LMFDB - Embedded newform 1976.1.o.b.493.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1976,1,Mod(493,1976)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1976, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1976.493");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1976 = 2^{3} \cdot 13 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1976.o (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:	$0.986152464963$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.7715442176.1

Embedding invariants

Embedding label			493.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	1976.493

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +0.445042 q^{5} -1.24698 q^{6} -1.00000 q^{8} +0.554958 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +0.445042 q^{5} -1.24698 q^{6} -1.00000 q^{8} +0.554958 q^{9} -0.445042 q^{10} +1.80194 q^{11} +1.24698 q^{12} +1.00000 q^{13} +0.554958 q^{15} +1.00000 q^{16} +1.24698 q^{17} -0.554958 q^{18} -1.00000 q^{19} +0.445042 q^{20} -1.80194 q^{22} -1.80194 q^{23} -1.24698 q^{24} -0.801938 q^{25} -1.00000 q^{26} -0.554958 q^{27} -1.80194 q^{29} -0.554958 q^{30} +0.445042 q^{31} -1.00000 q^{32} +2.24698 q^{33} -1.24698 q^{34} +0.554958 q^{36} +1.00000 q^{38} +1.24698 q^{39} -0.445042 q^{40} -1.24698 q^{41} +1.80194 q^{44} +0.246980 q^{45} +1.80194 q^{46} +1.24698 q^{48} +1.00000 q^{49} +0.801938 q^{50} +1.55496 q^{51} +1.00000 q^{52} -0.445042 q^{53} +0.554958 q^{54} +0.801938 q^{55} -1.24698 q^{57} +1.80194 q^{58} +0.554958 q^{60} -0.445042 q^{62} +1.00000 q^{64} +0.445042 q^{65} -2.24698 q^{66} +1.24698 q^{68} -2.24698 q^{69} +0.445042 q^{71} -0.554958 q^{72} -1.00000 q^{75} -1.00000 q^{76} -1.24698 q^{78} +0.445042 q^{80} -1.24698 q^{81} +1.24698 q^{82} -1.24698 q^{83} +0.554958 q^{85} -2.24698 q^{87} -1.80194 q^{88} +1.80194 q^{89} -0.246980 q^{90} -1.80194 q^{92} +0.554958 q^{93} -0.445042 q^{95} -1.24698 q^{96} +1.80194 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{5} + q^{6} - 3 q^{8} + 2 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^5 + q^6 - 3 * q^8 + 2 * q^9	$ 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{5} + q^{6} - 3 q^{8} + 2 q^{9} - q^{10} + q^{11} - q^{12} + 3 q^{13} + 2 q^{15} + 3 q^{16} - q^{17} - 2 q^{18} - 3 q^{19} + q^{20} - q^{22} - q^{23} + q^{24} + 2 q^{25} - 3 q^{26} - 2 q^{27} - q^{29} - 2 q^{30} + q^{31} - 3 q^{32} + 2 q^{33} + q^{34} + 2 q^{36} + 3 q^{38} - q^{39} - q^{40} + q^{41} + q^{44} - 4 q^{45} + q^{46} - q^{48} + 3 q^{49} - 2 q^{50} + 5 q^{51} + 3 q^{52} - q^{53} + 2 q^{54} - 2 q^{55} + q^{57} + q^{58} + 2 q^{60} - q^{62} + 3 q^{64} + q^{65} - 2 q^{66} - q^{68} - 2 q^{69} + q^{71} - 2 q^{72} - 3 q^{75} - 3 q^{76} + q^{78} + q^{80} + q^{81} - q^{82} + q^{83} + 2 q^{85} - 2 q^{87} - q^{88} + q^{89} + 4 q^{90} - q^{92} + 2 q^{93} - q^{95} + q^{96} + q^{97} - 3 q^{98} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^5 + q^6 - 3 * q^8 + 2 * q^9 - q^10 + q^11 - q^12 + 3 * q^13 + 2 * q^15 + 3 * q^16 - q^17 - 2 * q^18 - 3 * q^19 + q^20 - q^22 - q^23 + q^24 + 2 * q^25 - 3 * q^26 - 2 * q^27 - q^29 - 2 * q^30 + q^31 - 3 * q^32 + 2 * q^33 + q^34 + 2 * q^36 + 3 * q^38 - q^39 - q^40 + q^41 + q^44 - 4 * q^45 + q^46 - q^48 + 3 * q^49 - 2 * q^50 + 5 * q^51 + 3 * q^52 - q^53 + 2 * q^54 - 2 * q^55 + q^57 + q^58 + 2 * q^60 - q^62 + 3 * q^64 + q^65 - 2 * q^66 - q^68 - 2 * q^69 + q^71 - 2 * q^72 - 3 * q^75 - 3 * q^76 + q^78 + q^80 + q^81 - q^82 + q^83 + 2 * q^85 - 2 * q^87 - q^88 + q^89 + 4 * q^90 - q^92 + 2 * q^93 - q^95 + q^96 + q^97 - 3 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1976\mathbb{Z}\right)^\times$.

$n$	$457$	$495$	$989$	$1769$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−1.00000	−1.00000
$2$	−1.00000	−1.00000
$3$	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
$5$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$5$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$6$	−1.24698	−1.24698
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−1.00000	−1.00000
$9$	0.554958	0.554958
$10$	−0.445042	−0.445042
$11$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$11$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$12$	1.24698	1.24698
$13$	1.00000	1.00000
$13$	1.00000	1.00000
$14$	0	0
$15$	0.554958	0.554958
$16$	1.00000	1.00000
$17$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$17$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$18$	−0.554958	−0.554958
$19$	−1.00000	−1.00000
$19$	−1.00000	−1.00000
$20$	0.445042	0.445042
$21$	0	0
$22$	−1.80194	−1.80194
$23$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$23$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$24$	−1.24698	−1.24698
$25$	−0.801938	−0.801938
$26$	−1.00000	−1.00000
$27$	−0.554958	−0.554958
$28$	0	0
$29$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$29$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$30$	−0.554958	−0.554958
$31$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$31$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$32$	−1.00000	−1.00000
$33$	2.24698	2.24698
$34$	−1.24698	−1.24698
$35$	0	0
$36$	0.554958	0.554958
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.00000	1.00000
$39$	1.24698	1.24698
$40$	−0.445042	−0.445042
$41$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$41$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	1.80194	1.80194
$45$	0.246980	0.246980
$46$	1.80194	1.80194
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.24698	1.24698
$49$	1.00000	1.00000
$50$	0.801938	0.801938
$51$	1.55496	1.55496
$52$	1.00000	1.00000
$53$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$53$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$54$	0.554958	0.554958
$55$	0.801938	0.801938
$56$	0	0
$57$	−1.24698	−1.24698
$58$	1.80194	1.80194
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0.554958	0.554958
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−0.445042	−0.445042
$63$	0	0
$64$	1.00000	1.00000
$65$	0.445042	0.445042
$66$	−2.24698	−2.24698
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	1.24698	1.24698
$69$	−2.24698	−2.24698
$70$	0	0
$71$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$71$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$72$	−0.554958	−0.554958
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−1.00000	−1.00000
$76$	−1.00000	−1.00000
$77$	0	0
$78$	−1.24698	−1.24698
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.445042	0.445042
$81$	−1.24698	−1.24698
$82$	1.24698	1.24698
$83$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$83$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$84$	0	0
$85$	0.554958	0.554958
$86$	0	0
$87$	−2.24698	−2.24698
$88$	−1.80194	−1.80194
$89$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$89$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$90$	−0.246980	−0.246980
$91$	0	0
$92$	−1.80194	−1.80194
$93$	0.554958	0.554958
$94$	0	0
$95$	−0.445042	−0.445042
$96$	−1.24698	−1.24698
$97$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$97$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$98$	−1.00000	−1.00000
$99$	1.00000	1.00000
$100$	−0.801938	−0.801938
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−1.55496	−1.55496
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−1.00000	−1.00000
$105$	0	0
$106$	0.445042	0.445042
$107$	−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	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−0.801938	−0.801938
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	1.24698	1.24698
$115$	−0.801938	−0.801938
$116$	−1.80194	−1.80194
$117$	0.554958	0.554958
$118$	0	0
$119$	0	0
$120$	−0.554958	−0.554958
$121$	2.24698	2.24698
$122$	0	0
$123$	−1.55496	−1.55496
$124$	0.445042	0.445042
$125$	−0.801938	−0.801938
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	−0.445042	−0.445042
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	2.24698	2.24698
$133$	0	0
$134$	0	0
$135$	−0.246980	−0.246980
$136$	−1.24698	−1.24698
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	2.24698	2.24698
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	−0.445042	−0.445042
$143$	1.80194	1.80194
$144$	0.554958	0.554958
$145$	−0.801938	−0.801938
$146$	0	0
$147$	1.24698	1.24698
$148$	0	0
$149$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$149$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$150$	1.00000	1.00000
$151$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$151$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$152$	1.00000	1.00000
$153$	0.692021	0.692021
$154$	0	0
$155$	0.198062	0.198062
$156$	1.24698	1.24698
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−0.554958	−0.554958
$160$	−0.445042	−0.445042
$161$	0	0
$162$	1.24698	1.24698
$163$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$163$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$164$	−1.24698	−1.24698
$165$	1.00000	1.00000
$166$	1.24698	1.24698
$167$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$167$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	−0.554958	−0.554958
$171$	−0.554958	−0.554958
$172$	0	0
$173$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$173$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$174$	2.24698	2.24698
$175$	0	0
$176$	1.80194	1.80194
$177$	0	0
$178$	−1.80194	−1.80194
$179$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$179$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$180$	0.246980	0.246980
$181$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$181$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$182$	0	0
$183$	0	0
$184$	1.80194	1.80194
$185$	0	0
$186$	−0.554958	−0.554958
$187$	2.24698	2.24698
$188$	0	0
$189$	0	0
$190$	0.445042	0.445042
$191$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$191$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$192$	1.24698	1.24698
$193$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$193$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$194$	−1.80194	−1.80194
$195$	0.554958	0.554958
$196$	1.00000	1.00000
$197$	−2.00000	−2.00000	−1.00000			$\pi$
$197$	−2.00000	−2.00000	−1.00000			$\pi$
$198$	−1.00000	−1.00000
$199$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$199$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$200$	0.801938	0.801938
$201$	0	0
$202$	0	0
$203$	0	0
$204$	1.55496	1.55496
$205$	−0.554958	−0.554958
$206$	0	0
$207$	−1.00000	−1.00000
$208$	1.00000	1.00000
$209$	−1.80194	−1.80194
$210$	0	0
$211$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$211$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$212$	−0.445042	−0.445042
$213$	0.554958	0.554958
$214$	0.445042	0.445042
$215$	0	0
$216$	0.554958	0.554958
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0.801938	0.801938
$221$	1.24698	1.24698
$222$	0	0
$223$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$223$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$224$	0	0
$225$	−0.445042	−0.445042
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	−1.24698	−1.24698
$229$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$229$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$230$	0.801938	0.801938
$231$	0	0
$232$	1.80194	1.80194
$233$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$233$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$234$	−0.554958	−0.554958
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0.554958	0.554958
$241$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$241$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$242$	−2.24698	−2.24698
$243$	−1.00000	−1.00000
$244$	0	0
$245$	0.445042	0.445042
$246$	1.55496	1.55496
$247$	−1.00000	−1.00000
$248$	−0.445042	−0.445042
$249$	−1.55496	−1.55496
$250$	0.801938	0.801938
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	−3.24698	−3.24698
$254$	0	0
$255$	0.692021	0.692021
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0.445042	0.445042
$261$	−1.00000	−1.00000
$262$	0	0
$263$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$263$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$264$	−2.24698	−2.24698
$265$	−0.198062	−0.198062
$266$	0	0
$267$	2.24698	2.24698
$268$	0	0
$269$	2.00000	2.00000	1.00000			$0$
$269$	2.00000	2.00000	1.00000			$0$
$270$	0.246980	0.246980
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	1.24698	1.24698
$273$	0	0
$274$	0	0
$275$	−1.44504	−1.44504
$276$	−2.24698	−2.24698
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0.246980	0.246980
$280$	0	0
$281$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$281$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0.445042	0.445042
$285$	−0.554958	−0.554958
$286$	−1.80194	−1.80194
$287$	0	0
$288$	−0.554958	−0.554958
$289$	0.554958	0.554958
$290$	0.801938	0.801938
$291$	2.24698	2.24698
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.24698	−1.24698
$295$	0	0
$296$	0	0
$297$	−1.00000	−1.00000
$298$	1.24698	1.24698
$299$	−1.80194	−1.80194
$300$	−1.00000	−1.00000
$301$	0	0
$302$	−1.80194	−1.80194
$303$	0	0
$304$	−1.00000	−1.00000
$305$	0	0
$306$	−0.692021	−0.692021
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	−0.198062	−0.198062
$311$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$311$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$312$	−1.24698	−1.24698
$313$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$313$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.554958	0.554958
$319$	−3.24698	−3.24698
$320$	0.445042	0.445042
$321$	−0.554958	−0.554958
$322$	0	0
$323$	−1.24698	−1.24698
$324$	−1.24698	−1.24698
$325$	−0.801938	−0.801938
$326$	1.24698	1.24698
$327$	0	0
$328$	1.24698	1.24698
$329$	0	0
$330$	−1.00000	−1.00000
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−1.24698	−1.24698
$333$	0	0
$334$	1.24698	1.24698
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.00000	−1.00000
$339$	0	0
$340$	0.554958	0.554958
$341$	0.801938	0.801938
$342$	0.554958	0.554958
$343$	0	0
$344$	0	0
$345$	−1.00000	−1.00000
$346$	−1.24698	−1.24698
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−2.24698	−2.24698
$349$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$349$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$350$	0	0
$351$	−0.554958	−0.554958
$352$	−1.80194	−1.80194
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0.198062	0.198062
$356$	1.80194	1.80194
$357$	0	0
$358$	0.445042	0.445042
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−0.246980	−0.246980
$361$	1.00000	1.00000
$362$	−1.24698	−1.24698
$363$	2.80194	2.80194
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$367$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$368$	−1.80194	−1.80194
$369$	−0.692021	−0.692021
$370$	0	0
$371$	0	0
$372$	0.554958	0.554958
$373$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$373$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$374$	−2.24698	−2.24698
$375$	−1.00000	−1.00000
$376$	0	0
$377$	−1.80194	−1.80194
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	−0.445042	−0.445042
$381$	0	0
$382$	0.445042	0.445042
$383$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$383$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$384$	−1.24698	−1.24698
$385$	0	0
$386$	1.24698	1.24698
$387$	0	0
$388$	1.80194	1.80194
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	−0.554958	−0.554958
$391$	−2.24698	−2.24698
$392$	−1.00000	−1.00000
$393$	0	0
$394$	2.00000	2.00000
$395$	0	0
$396$	1.00000	1.00000
$397$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$397$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$398$	1.80194	1.80194
$399$	0	0
$400$	−0.801938	−0.801938
$401$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$401$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$402$	0	0
$403$	0.445042	0.445042
$404$	0	0
$405$	−0.554958	−0.554958
$406$	0	0
$407$	0	0
$408$	−1.55496	−1.55496
$409$	−2.00000	−2.00000	−1.00000			$\pi$
$409$	−2.00000	−2.00000	−1.00000			$\pi$
$410$	0.554958	0.554958
$411$	0	0
$412$	0	0
$413$	0	0
$414$	1.00000	1.00000
$415$	−0.554958	−0.554958
$416$	−1.00000	−1.00000
$417$	0	0
$418$	1.80194	1.80194
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	1.80194	1.80194
$423$	0	0
$424$	0.445042	0.445042
$425$	−1.00000	−1.00000
$426$	−0.554958	−0.554958
$427$	0	0
$428$	−0.445042	−0.445042
$429$	2.24698	2.24698
$430$	0	0
$431$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$431$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$432$	−0.554958	−0.554958
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	−1.00000	−1.00000
$436$	0	0
$437$	1.80194	1.80194
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−0.801938	−0.801938
$441$	0.554958	0.554958
$442$	−1.24698	−1.24698
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0.801938	0.801938
$446$	1.24698	1.24698
$447$	−1.55496	−1.55496
$448$	0	0
$449$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$449$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$450$	0.445042	0.445042
$451$	−2.24698	−2.24698
$452$	0	0
$453$	2.24698	2.24698
$454$	0	0
$455$	0	0
$456$	1.24698	1.24698
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−1.80194	−1.80194
$459$	−0.692021	−0.692021
$460$	−0.801938	−0.801938
$461$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$461$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−1.80194	−1.80194
$465$	0.246980	0.246980
$466$	0.445042	0.445042
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0.554958	0.554958
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.801938	0.801938
$476$	0	0
$477$	−0.246980	−0.246980
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−0.554958	−0.554958
$481$	0	0
$482$	−0.445042	−0.445042
$483$	0	0
$484$	2.24698	2.24698
$485$	0.801938	0.801938
$486$	1.00000	1.00000
$487$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$487$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$488$	0	0
$489$	−1.55496	−1.55496
$490$	−0.445042	−0.445042
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−1.55496	−1.55496
$493$	−2.24698	−2.24698
$494$	1.00000	1.00000
$495$	0.445042	0.445042
$496$	0.445042	0.445042
$497$	0	0
$498$	1.55496	1.55496
$499$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$499$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$500$	−0.801938	−0.801938
$501$	−1.55496	−1.55496
$502$	0	0
$503$	2.00000	2.00000	1.00000			$0$
$503$	2.00000	2.00000	1.00000			$0$
$504$	0	0
$505$	0	0
$506$	3.24698	3.24698
$507$	1.24698	1.24698
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−0.692021	−0.692021
$511$	0	0
$512$	−1.00000	−1.00000
$513$	0.554958	0.554958
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1.55496	1.55496
$520$	−0.445042	−0.445042
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	1.00000	1.00000
$523$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$523$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$524$	0	0
$525$	0	0
$526$	−1.24698	−1.24698
$527$	0.554958	0.554958
$528$	2.24698	2.24698
$529$	2.24698	2.24698
$530$	0.198062	0.198062
$531$	0	0
$532$	0	0
$533$	−1.24698	−1.24698
$534$	−2.24698	−2.24698
$535$	−0.198062	−0.198062
$536$	0	0
$537$	−0.554958	−0.554958
$538$	−2.00000	−2.00000
$539$	1.80194	1.80194
$540$	−0.246980	−0.246980
$541$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$541$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$542$	0	0
$543$	1.55496	1.55496
$544$	−1.24698	−1.24698
$545$	0	0
$546$	0	0
$547$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$547$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$548$	0	0
$549$	0	0
$550$	1.44504	1.44504
$551$	1.80194	1.80194
$552$	2.24698	2.24698
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$557$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$558$	−0.246980	−0.246980
$559$	0	0
$560$	0	0
$561$	2.80194	2.80194
$562$	−1.80194	−1.80194
$563$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$563$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−0.445042	−0.445042
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0.554958	0.554958
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	1.80194	1.80194
$573$	−0.554958	−0.554958
$574$	0	0
$575$	1.44504	1.44504
$576$	0.554958	0.554958
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−0.554958	−0.554958
$579$	−1.55496	−1.55496
$580$	−0.801938	−0.801938
$581$	0	0
$582$	−2.24698	−2.24698
$583$	−0.801938	−0.801938
$584$	0	0
$585$	0.246980	0.246980
$586$	0	0
$587$	−2.00000	−2.00000	−1.00000			$\pi$
$587$	−2.00000	−2.00000	−1.00000			$\pi$
$588$	1.24698	1.24698
$589$	−0.445042	−0.445042
$590$	0	0
$591$	−2.49396	−2.49396
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	1.00000	1.00000
$595$	0	0
$596$	−1.24698	−1.24698
$597$	−2.24698	−2.24698
$598$	1.80194	1.80194
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	1.00000	1.00000
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	1.80194	1.80194
$605$	1.00000	1.00000
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.00000	1.00000
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0.692021	0.692021
$613$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$613$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$614$	0	0
$615$	−0.692021	−0.692021
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$619$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$620$	0.198062	0.198062
$621$	1.00000	1.00000
$622$	−1.24698	−1.24698
$623$	0	0
$624$	1.24698	1.24698
$625$	0.445042	0.445042
$626$	0.445042	0.445042
$627$	−2.24698	−2.24698
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−2.24698	−2.24698
$634$	0	0
$635$	0	0
$636$	−0.554958	−0.554958
$637$	1.00000	1.00000
$638$	3.24698	3.24698
$639$	0.246980	0.246980
$640$	−0.445042	−0.445042
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0.554958	0.554958
$643$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$643$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$644$	0	0
$645$	0	0
$646$	1.24698	1.24698
$647$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$647$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$648$	1.24698	1.24698
$649$	0	0
$650$	0.801938	0.801938
$651$	0	0
$652$	−1.24698	−1.24698
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−1.24698	−1.24698
$657$	0	0
$658$	0	0
$659$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$659$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$660$	1.00000	1.00000
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	1.55496	1.55496
$664$	1.24698	1.24698
$665$	0	0
$666$	0	0
$667$	3.24698	3.24698
$668$	−1.24698	−1.24698
$669$	−1.55496	−1.55496
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0.445042	0.445042
$676$	1.00000	1.00000
$677$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$677$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$678$	0	0
$679$	0	0
$680$	−0.554958	−0.554958
$681$	0	0
$682$	−0.801938	−0.801938
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−0.554958	−0.554958
$685$	0	0
$686$	0	0
$687$	2.24698	2.24698
$688$	0	0
$689$	−0.445042	−0.445042
$690$	1.00000	1.00000
$691$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$691$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$692$	1.24698	1.24698
$693$	0	0
$694$	0	0
$695$	0	0
$696$	2.24698	2.24698
$697$	−1.55496	−1.55496
$698$	1.24698	1.24698
$699$	−0.554958	−0.554958
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0.554958	0.554958
$703$	0	0
$704$	1.80194	1.80194
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$710$	−0.198062	−0.198062
$711$	0	0
$712$	−1.80194	−1.80194
$713$	−0.801938	−0.801938
$714$	0	0
$715$	0.801938	0.801938
$716$	−0.445042	−0.445042
$717$	0	0
$718$	0	0
$719$	2.00000	2.00000	1.00000			$0$
$719$	2.00000	2.00000	1.00000			$0$
$720$	0.246980	0.246980
$721$	0	0
$722$	−1.00000	−1.00000
$723$	0.554958	0.554958
$724$	1.24698	1.24698
$725$	1.44504	1.44504
$726$	−2.80194	−2.80194
$727$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$727$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$733$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$734$	0.445042	0.445042
$735$	0.554958	0.554958
$736$	1.80194	1.80194
$737$	0	0
$738$	0.692021	0.692021
$739$	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
$741$	−1.24698	−1.24698
$742$	0	0
$743$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$743$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$744$	−0.554958	−0.554958
$745$	−0.554958	−0.554958
$746$	−1.24698	−1.24698
$747$	−0.692021	−0.692021
$748$	2.24698	2.24698
$749$	0	0
$750$	1.00000	1.00000
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	1.80194	1.80194
$755$	0.801938	0.801938
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	−4.04892	−4.04892
$760$	0.445042	0.445042
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−0.445042	−0.445042
$765$	0.307979	0.307979
$766$	−0.445042	−0.445042
$767$	0	0
$768$	1.24698	1.24698
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	−1.24698	−1.24698
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−0.356896	−0.356896
$776$	−1.80194	−1.80194
$777$	0	0
$778$	0	0
$779$	1.24698	1.24698
$780$	0.554958	0.554958
$781$	0.801938	0.801938
$782$	2.24698	2.24698
$783$	1.00000	1.00000
$784$	1.00000	1.00000
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−2.00000	−2.00000
$789$	1.55496	1.55496
$790$	0	0
$791$	0	0
$792$	−1.00000	−1.00000
$793$	0	0
$794$	−0.445042	−0.445042
$795$	−0.246980	−0.246980
$796$	−1.80194	−1.80194
$797$	−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
$799$	0	0
$800$	0.801938	0.801938
$801$	1.00000	1.00000
$802$	−0.445042	−0.445042
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−0.445042	−0.445042
$807$	2.49396	2.49396
$808$	0	0
$809$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$809$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$810$	0.554958	0.554958
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.554958	−0.554958
$816$	1.55496	1.55496
$817$	0	0
$818$	2.00000	2.00000
$819$	0	0
$820$	−0.554958	−0.554958
$821$	−2.00000	−2.00000	−1.00000			$\pi$
$821$	−2.00000	−2.00000	−1.00000			$\pi$
$822$	0	0
$823$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$823$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$824$	0	0
$825$	−1.80194	−1.80194
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−1.00000	−1.00000
$829$	−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
$832$	1.00000	1.00000
$833$	1.24698	1.24698
$834$	0	0
$835$	−0.554958	−0.554958
$836$	−1.80194	−1.80194
$837$	−0.246980	−0.246980
$838$	0	0
$839$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$839$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$840$	0	0
$841$	2.24698	2.24698
$842$	0	0
$843$	2.24698	2.24698
$844$	−1.80194	−1.80194
$845$	0.445042	0.445042
$846$	0	0
$847$	0	0
$848$	−0.445042	−0.445042
$849$	0	0
$850$	1.00000	1.00000
$851$	0	0
$852$	0.554958	0.554958
$853$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$853$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$854$	0	0
$855$	−0.246980	−0.246980
$856$	0.445042	0.445042
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	−2.24698	−2.24698
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	1.24698	1.24698
$863$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$863$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$864$	0.554958	0.554958
$865$	0.554958	0.554958
$866$	0	0
$867$	0.692021	0.692021
$868$	0	0
$869$	0	0
$870$	1.00000	1.00000
$871$	0	0
$872$	0	0
$873$	1.00000	1.00000
$874$	−1.80194	−1.80194
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0.801938	0.801938
$881$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$881$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$882$	−0.554958	−0.554958
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	1.24698	1.24698
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	−0.801938	−0.801938
$891$	−2.24698	−2.24698
$892$	−1.24698	−1.24698
$893$	0	0
$894$	1.55496	1.55496
$895$	−0.198062	−0.198062
$896$	0	0
$897$	−2.24698	−2.24698
$898$	1.24698	1.24698
$899$	−0.801938	−0.801938
$900$	−0.445042	−0.445042
$901$	−0.554958	−0.554958
$902$	2.24698	2.24698
$903$	0	0
$904$	0	0
$905$	0.554958	0.554958
$906$	−2.24698	−2.24698
$907$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$907$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−1.24698	−1.24698
$913$	−2.24698	−2.24698
$914$	0	0
$915$	0	0
$916$	1.80194	1.80194
$917$	0	0
$918$	0.692021	0.692021
$919$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$919$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$920$	0.801938	0.801938
$921$	0	0
$922$	−1.80194	−1.80194
$923$	0.445042	0.445042
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	1.80194	1.80194
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	−0.246980	−0.246980
$931$	−1.00000	−1.00000
$932$	−0.445042	−0.445042
$933$	1.55496	1.55496
$934$	0	0
$935$	1.00000	1.00000
$936$	−0.554958	−0.554958
$937$	−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
$939$	−0.554958	−0.554958
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	2.24698	2.24698
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$947$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$948$	0	0
$949$	0	0
$950$	−0.801938	−0.801938
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0.246980	0.246980
$955$	−0.198062	−0.198062
$956$	0	0
$957$	−4.04892	−4.04892
$958$	0	0
$959$	0	0
$960$	0.554958	0.554958
$961$	−0.801938	−0.801938
$962$	0	0
$963$	−0.246980	−0.246980
$964$	0.445042	0.445042
$965$	−0.554958	−0.554958
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.24698	−2.24698
$969$	−1.55496	−1.55496
$970$	−0.801938	−0.801938
$971$	−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
$974$	−1.80194	−1.80194
$975$	−1.00000	−1.00000
$976$	0	0
$977$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$977$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$978$	1.55496	1.55496
$979$	3.24698	3.24698
$980$	0.445042	0.445042
$981$	0	0
$982$	0	0
$983$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$983$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$984$	1.55496	1.55496
$985$	−0.890084	−0.890084
$986$	2.24698	2.24698
$987$	0	0
$988$	−1.00000	−1.00000
$989$	0	0
$990$	−0.445042	−0.445042
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−0.445042	−0.445042
$993$	0	0
$994$	0	0
$995$	−0.801938	−0.801938
$996$	−1.55496	−1.55496
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.24698	1.24698
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1976.1.o.b.493.3	✓	3
8.5	even	2		1976.1.o.c.493.1	yes	3
13.12	even	2		1976.1.o.d.493.3	yes	3
19.18	odd	2		1976.1.o.e.493.1	yes	3
104.77	even	2		1976.1.o.e.493.1	yes	3
152.37	odd	2		1976.1.o.d.493.3	yes	3
247.246	odd	2		1976.1.o.c.493.1	yes	3
1976.493	odd	2	CM	1976.1.o.b.493.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1976.1.o.b.493.3	✓	3	1.1	even	1	trivial
1976.1.o.b.493.3	✓	3	1976.493	odd	2	CM
1976.1.o.c.493.1	yes	3	8.5	even	2
1976.1.o.c.493.1	yes	3	247.246	odd	2
1976.1.o.d.493.3	yes	3	13.12	even	2
1976.1.o.d.493.3	yes	3	152.37	odd	2
1976.1.o.e.493.1	yes	3	19.18	odd	2
1976.1.o.e.493.1	yes	3	104.77	even	2

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:	\( N \)	\(=\)	\( 1976 = 2^{3} \cdot 13 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1976.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +0.445042 q^{5} -1.24698 q^{6} -1.00000 q^{8} +0.554958 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +0.445042 q^{5} -1.24698 q^{6} -1.00000 q^{8} +0.554958 q^{9} -0.445042 q^{10} +1.80194 q^{11} +1.24698 q^{12} +1.00000 q^{13} +0.554958 q^{15} +1.00000 q^{16} +1.24698 q^{17} -0.554958 q^{18} -1.00000 q^{19} +0.445042 q^{20} -1.80194 q^{22} -1.80194 q^{23} -1.24698 q^{24} -0.801938 q^{25} -1.00000 q^{26} -0.554958 q^{27} -1.80194 q^{29} -0.554958 q^{30} +0.445042 q^{31} -1.00000 q^{32} +2.24698 q^{33} -1.24698 q^{34} +0.554958 q^{36} +1.00000 q^{38} +1.24698 q^{39} -0.445042 q^{40} -1.24698 q^{41} +1.80194 q^{44} +0.246980 q^{45} +1.80194 q^{46} +1.24698 q^{48} +1.00000 q^{49} +0.801938 q^{50} +1.55496 q^{51} +1.00000 q^{52} -0.445042 q^{53} +0.554958 q^{54} +0.801938 q^{55} -1.24698 q^{57} +1.80194 q^{58} +0.554958 q^{60} -0.445042 q^{62} +1.00000 q^{64} +0.445042 q^{65} -2.24698 q^{66} +1.24698 q^{68} -2.24698 q^{69} +0.445042 q^{71} -0.554958 q^{72} -1.00000 q^{75} -1.00000 q^{76} -1.24698 q^{78} +0.445042 q^{80} -1.24698 q^{81} +1.24698 q^{82} -1.24698 q^{83} +0.554958 q^{85} -2.24698 q^{87} -1.80194 q^{88} +1.80194 q^{89} -0.246980 q^{90} -1.80194 q^{92} +0.554958 q^{93} -0.445042 q^{95} -1.24698 q^{96} +1.80194 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{5} + q^{6} - 3 q^{8} + 2 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^5 + q^6 - 3 * q^8 + 2 * q^9	\( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{5} + q^{6} - 3 q^{8} + 2 q^{9} - q^{10} + q^{11} - q^{12} + 3 q^{13} + 2 q^{15} + 3 q^{16} - q^{17} - 2 q^{18} - 3 q^{19} + q^{20} - q^{22} - q^{23} + q^{24} + 2 q^{25} - 3 q^{26} - 2 q^{27} - q^{29} - 2 q^{30} + q^{31} - 3 q^{32} + 2 q^{33} + q^{34} + 2 q^{36} + 3 q^{38} - q^{39} - q^{40} + q^{41} + q^{44} - 4 q^{45} + q^{46} - q^{48} + 3 q^{49} - 2 q^{50} + 5 q^{51} + 3 q^{52} - q^{53} + 2 q^{54} - 2 q^{55} + q^{57} + q^{58} + 2 q^{60} - q^{62} + 3 q^{64} + q^{65} - 2 q^{66} - q^{68} - 2 q^{69} + q^{71} - 2 q^{72} - 3 q^{75} - 3 q^{76} + q^{78} + q^{80} + q^{81} - q^{82} + q^{83} + 2 q^{85} - 2 q^{87} - q^{88} + q^{89} + 4 q^{90} - q^{92} + 2 q^{93} - q^{95} + q^{96} + q^{97} - 3 q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^5 + q^6 - 3 * q^8 + 2 * q^9 - q^10 + q^11 - q^12 + 3 * q^13 + 2 * q^15 + 3 * q^16 - q^17 - 2 * q^18 - 3 * q^19 + q^20 - q^22 - q^23 + q^24 + 2 * q^25 - 3 * q^26 - 2 * q^27 - q^29 - 2 * q^30 + q^31 - 3 * q^32 + 2 * q^33 + q^34 + 2 * q^36 + 3 * q^38 - q^39 - q^40 + q^41 + q^44 - 4 * q^45 + q^46 - q^48 + 3 * q^49 - 2 * q^50 + 5 * q^51 + 3 * q^52 - q^53 + 2 * q^54 - 2 * q^55 + q^57 + q^58 + 2 * q^60 - q^62 + 3 * q^64 + q^65 - 2 * q^66 - q^68 - 2 * q^69 + q^71 - 2 * q^72 - 3 * q^75 - 3 * q^76 + q^78 + q^80 + q^81 - q^82 + q^83 + 2 * q^85 - 2 * q^87 - q^88 + q^89 + 4 * q^90 - q^92 + 2 * q^93 - q^95 + q^96 + q^97 - 3 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more