1.55496	1.55496
$691$	−2.00000	−2.00000	−1.00000			$\pi$
$691$	−2.00000	−2.00000	−1.00000			$\pi$
$692$	−0.692021	−0.692021
$693$	0	0
$694$	2.24698	2.24698
$695$	1.24698	1.24698
$696$	0	0
$697$	0	0
$698$	0.554958	0.554958
$699$	−0.445042	−0.445042
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0.445042	0.445042
$706$	1.55496	1.55496
$707$	0	0
$708$	0	0
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$710$	0	0
$711$	1.24698	1.24698
$712$	0	0
$713$	−2.24698	−2.24698
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−1.24698	−1.24698
$721$	0	0
$722$	−1.00000	−1.00000
$723$	1.24698	1.24698
$724$	−0.246980	−0.246980
$725$	0	0
$726$	−1.24698	−1.24698
$727$	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
$736$	1.24698	1.24698
$737$	0	0
$738$	0	0
$739$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$739$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$743$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$744$	1.00000	1.00000
$745$	0	0
$746$	0	0
$747$	−1.80194	−1.80194
$748$	0	0
$749$	0	0
$750$	−1.24698	−1.24698
$751$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$751$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$752$	0.554958	0.554958
$753$	0	0
$754$	0	0
$755$	−1.24698	−1.24698
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−1.55496	−1.55496
$759$	0	0
$760$	−0.246980	−0.246980
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.80194	1.80194
$766$	1.55496	1.55496
$767$	0	0
$768$	−1.24698	−1.24698
$769$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$769$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$770$	0	0
$771$	−0.445042	−0.445042
$772$	0	0
$773$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$773$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$774$	0	0
$775$	1.80194	1.80194
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−2.80194	−2.80194
$783$	0	0
$784$	−1.24698	−1.24698
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−0.246980	−0.246980
$789$	−1.80194	−1.80194
$790$	1.55496	1.55496
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−0.445042	−0.445042
$796$	−1.00000	−1.00000
$797$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$797$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$798$	0	0
$799$	−0.801938	−0.801938
$800$	−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			$\pi$
$810$	1.24698	1.24698
$811$	−2.00000	−2.00000	−1.00000			$\pi$
$811$	−2.00000	−2.00000	−1.00000			$\pi$
$812$	0	0
$813$	1.24698	1.24698
$814$	0	0
$815$	0	0
$816$	2.24698	2.24698
$817$	0	0
$818$	−2.49396	−2.49396
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	2.24698	2.24698
$823$	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			$0$
$828$	−0.692021	−0.692021
$829$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$829$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$830$	−2.24698	−2.24698
$831$	0	0
$832$	0	0
$833$	1.80194	1.80194
$834$	−1.55496	−1.55496
$835$	−1.80194	−1.80194
$836$	0	0
$837$	−1.80194	−1.80194
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	−1.55496	−1.55496
$843$	0	0
$844$	−0.246980	−0.246980
$845$	0	0
$846$	−0.554958	−0.554958
$847$	0	0
$848$	−0.554958	−0.554958
$849$	0	0
$850$	2.24698	2.24698
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0.445042	0.445042
$856$	−0.246980	−0.246980
$857$	−2.00000	−2.00000	−1.00000			$\pi$
$857$	−2.00000	−2.00000	−1.00000			$\pi$
$858$	0	0
$859$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$859$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$863$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$864$	1.00000	1.00000
$865$	−1.24698	−1.24698
$866$	0	0
$867$	−2.24698	−2.24698
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0.692021	0.692021
$873$	0	0
$874$	−0.692021	−0.692021
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.554958	−0.554958
$879$	0.445042	0.445042
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.24698	1.24698
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−1.55496	−1.55496
$887$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$887$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.198062	−0.198062
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0.554958	0.554958
$901$	0.801938	0.801938
$902$	0	0
$903$	0	0
$904$	0.692021	0.692021
$905$	−0.445042	−0.445042
$906$	1.55496	1.55496
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−0.246980	−0.246980
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.554958	0.554958
$913$	0	0
$914$	0	0
$915$	1.80194	1.80194
$916$	−1.10992	−1.10992
$917$	0	0
$918$	−2.24698	−2.24698
$919$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$919$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$920$	0.692021	0.692021
$921$	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			$\pi$
$930$	−2.24698	−2.24698
$931$	0.445042	0.445042
$932$	0.246980	0.246980
$933$	0	0
$934$	0.554958	0.554958
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	−0.246980	−0.246980
$941$	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
$947$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$947$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$948$	−0.692021	−0.692021
$949$	0	0
$950$	0.554958	0.554958
$951$	−1.24698	−1.24698
$952$	0	0
$953$	−2.00000	−2.00000	−1.00000			$\pi$
$953$	−2.00000	−2.00000	−1.00000			$\pi$
$954$	0.554958	0.554958
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.24698	2.24698
$962$	0	0
$963$	0.445042	0.445042
$964$	−0.692021	−0.692021
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.554958	−0.554958
$969$	−0.801938	−0.801938
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.554958	−0.554958
$973$	0	0
$974$	0	0
$975$	0	0
$976$	2.24698	2.24698
$977$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$977$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$978$	0	0
$979$	0	0
$980$	0.554958	0.554958
$981$	−1.24698	−1.24698
$982$	0	0
$983$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$983$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$984$	0	0
$985$	−0.445042	−0.445042
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$991$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$992$	−1.80194	−1.80194
$993$	−1.80194	−1.80194
$994$	0	0
$995$	−1.80194	−1.80194
$996$	1.00000	1.00000
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	2.24698	2.24698
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2535.1.f.a.1184.3	✓	3
3.2	odd	2		2535.1.f.d.1184.1	yes	3
5.4	even	2		2535.1.f.d.1184.1	yes	3
13.2	odd	12		2535.1.y.c.1499.5		12
13.3	even	3		2535.1.x.d.2219.1		6
13.4	even	6		2535.1.x.b.2174.3		6
13.5	odd	4		2535.1.e.a.2534.2		6
13.6	odd	12		2535.1.y.c.1544.2		12
13.7	odd	12		2535.1.y.c.1544.5		12
13.8	odd	4		2535.1.e.a.2534.5		6
13.9	even	3		2535.1.x.d.2174.1		6
13.10	even	6		2535.1.x.b.2219.3		6
13.11	odd	12		2535.1.y.c.1499.2		12
13.12	even	2		2535.1.f.c.1184.1	yes	3
15.14	odd	2	CM	2535.1.f.a.1184.3	✓	3
39.2	even	12		2535.1.y.b.1499.2		12
39.5	even	4		2535.1.e.b.2534.5		6
39.8	even	4		2535.1.e.b.2534.2		6
39.11	even	12		2535.1.y.b.1499.5		12
39.17	odd	6		2535.1.x.c.2174.1		6
39.20	even	12		2535.1.y.b.1544.2		12
39.23	odd	6		2535.1.x.c.2219.1		6
39.29	odd	6		2535.1.x.a.2219.3		6
39.32	even	12		2535.1.y.b.1544.5		12
39.35	odd	6		2535.1.x.a.2174.3		6
39.38	odd	2		2535.1.f.b.1184.3	yes	3
65.4	even	6		2535.1.x.c.2174.1		6
65.9	even	6		2535.1.x.a.2174.3		6
65.19	odd	12		2535.1.y.b.1544.5		12
65.24	odd	12		2535.1.y.b.1499.5		12
65.29	even	6		2535.1.x.a.2219.3		6
65.34	odd	4		2535.1.e.b.2534.2		6
65.44	odd	4		2535.1.e.b.2534.5		6
65.49	even	6		2535.1.x.c.2219.1		6
65.54	odd	12		2535.1.y.b.1499.2		12
65.59	odd	12		2535.1.y.b.1544.2		12
65.64	even	2		2535.1.f.b.1184.3	yes	3
195.29	odd	6		2535.1.x.d.2219.1		6
195.44	even	4		2535.1.e.a.2534.2		6
195.59	even	12		2535.1.y.c.1544.5		12
195.74	odd	6		2535.1.x.d.2174.1		6
195.89	even	12		2535.1.y.c.1499.2		12
195.119	even	12		2535.1.y.c.1499.5		12
195.134	odd	6		2535.1.x.b.2174.3		6
195.149	even	12		2535.1.y.c.1544.2		12
195.164	even	4		2535.1.e.a.2534.5		6
195.179	odd	6		2535.1.x.b.2219.3		6
195.194	odd	2		2535.1.f.c.1184.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2535.1.e.a.2534.2		6	13.5	odd	4
2535.1.e.a.2534.2		6	195.44	even	4
2535.1.e.a.2534.5		6	13.8	odd	4
2535.1.e.a.2534.5		6	195.164	even	4
2535.1.e.b.2534.2		6	39.8	even	4
2535.1.e.b.2534.2		6	65.34	odd	4
2535.1.e.b.2534.5		6	39.5	even	4
2535.1.e.b.2534.5		6	65.44	odd	4
2535.1.f.a.1184.3	✓	3	1.1	even	1	trivial
2535.1.f.a.1184.3	✓	3	15.14	odd	2	CM
2535.1.f.b.1184.3	yes	3	39.38	odd	2
2535.1.f.b.1184.3	yes	3	65.64	even	2
2535.1.f.c.1184.1	yes	3	13.12	even	2
2535.1.f.c.1184.1	yes	3	195.194	odd	2
2535.1.f.d.1184.1	yes	3	3.2	odd	2
2535.1.f.d.1184.1	yes	3	5.4	even	2
2535.1.x.a.2174.3		6	39.35	odd	6
2535.1.x.a.2174.3		6	65.9	even	6
2535.1.x.a.2219.3		6	39.29	odd	6
2535.1.x.a.2219.3		6	65.29	even	6
2535.1.x.b.2174.3		6	13.4	even	6
2535.1.x.b.2174.3		6	195.134	odd	6
2535.1.x.b.2219.3		6	13.10	even	6
2535.1.x.b.2219.3		6	195.179	odd	6
2535.1.x.c.2174.1		6	39.17	odd	6
2535.1.x.c.2174.1		6	65.4	even	6
2535.1.x.c.2219.1		6	39.23	odd	6
2535.1.x.c.2219.1		6	65.49	even	6
2535.1.x.d.2174.1		6	13.9	even	3
2535.1.x.d.2174.1		6	195.74	odd	6
2535.1.x.d.2219.1		6	13.3	even	3
2535.1.x.d.2219.1		6	195.29	odd	6
2535.1.y.b.1499.2		12	39.2	even	12
2535.1.y.b.1499.2		12	65.54	odd	12
2535.1.y.b.1499.5		12	39.11	even	12
2535.1.y.b.1499.5		12	65.24	odd	12
2535.1.y.b.1544.2		12	39.20	even	12
2535.1.y.b.1544.2		12	65.59	odd	12
2535.1.y.b.1544.5		12	39.32	even	12
2535.1.y.b.1544.5		12	65.19	odd	12
2535.1.y.c.1499.2		12	13.11	odd	12
2535.1.y.c.1499.2		12	195.89	even	12
2535.1.y.c.1499.5		12	13.2	odd	12
2535.1.y.c.1499.5		12	195.119	even	12
2535.1.y.c.1544.2		12	13.6	odd	12
2535.1.y.c.1544.2		12	195.149	even	12
2535.1.y.c.1544.5		12	13.7	odd	12
2535.1.y.c.1544.5		12	195.59	even	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)

\(f(q)\)	\(=\)	\(q+1.24698 q^{2} -1.00000 q^{3} +0.554958 q^{4} +1.00000 q^{5} -1.24698 q^{6} -0.554958 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.24698 q^{2} -1.00000 q^{3} +0.554958 q^{4} +1.00000 q^{5} -1.24698 q^{6} -0.554958 q^{8} +1.00000 q^{9} +1.24698 q^{10} -0.554958 q^{12} -1.00000 q^{15} -1.24698 q^{16} +1.80194 q^{17} +1.24698 q^{18} +0.445042 q^{19} +0.554958 q^{20} -1.24698 q^{23} +0.554958 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.24698 q^{30} +1.80194 q^{31} -1.00000 q^{32} +2.24698 q^{34} +0.554958 q^{36} +0.554958 q^{38} -0.554958 q^{40} +1.00000 q^{45} -1.55496 q^{46} -0.445042 q^{47} +1.24698 q^{48} +1.00000 q^{49} +1.24698 q^{50} -1.80194 q^{51} +0.445042 q^{53} -1.24698 q^{54} -0.445042 q^{57} -0.554958 q^{60} -1.80194 q^{61} +2.24698 q^{62} +1.00000 q^{68} +1.24698 q^{69} -0.554958 q^{72} -1.00000 q^{75} +0.246980 q^{76} +1.24698 q^{79} -1.24698 q^{80} +1.00000 q^{81} -1.80194 q^{83} +1.80194 q^{85} +1.24698 q^{90} -0.692021 q^{92} -1.80194 q^{93} -0.554958 q^{94} +0.445042 q^{95} +1.00000 q^{96} +1.24698 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 3 q^{3} + 2 q^{4} + 3 q^{5} + q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more