LMFDB - Embedded newform 3743.1.d.d.3742.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3743,1,Mod(3742,3743)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3743, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3743.3742");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3743 = 19 \cdot 197 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3743.d (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:	$1.86800034229$
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 7.1.52439613407.1
Artin image:	$D_{14}$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{14} - \cdots)$

Embedding invariants

Embedding label			3742.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	3743.3742

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.445042 q^{2} +1.80194 q^{3} -0.801938 q^{4} +0.801938 q^{6} -0.445042 q^{7} -0.801938 q^{8} +2.24698 q^{9} +O(q^{10})$	$q+0.445042 q^{2} +1.80194 q^{3} -0.801938 q^{4} +0.801938 q^{6} -0.445042 q^{7} -0.801938 q^{8} +2.24698 q^{9} -1.44504 q^{12} +0.445042 q^{13} -0.198062 q^{14} +0.445042 q^{16} +1.00000 q^{18} +1.00000 q^{19} -0.801938 q^{21} -1.80194 q^{23} -1.44504 q^{24} +1.00000 q^{25} +0.198062 q^{26} +2.24698 q^{27} +0.356896 q^{28} +1.80194 q^{31} +1.00000 q^{32} -1.80194 q^{36} +0.445042 q^{38} +0.801938 q^{39} -0.356896 q^{42} +2.00000 q^{43} -0.801938 q^{46} -0.445042 q^{47} +0.801938 q^{48} -0.801938 q^{49} +0.445042 q^{50} -0.356896 q^{52} +1.00000 q^{54} +0.356896 q^{56} +1.80194 q^{57} -1.80194 q^{61} +0.801938 q^{62} -1.00000 q^{63} +1.80194 q^{67} -3.24698 q^{69} -1.24698 q^{71} -1.80194 q^{72} +1.80194 q^{75} -0.801938 q^{76} +0.356896 q^{78} -1.24698 q^{79} +1.80194 q^{81} -1.80194 q^{83} +0.643104 q^{84} +0.890084 q^{86} -2.00000 q^{89} -0.198062 q^{91} +1.44504 q^{92} +3.24698 q^{93} -0.198062 q^{94} +1.80194 q^{96} -0.356896 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 3 * q + q^2 + q^3 + 2 * q^4 - 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9	$ 3 q + q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{12} + q^{13} - 5 q^{14} + q^{16} + 3 q^{18} + 3 q^{19} + 2 q^{21} - q^{23} - 4 q^{24} + 3 q^{25} + 5 q^{26} + 2 q^{27} - 3 q^{28} + q^{31} + 3 q^{32} - q^{36} + q^{38} - 2 q^{39} + 3 q^{42} + 6 q^{43} + 2 q^{46} - q^{47} - 2 q^{48} + 2 q^{49} + q^{50} + 3 q^{52} + 3 q^{54} - 3 q^{56} + q^{57} - q^{61} - 2 q^{62} - 3 q^{63} + q^{67} - 5 q^{69} + q^{71} - q^{72} + q^{75} + 2 q^{76} - 3 q^{78} + q^{79} + q^{81} - q^{83} + 6 q^{84} + 2 q^{86} - 6 q^{89} - 5 q^{91} + 4 q^{92} + 5 q^{93} - 5 q^{94} + q^{96} + 3 q^{98}+O(q^{100}) $ 3 * q + q^2 + q^3 + 2 * q^4 - 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 - 4 * q^12 + q^13 - 5 * q^14 + q^16 + 3 * q^18 + 3 * q^19 + 2 * q^21 - q^23 - 4 * q^24 + 3 * q^25 + 5 * q^26 + 2 * q^27 - 3 * q^28 + q^31 + 3 * q^32 - q^36 + q^38 - 2 * q^39 + 3 * q^42 + 6 * q^43 + 2 * q^46 - q^47 - 2 * q^48 + 2 * q^49 + q^50 + 3 * q^52 + 3 * q^54 - 3 * q^56 + q^57 - q^61 - 2 * q^62 - 3 * q^63 + q^67 - 5 * q^69 + q^71 - q^72 + q^75 + 2 * q^76 - 3 * q^78 + q^79 + q^81 - q^83 + 6 * q^84 + 2 * q^86 - 6 * q^89 - 5 * q^91 + 4 * q^92 + 5 * q^93 - 5 * q^94 + q^96 + 3 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3743\mathbb{Z}\right)^\times$.

$n$	$1578$	$2168$
$\chi(n)$	$-1$	$-1$

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	$-0.428571\pi$
$2$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$3$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$3$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$4$	−0.801938	−0.801938
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0.801938	0.801938
$7$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$7$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$8$	−0.801938	−0.801938
$9$	2.24698	2.24698
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.44504	−1.44504
$13$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$13$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$14$	−0.198062	−0.198062
$15$	0	0
$16$	0.445042	0.445042
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.00000	1.00000
$19$	1.00000	1.00000
$19$	1.00000	1.00000
$20$	0	0
$21$	−0.801938	−0.801938
$22$	0	0
$23$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$23$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$24$	−1.44504	−1.44504
$25$	1.00000	1.00000
$26$	0.198062	0.198062
$27$	2.24698	2.24698
$28$	0.356896	0.356896
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	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
$34$	0	0
$35$	0	0
$36$	−1.80194	−1.80194
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0.445042	0.445042
$39$	0.801938	0.801938
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−0.356896	−0.356896
$43$	2.00000	2.00000	1.00000			$0$
$43$	2.00000	2.00000	1.00000			$0$
$44$	0	0
$45$	0	0
$46$	−0.801938	−0.801938
$47$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$47$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$48$	0.801938	0.801938
$49$	−0.801938	−0.801938
$50$	0.445042	0.445042
$51$	0	0
$52$	−0.356896	−0.356896
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	1.00000	1.00000
$55$	0	0
$56$	0.356896	0.356896
$57$	1.80194	1.80194
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$61$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$62$	0.801938	0.801938
$63$	−1.00000	−1.00000
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$67$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$68$	0	0
$69$	−3.24698	−3.24698
$70$	0	0
$71$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$71$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$72$	−1.80194	−1.80194
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	1.80194	1.80194
$76$	−0.801938	−0.801938
$77$	0	0
$78$	0.356896	0.356896
$79$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$79$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$80$	0	0
$81$	1.80194	1.80194
$82$	0	0
$83$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$83$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$84$	0.643104	0.643104
$85$	0	0
$86$	0.890084	0.890084
$87$	0	0
$88$	0	0
$89$	−2.00000	−2.00000	−1.00000			$\pi$
$89$	−2.00000	−2.00000	−1.00000			$\pi$
$90$	0	0
$91$	−0.198062	−0.198062
$92$	1.44504	1.44504
$93$	3.24698	3.24698
$94$	−0.198062	−0.198062
$95$	0	0
$96$	1.80194	1.80194
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−0.356896	−0.356896
$99$	0	0
$100$	−0.801938	−0.801938
$101$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$101$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$102$	0	0
$103$	−2.00000	−2.00000	−1.00000			$\pi$
$103$	−2.00000	−2.00000	−1.00000			$\pi$
$104$	−0.356896	−0.356896
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−1.80194	−1.80194
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	−0.198062	−0.198062
$113$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$113$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$114$	0.801938	0.801938
$115$	0	0
$116$	0	0
$117$	1.00000	1.00000
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	−0.801938	−0.801938
$123$	0	0
$124$	−1.44504	−1.44504
$125$	0	0
$126$	−0.445042	−0.445042
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	3.60388	3.60388
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−0.445042	−0.445042
$134$	0.801938	0.801938
$135$	0	0
$136$	0	0
$137$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$137$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$138$	−1.44504	−1.44504
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−0.801938	−0.801938
$142$	−0.554958	−0.554958
$143$	0	0
$144$	1.00000	1.00000
$145$	0	0
$146$	0	0
$147$	−1.44504	−1.44504
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.801938	0.801938
$151$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$151$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$152$	−0.801938	−0.801938
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−0.643104	−0.643104
$157$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$157$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$158$	−0.554958	−0.554958
$159$	0	0
$160$	0	0
$161$	0.801938	0.801938
$162$	0.801938	0.801938
$163$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$163$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$164$	0	0
$165$	0	0
$166$	−0.801938	−0.801938
$167$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$167$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$168$	0.643104	0.643104
$169$	−0.801938	−0.801938
$170$	0	0
$171$	2.24698	2.24698
$172$	−1.60388	−1.60388
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−0.445042	−0.445042
$176$	0	0
$177$	0	0
$178$	−0.890084	−0.890084
$179$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$179$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−0.0881460	−0.0881460
$183$	−3.24698	−3.24698
$184$	1.44504	1.44504
$185$	0	0
$186$	1.44504	1.44504
$187$	0	0
$188$	0.356896	0.356896
$189$	−1.00000	−1.00000
$190$	0	0
$191$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$191$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0.643104	0.643104
$197$	1.00000	1.00000
$197$	1.00000	1.00000
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−0.801938	−0.801938
$201$	3.24698	3.24698
$202$	0.554958	0.554958
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−0.890084	−0.890084
$207$	−4.04892	−4.04892
$208$	0.198062	0.198062
$209$	0	0
$210$	0	0
$211$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$211$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$212$	0	0
$213$	−2.24698	−2.24698
$214$	0	0
$215$	0	0
$216$	−1.80194	−1.80194
$217$	−0.801938	−0.801938
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−0.445042	−0.445042
$225$	2.24698	2.24698
$226$	−0.554958	−0.554958
$227$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$227$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$228$	−1.44504	−1.44504
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$233$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$234$	0.445042	0.445042
$235$	0	0
$236$	0	0
$237$	−2.24698	−2.24698
$238$	0	0
$239$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$239$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$240$	0	0
$241$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$241$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$242$	0.445042	0.445042
$243$	1.00000	1.00000
$244$	1.44504	1.44504
$245$	0	0
$246$	0	0
$247$	0.445042	0.445042
$248$	−1.44504	−1.44504
$249$	−3.24698	−3.24698
$250$	0	0
$251$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$251$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$252$	0.801938	0.801938
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.445042	−0.445042
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	1.60388	1.60388
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−0.198062	−0.198062
$267$	−3.60388	−3.60388
$268$	−1.44504	−1.44504
$269$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$269$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	−0.356896	−0.356896
$274$	−0.198062	−0.198062
$275$	0	0
$276$	2.60388	2.60388
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	4.04892	4.04892
$280$	0	0
$281$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$281$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$282$	−0.356896	−0.356896
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	1.00000	1.00000
$285$	0	0
$286$	0	0
$287$	0	0
$288$	2.24698	2.24698
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−0.643104	−0.643104
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.801938	−0.801938
$300$	−1.44504	−1.44504
$301$	−0.890084	−0.890084
$302$	−0.554958	−0.554958
$303$	2.24698	2.24698
$304$	0.445042	0.445042
$305$	0	0
$306$	0	0
$307$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$307$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$308$	0	0
$309$	−3.60388	−3.60388
$310$	0	0
$311$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$311$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$312$	−0.643104	−0.643104
$313$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$313$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$314$	0.554958	0.554958
$315$	0	0
$316$	1.00000	1.00000
$317$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$317$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0.356896	0.356896
$323$	0	0
$324$	−1.44504	−1.44504
$325$	0.445042	0.445042
$326$	−0.198062	−0.198062
$327$	0	0
$328$	0	0
$329$	0.198062	0.198062
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	1.44504	1.44504
$333$	0	0
$334$	0.801938	0.801938
$335$	0	0
$336$	−0.356896	−0.356896
$337$	−2.00000	−2.00000	−1.00000			$\pi$
$337$	−2.00000	−2.00000	−1.00000			$\pi$
$338$	−0.356896	−0.356896
$339$	−2.24698	−2.24698
$340$	0	0
$341$	0	0
$342$	1.00000	1.00000
$343$	0.801938	0.801938
$344$	−1.60388	−1.60388
$345$	0	0
$346$	0	0
$347$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$347$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−0.198062	−0.198062
$351$	1.00000	1.00000
$352$	0	0
$353$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$353$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$354$	0	0
$355$	0	0
$356$	1.60388	1.60388
$357$	0	0
$358$	0.198062	0.198062
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	0	0
$363$	1.80194	1.80194
$364$	0.158834	0.158834
$365$	0	0
$366$	−1.44504	−1.44504
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−0.801938	−0.801938
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−2.60388	−2.60388
$373$	−2.00000	−2.00000	−1.00000			$\pi$
$373$	−2.00000	−2.00000	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0.356896	0.356896
$377$	0	0
$378$	−0.445042	−0.445042
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0.554958	0.554958
$383$	−2.00000	−2.00000	−1.00000			$\pi$
$383$	−2.00000	−2.00000	−1.00000			$\pi$
$384$	−1.80194	−1.80194
$385$	0	0
$386$	0	0
$387$	4.49396	4.49396
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0.643104	0.643104
$393$	0	0
$394$	0.445042	0.445042
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	−0.801938	−0.801938
$400$	0.445042	0.445042
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	1.44504	1.44504
$403$	0.801938	0.801938
$404$	−1.00000	−1.00000
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−0.801938	−0.801938
$412$	1.60388	1.60388
$413$	0	0
$414$	−1.80194	−1.80194
$415$	0	0
$416$	0.445042	0.445042
$417$	0	0
$418$	0	0
$419$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$419$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$420$	0	0
$421$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$421$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$422$	−0.554958	−0.554958
$423$	−1.00000	−1.00000
$424$	0	0
$425$	0	0
$426$	−1.00000	−1.00000
$427$	0.801938	0.801938
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.00000	1.00000
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−0.356896	−0.356896
$435$	0	0
$436$	0	0
$437$	−1.80194	−1.80194
$438$	0	0
$439$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$439$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$440$	0	0
$441$	−1.80194	−1.80194
$442$	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			$\pi$
$450$	1.00000	1.00000
$451$	0	0
$452$	1.00000	1.00000
$453$	−2.24698	−2.24698
$454$	−0.554958	−0.554958
$455$	0	0
$456$	−1.44504	−1.44504
$457$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$457$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	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
$465$	0	0
$466$	0.554958	0.554958
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−0.801938	−0.801938
$469$	−0.801938	−0.801938
$470$	0	0
$471$	2.24698	2.24698
$472$	0	0
$473$	0	0
$474$	−1.00000	−1.00000
$475$	1.00000	1.00000
$476$	0	0
$477$	0	0
$478$	−0.801938	−0.801938
$479$	2.00000	2.00000	1.00000			$0$
$479$	2.00000	2.00000	1.00000			$0$
$480$	0	0
$481$	0	0
$482$	−0.554958	−0.554958
$483$	1.44504	1.44504
$484$	−0.801938	−0.801938
$485$	0	0
$486$	0.445042	0.445042
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	1.44504	1.44504
$489$	−0.801938	−0.801938
$490$	0	0
$491$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$491$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$492$	0	0
$493$	0	0
$494$	0.198062	0.198062
$495$	0	0
$496$	0.801938	0.801938
$497$	0.554958	0.554958
$498$	−1.44504	−1.44504
$499$	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	0
$501$	3.24698	3.24698
$502$	0.554958	0.554958
$503$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$503$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$504$	0.801938	0.801938
$505$	0	0
$506$	0	0
$507$	−1.44504	−1.44504
$508$	0	0
$509$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$509$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$510$	0	0
$511$	0	0
$512$	0.801938	0.801938
$513$	2.24698	2.24698
$514$	0	0
$515$	0	0
$516$	−2.89008	−2.89008
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$523$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$524$	0	0
$525$	−0.801938	−0.801938
$526$	0	0
$527$	0	0
$528$	0	0
$529$	2.24698	2.24698
$530$	0	0
$531$	0	0
$532$	0.356896	0.356896
$533$	0	0
$534$	−1.60388	−1.60388
$535$	0	0
$536$	−1.44504	−1.44504
$537$	0.801938	0.801938
$538$	0.801938	0.801938
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	−0.158834	−0.158834
$547$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$547$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$548$	0.356896	0.356896
$549$	−4.04892	−4.04892
$550$	0	0
$551$	0	0
$552$	2.60388	2.60388
$553$	0.554958	0.554958
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$557$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$558$	1.80194	1.80194
$559$	0.890084	0.890084
$560$	0	0
$561$	0	0
$562$	0.198062	0.198062
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0.643104	0.643104
$565$	0	0
$566$	0	0
$567$	−0.801938	−0.801938
$568$	1.00000	1.00000
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	2.24698	2.24698
$574$	0	0
$575$	−1.80194	−1.80194
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0.445042	0.445042
$579$	0	0
$580$	0	0
$581$	0.801938	0.801938
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$587$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$588$	1.15883	1.15883
$589$	1.80194	1.80194
$590$	0	0
$591$	1.80194	1.80194
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−0.356896	−0.356896
$599$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$599$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$600$	−1.44504	−1.44504
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−0.396125	−0.396125
$603$	4.04892	4.04892
$604$	1.00000	1.00000
$605$	0	0
$606$	1.00000	1.00000
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.00000	1.00000
$609$	0	0
$610$	0	0
$611$	−0.198062	−0.198062
$612$	0	0
$613$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$613$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$614$	−0.554958	−0.554958
$615$	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	$0.571429\pi$
$618$	−1.60388	−1.60388
$619$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$619$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$620$	0	0
$621$	−4.04892	−4.04892
$622$	−0.198062	−0.198062
$623$	0.890084	0.890084
$624$	0.356896	0.356896
$625$	1.00000	1.00000
$626$	0.554958	0.554958
$627$	0	0
$628$	−1.00000	−1.00000
$629$	0	0
$630$	0	0
$631$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$631$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$632$	1.00000	1.00000
$633$	−2.24698	−2.24698
$634$	0.801938	0.801938
$635$	0	0
$636$	0	0
$637$	−0.356896	−0.356896
$638$	0	0
$639$	−2.80194	−2.80194
$640$	0	0
$641$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$641$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−0.643104	−0.643104
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.44504	−1.44504
$649$	0	0
$650$	0.198062	0.198062
$651$	−1.44504	−1.44504
$652$	0.356896	0.356896
$653$	2.00000	2.00000	1.00000			$0$
$653$	2.00000	2.00000	1.00000			$0$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0.0881460	0.0881460
$659$	−2.00000	−2.00000	−1.00000			$\pi$
$659$	−2.00000	−2.00000	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	1.44504	1.44504
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−1.44504	−1.44504
$669$	0	0
$670$	0	0
$671$	0	0
$672$	−0.801938	−0.801938
$673$	−2.00000	−2.00000	−1.00000			$\pi$
$673$	−2.00000	−2.00000	−1.00000			$\pi$
$674$	−0.890084	−0.890084
$675$	2.24698	2.24698
$676$	0.643104	0.643104
$677$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$677$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$678$	−1.00000	−1.00000
$679$	0	0
$680$	0	0
$681$	−2.24698	−2.24698
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−1.80194	−1.80194
$685$	0	0
$686$	0.356896	0.356896
$687$	0	0
$688$	0.890084	0.890084
$689$	0	0
$690$	0	0
$691$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$691$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$692$	0	0
$693$	0	0
$694$	−0.801938	−0.801938
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	2.24698	2.24698
$700$	0.356896	0.356896
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0.445042	0.445042
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−0.801938	−0.801938
$707$	−0.554958	−0.554958
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−2.80194	−2.80194
$712$	1.60388	1.60388
$713$	−3.24698	−3.24698
$714$	0	0
$715$	0	0
$716$	−0.356896	−0.356896
$717$	−3.24698	−3.24698
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0.890084	0.890084
$722$	0.445042	0.445042
$723$	−2.24698	−2.24698
$724$	0	0
$725$	0	0
$726$	0.801938	0.801938
$727$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$727$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$728$	0.158834	0.158834
$729$	0	0
$730$	0	0
$731$	0	0
$732$	2.60388	2.60388
$733$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$733$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$734$	0	0
$735$	0	0
$736$	−1.80194	−1.80194
$737$	0	0
$738$	0	0
$739$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$739$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$740$	0	0
$741$	0.801938	0.801938
$742$	0	0
$743$	−2.00000	−2.00000	−1.00000			$\pi$
$743$	−2.00000	−2.00000	−1.00000			$\pi$
$744$	−2.60388	−2.60388
$745$	0	0
$746$	−0.890084	−0.890084
$747$	−4.04892	−4.04892
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−0.198062	−0.198062
$753$	2.24698	2.24698
$754$	0	0
$755$	0	0
$756$	0.801938	0.801938
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−1.00000	−1.00000
$765$	0	0
$766$	−0.890084	−0.890084
$767$	0	0
$768$	−0.801938	−0.801938
$769$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$769$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	2.00000	2.00000
$775$	1.80194	1.80194
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−0.356896	−0.356896
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−0.801938	−0.801938
$789$	0	0
$790$	0	0
$791$	0.554958	0.554958
$792$	0	0
$793$	−0.801938	−0.801938
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	−0.356896	−0.356896
$799$	0	0
$800$	1.00000	1.00000
$801$	−4.49396	−4.49396
$802$	0	0
$803$	0	0
$804$	−2.60388	−2.60388
$805$	0	0
$806$	0.356896	0.356896
$807$	3.24698	3.24698
$808$	−1.00000	−1.00000
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.00000	2.00000
$818$	0	0
$819$	−0.445042	−0.445042
$820$	0	0
$821$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$821$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$822$	−0.356896	−0.356896
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	1.60388	1.60388
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	3.24698	3.24698
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.04892	4.04892
$838$	−0.198062	−0.198062
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0.801938	0.801938
$843$	0.801938	0.801938
$844$	1.00000	1.00000
$845$	0	0
$846$	−0.445042	−0.445042
$847$	−0.445042	−0.445042
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	1.80194	1.80194
$853$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$853$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$854$	0.356896	0.356896
$855$	0	0
$856$	0	0
$857$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$857$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$863$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$864$	2.24698	2.24698
$865$	0	0
$866$	0	0
$867$	1.80194	1.80194
$868$	0.643104	0.643104
$869$	0	0
$870$	0	0
$871$	0.801938	0.801938
$872$	0	0
$873$	0	0
$874$	−0.801938	−0.801938
$875$	0	0
$876$	0	0
$877$	−2.00000	−2.00000	−1.00000			$\pi$
$877$	−2.00000	−2.00000	−1.00000			$\pi$
$878$	−0.554958	−0.554958
$879$	0	0
$880$	0	0
$881$	2.00000	2.00000	1.00000			$0$
$881$	2.00000	2.00000	1.00000			$0$
$882$	−0.801938	−0.801938
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−0.198062	−0.198062
$887$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$887$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.445042	−0.445042
$894$	0	0
$895$	0	0
$896$	0.445042	0.445042
$897$	−1.44504	−1.44504
$898$	0	0
$899$	0	0
$900$	−1.80194	−1.80194
$901$	0	0
$902$	0	0
$903$	−1.60388	−1.60388
$904$	1.00000	1.00000
$905$	0	0
$906$	−1.00000	−1.00000
$907$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$907$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$908$	1.00000	1.00000
$909$	2.80194	2.80194
$910$	0	0
$911$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$911$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$912$	0.801938	0.801938
$913$	0	0
$914$	−0.198062	−0.198062
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−2.24698	−2.24698
$922$	0	0
$923$	−0.554958	−0.554958
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.49396	−4.49396
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−0.801938	−0.801938
$932$	−1.00000	−1.00000
$933$	−0.801938	−0.801938
$934$	0	0
$935$	0	0
$936$	−0.801938	−0.801938
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−0.356896	−0.356896
$939$	2.24698	2.24698
$940$	0	0
$941$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$941$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$942$	1.00000	1.00000
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	1.80194	1.80194
$949$	0	0
$950$	0.445042	0.445042
$951$	3.24698	3.24698
$952$	0	0
$953$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$953$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$954$	0	0
$955$	0	0
$956$	1.44504	1.44504
$957$	0	0
$958$	0.890084	0.890084
$959$	0.198062	0.198062
$960$	0	0
$961$	2.24698	2.24698
$962$	0	0
$963$	0	0
$964$	1.00000	1.00000
$965$	0	0
$966$	0.643104	0.643104
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.801938	−0.801938
$969$	0	0
$970$	0	0
$971$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$971$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$972$	−0.801938	−0.801938
$973$	0	0
$974$	0	0
$975$	0.801938	0.801938
$976$	−0.801938	−0.801938
$977$	−2.00000	−2.00000	−1.00000			$\pi$
$977$	−2.00000	−2.00000	−1.00000			$\pi$
$978$	−0.356896	−0.356896
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0.554958	0.554958
$983$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$983$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.356896	0.356896
$988$	−0.356896	−0.356896
$989$	−3.60388	−3.60388
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	1.80194	1.80194
$993$	0	0
$994$	0.246980	0.246980
$995$	0	0
$996$	2.60388	2.60388
$997$	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	3743.1.d.d.3742.2	yes	3
19.18	odd	2		3743.1.d.c.3742.2	✓	3
197.196	even	2		3743.1.d.c.3742.2	✓	3
3743.3742	odd	2	CM	3743.1.d.d.3742.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3743.1.d.c.3742.2	✓	3	19.18	odd	2
3743.1.d.c.3742.2	✓	3	197.196	even	2
3743.1.d.d.3742.2	yes	3	1.1	even	1	trivial
3743.1.d.d.3742.2	yes	3	3743.3742	odd	2	CM

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:	\( N \)	\(=\)	\( 3743 = 19 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3743.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.445042 q^{2} +1.80194 q^{3} -0.801938 q^{4} +0.801938 q^{6} -0.445042 q^{7} -0.801938 q^{8} +2.24698 q^{9} +O(q^{10})\)	\(q+0.445042 q^{2} +1.80194 q^{3} -0.801938 q^{4} +0.801938 q^{6} -0.445042 q^{7} -0.801938 q^{8} +2.24698 q^{9} -1.44504 q^{12} +0.445042 q^{13} -0.198062 q^{14} +0.445042 q^{16} +1.00000 q^{18} +1.00000 q^{19} -0.801938 q^{21} -1.80194 q^{23} -1.44504 q^{24} +1.00000 q^{25} +0.198062 q^{26} +2.24698 q^{27} +0.356896 q^{28} +1.80194 q^{31} +1.00000 q^{32} -1.80194 q^{36} +0.445042 q^{38} +0.801938 q^{39} -0.356896 q^{42} +2.00000 q^{43} -0.801938 q^{46} -0.445042 q^{47} +0.801938 q^{48} -0.801938 q^{49} +0.445042 q^{50} -0.356896 q^{52} +1.00000 q^{54} +0.356896 q^{56} +1.80194 q^{57} -1.80194 q^{61} +0.801938 q^{62} -1.00000 q^{63} +1.80194 q^{67} -3.24698 q^{69} -1.24698 q^{71} -1.80194 q^{72} +1.80194 q^{75} -0.801938 q^{76} +0.356896 q^{78} -1.24698 q^{79} +1.80194 q^{81} -1.80194 q^{83} +0.643104 q^{84} +0.890084 q^{86} -2.00000 q^{89} -0.198062 q^{91} +1.44504 q^{92} +3.24698 q^{93} -0.198062 q^{94} +1.80194 q^{96} -0.356896 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 3 * q + q^2 + q^3 + 2 * q^4 - 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9	\( 3 q + q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{12} + q^{13} - 5 q^{14} + q^{16} + 3 q^{18} + 3 q^{19} + 2 q^{21} - q^{23} - 4 q^{24} + 3 q^{25} + 5 q^{26} + 2 q^{27} - 3 q^{28} + q^{31} + 3 q^{32} - q^{36} + q^{38} - 2 q^{39} + 3 q^{42} + 6 q^{43} + 2 q^{46} - q^{47} - 2 q^{48} + 2 q^{49} + q^{50} + 3 q^{52} + 3 q^{54} - 3 q^{56} + q^{57} - q^{61} - 2 q^{62} - 3 q^{63} + q^{67} - 5 q^{69} + q^{71} - q^{72} + q^{75} + 2 q^{76} - 3 q^{78} + q^{79} + q^{81} - q^{83} + 6 q^{84} + 2 q^{86} - 6 q^{89} - 5 q^{91} + 4 q^{92} + 5 q^{93} - 5 q^{94} + q^{96} + 3 q^{98}+O(q^{100}) \) 3 * q + q^2 + q^3 + 2 * q^4 - 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 - 4 * q^12 + q^13 - 5 * q^14 + q^16 + 3 * q^18 + 3 * q^19 + 2 * q^21 - q^23 - 4 * q^24 + 3 * q^25 + 5 * q^26 + 2 * q^27 - 3 * q^28 + q^31 + 3 * q^32 - q^36 + q^38 - 2 * q^39 + 3 * q^42 + 6 * q^43 + 2 * q^46 - q^47 - 2 * q^48 + 2 * q^49 + q^50 + 3 * q^52 + 3 * q^54 - 3 * q^56 + q^57 - q^61 - 2 * q^62 - 3 * q^63 + q^67 - 5 * q^69 + q^71 - q^72 + q^75 + 2 * q^76 - 3 * q^78 + q^79 + q^81 - q^83 + 6 * q^84 + 2 * q^86 - 6 * q^89 - 5 * q^91 + 4 * q^92 + 5 * q^93 - 5 * q^94 + q^96 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more