LMFDB - Embedded newform 3800.1.o.f.1101.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(1101,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3800.1101");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3800.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:	$1.89644704801$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.8340544000000.1
Artin image:	$D_9$
Artin field:	Galois closure of 9.1.8340544000000.1

Embedding invariants

Embedding label			1101.2
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	3800.1101

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} +0.347296 q^{6} -1.87939 q^{7} +1.00000 q^{8} -0.879385 q^{9} +O(q^{10})$	$q+1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} +0.347296 q^{6} -1.87939 q^{7} +1.00000 q^{8} -0.879385 q^{9} +0.347296 q^{12} +1.53209 q^{13} -1.87939 q^{14} +1.00000 q^{16} +1.53209 q^{17} -0.879385 q^{18} +1.00000 q^{19} -0.652704 q^{21} +0.347296 q^{23} +0.347296 q^{24} +1.53209 q^{26} -0.652704 q^{27} -1.87939 q^{28} +1.53209 q^{29} +1.00000 q^{32} +1.53209 q^{34} -0.879385 q^{36} -1.00000 q^{37} +1.00000 q^{38} +0.532089 q^{39} -0.652704 q^{42} +0.347296 q^{46} -1.00000 q^{47} +0.347296 q^{48} +2.53209 q^{49} +0.532089 q^{51} +1.53209 q^{52} -1.87939 q^{53} -0.652704 q^{54} -1.87939 q^{56} +0.347296 q^{57} +1.53209 q^{58} +0.347296 q^{59} +1.65270 q^{63} +1.00000 q^{64} -1.87939 q^{67} +1.53209 q^{68} +0.120615 q^{69} -0.879385 q^{72} +0.347296 q^{73} -1.00000 q^{74} +1.00000 q^{76} +0.532089 q^{78} +0.652704 q^{81} -0.652704 q^{84} +0.532089 q^{87} -2.87939 q^{91} +0.347296 q^{92} -1.00000 q^{94} +0.347296 q^{96} +2.53209 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9} + 3 q^{16} + 3 q^{18} + 3 q^{19} - 3 q^{21} - 3 q^{27} + 3 q^{32} + 3 q^{36} - 3 q^{37} + 3 q^{38} - 3 q^{39} - 3 q^{42} - 3 q^{47} + 3 q^{49} - 3 q^{51} - 3 q^{54} + 6 q^{63} + 3 q^{64} + 6 q^{69} + 3 q^{72} - 3 q^{74} + 3 q^{76} - 3 q^{78} + 3 q^{81} - 3 q^{84} - 3 q^{87} - 3 q^{91} - 3 q^{94} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^9 + 3 * q^16 + 3 * q^18 + 3 * q^19 - 3 * q^21 - 3 * q^27 + 3 * q^32 + 3 * q^36 - 3 * q^37 + 3 * q^38 - 3 * q^39 - 3 * q^42 - 3 * q^47 + 3 * q^49 - 3 * q^51 - 3 * q^54 + 6 * q^63 + 3 * q^64 + 6 * q^69 + 3 * q^72 - 3 * q^74 + 3 * q^76 - 3 * q^78 + 3 * q^81 - 3 * q^84 - 3 * q^87 - 3 * q^91 - 3 * q^94 + 3 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$3$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$4$	1.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0.347296	0.347296
$7$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$7$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$8$	1.00000	1.00000
$9$	−0.879385	−0.879385
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0.347296	0.347296
$13$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$13$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$14$	−1.87939	−1.87939
$15$	0	0
$16$	1.00000	1.00000
$17$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$17$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$18$	−0.879385	−0.879385
$19$	1.00000	1.00000
$19$	1.00000	1.00000
$20$	0	0
$21$	−0.652704	−0.652704
$22$	0	0
$23$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$23$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$24$	0.347296	0.347296
$25$	0	0
$26$	1.53209	1.53209
$27$	−0.652704	−0.652704
$28$	−1.87939	−1.87939
$29$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$29$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.00000	1.00000
$33$	0	0
$34$	1.53209	1.53209
$35$	0	0
$36$	−0.879385	−0.879385
$37$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$37$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$38$	1.00000	1.00000
$39$	0.532089	0.532089
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−0.652704	−0.652704
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0.347296	0.347296
$47$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$47$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$48$	0.347296	0.347296
$49$	2.53209	2.53209
$50$	0	0
$51$	0.532089	0.532089
$52$	1.53209	1.53209
$53$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$53$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$54$	−0.652704	−0.652704
$55$	0	0
$56$	−1.87939	−1.87939
$57$	0.347296	0.347296
$58$	1.53209	1.53209
$59$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$59$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	1.65270	1.65270
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$67$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$68$	1.53209	1.53209
$69$	0.120615	0.120615
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.879385	−0.879385
$73$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$73$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$74$	−1.00000	−1.00000
$75$	0	0
$76$	1.00000	1.00000
$77$	0	0
$78$	0.532089	0.532089
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0.652704	0.652704
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−0.652704	−0.652704
$85$	0	0
$86$	0	0
$87$	0.532089	0.532089
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−2.87939	−2.87939
$92$	0.347296	0.347296
$93$	0	0
$94$	−1.00000	−1.00000
$95$	0	0
$96$	0.347296	0.347296
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	2.53209	2.53209
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0.532089	0.532089
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	1.53209	1.53209
$105$	0	0
$106$	−1.87939	−1.87939
$107$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$107$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$108$	−0.652704	−0.652704
$109$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$109$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$110$	0	0
$111$	−0.347296	−0.347296
$112$	−1.87939	−1.87939
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0.347296	0.347296
$115$	0	0
$116$	1.53209	1.53209
$117$	−1.34730	−1.34730
$118$	0.347296	0.347296
$119$	−2.87939	−2.87939
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	1.65270	1.65270
$127$	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
$133$	−1.87939	−1.87939
$134$	−1.87939	−1.87939
$135$	0	0
$136$	1.53209	1.53209
$137$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$137$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$138$	0.120615	0.120615
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−0.347296	−0.347296
$142$	0	0
$143$	0	0
$144$	−0.879385	−0.879385
$145$	0	0
$146$	0.347296	0.347296
$147$	0.879385	0.879385
$148$	−1.00000	−1.00000
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	1.00000	1.00000
$153$	−1.34730	−1.34730
$154$	0	0
$155$	0	0
$156$	0.532089	0.532089
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−0.652704	−0.652704
$160$	0	0
$161$	−0.652704	−0.652704
$162$	0.652704	0.652704
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−0.652704	−0.652704
$169$	1.34730	1.34730
$170$	0	0
$171$	−0.879385	−0.879385
$172$	0	0
$173$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$173$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$174$	0.532089	0.532089
$175$	0	0
$176$	0	0
$177$	0.120615	0.120615
$178$	0	0
$179$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$179$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$180$	0	0
$181$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$182$	−2.87939	−2.87939
$183$	0	0
$184$	0.347296	0.347296
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−1.00000	−1.00000
$189$	1.22668	1.22668
$190$	0	0
$191$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$191$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$192$	0.347296	0.347296
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	2.53209	2.53209
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$199$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$200$	0	0
$201$	−0.652704	−0.652704
$202$	0	0
$203$	−2.87939	−2.87939
$204$	0.532089	0.532089
$205$	0	0
$206$	0	0
$207$	−0.305407	−0.305407
$208$	1.53209	1.53209
$209$	0	0
$210$	0	0
$211$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$211$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$212$	−1.87939	−1.87939
$213$	0	0
$214$	1.53209	1.53209
$215$	0	0
$216$	−0.652704	−0.652704
$217$	0	0
$218$	−1.87939	−1.87939
$219$	0.120615	0.120615
$220$	0	0
$221$	2.34730	2.34730
$222$	−0.347296	−0.347296
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−1.87939	−1.87939
$225$	0	0
$226$	0	0
$227$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$227$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$228$	0.347296	0.347296
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	1.53209	1.53209
$233$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$233$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$234$	−1.34730	−1.34730
$235$	0	0
$236$	0.347296	0.347296
$237$	0	0
$238$	−2.87939	−2.87939
$239$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$239$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.00000	1.00000
$243$	0.879385	0.879385
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.53209	1.53209
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	1.65270	1.65270
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	1.87939	1.87939
$260$	0	0
$261$	−1.34730	−1.34730
$262$	0	0
$263$	2.00000	2.00000	1.00000			$0$
$263$	2.00000	2.00000	1.00000			$0$
$264$	0	0
$265$	0	0
$266$	−1.87939	−1.87939
$267$	0	0
$268$	−1.87939	−1.87939
$269$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$270$	0	0
$271$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$271$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$272$	1.53209	1.53209
$273$	−1.00000	−1.00000
$274$	−1.87939	−1.87939
$275$	0	0
$276$	0.120615	0.120615
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−0.347296	−0.347296
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−0.879385	−0.879385
$289$	1.34730	1.34730
$290$	0	0
$291$	0	0
$292$	0.347296	0.347296
$293$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$293$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$294$	0.879385	0.879385
$295$	0	0
$296$	−1.00000	−1.00000
$297$	0	0
$298$	0	0
$299$	0.532089	0.532089
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	1.00000	1.00000
$305$	0	0
$306$	−1.34730	−1.34730
$307$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$307$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$311$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$312$	0.532089	0.532089
$313$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$313$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$317$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$318$	−0.652704	−0.652704
$319$	0	0
$320$	0	0
$321$	0.532089	0.532089
$322$	−0.652704	−0.652704
$323$	1.53209	1.53209
$324$	0.652704	0.652704
$325$	0	0
$326$	0	0
$327$	−0.652704	−0.652704
$328$	0	0
$329$	1.87939	1.87939
$330$	0	0
$331$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$331$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$332$	0	0
$333$	0.879385	0.879385
$334$	0	0
$335$	0	0
$336$	−0.652704	−0.652704
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	1.34730	1.34730
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−0.879385	−0.879385
$343$	−2.87939	−2.87939
$344$	0	0
$345$	0	0
$346$	−1.00000	−1.00000
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0.532089	0.532089
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	−1.00000	−1.00000
$352$	0	0
$353$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$353$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$354$	0.120615	0.120615
$355$	0	0
$356$	0	0
$357$	−1.00000	−1.00000
$358$	−1.00000	−1.00000
$359$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$359$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	−1.00000	−1.00000
$363$	0.347296	0.347296
$364$	−2.87939	−2.87939
$365$	0	0
$366$	0	0
$367$	2.00000	2.00000	1.00000			$0$
$367$	2.00000	2.00000	1.00000			$0$
$368$	0.347296	0.347296
$369$	0	0
$370$	0	0
$371$	3.53209	3.53209
$372$	0	0
$373$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$373$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$374$	0	0
$375$	0	0
$376$	−1.00000	−1.00000
$377$	2.34730	2.34730
$378$	1.22668	1.22668
$379$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$379$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$380$	0	0
$381$	0	0
$382$	0.347296	0.347296
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0.347296	0.347296
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0.532089	0.532089
$392$	2.53209	2.53209
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−1.87939	−1.87939
$399$	−0.652704	−0.652704
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−0.652704	−0.652704
$403$	0	0
$404$	0	0
$405$	0	0
$406$	−2.87939	−2.87939
$407$	0	0
$408$	0.532089	0.532089
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−0.652704	−0.652704
$412$	0	0
$413$	−0.652704	−0.652704
$414$	−0.305407	−0.305407
$415$	0	0
$416$	1.53209	1.53209
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$421$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$422$	0.347296	0.347296
$423$	0.879385	0.879385
$424$	−1.87939	−1.87939
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.53209	1.53209
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.652704	−0.652704
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	−1.87939	−1.87939
$437$	0.347296	0.347296
$438$	0.120615	0.120615
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−2.22668	−2.22668
$442$	2.34730	2.34730
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−0.347296	−0.347296
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−1.87939	−1.87939
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−1.87939	−1.87939
$455$	0	0
$456$	0.347296	0.347296
$457$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$457$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$458$	0	0
$459$	−1.00000	−1.00000
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$464$	1.53209	1.53209
$465$	0	0
$466$	−1.00000	−1.00000
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−1.34730	−1.34730
$469$	3.53209	3.53209
$470$	0	0
$471$	0	0
$472$	0.347296	0.347296
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−2.87939	−2.87939
$477$	1.65270	1.65270
$478$	0.347296	0.347296
$479$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$480$	0	0
$481$	−1.53209	−1.53209
$482$	0	0
$483$	−0.226682	−0.226682
$484$	1.00000	1.00000
$485$	0	0
$486$	0.879385	0.879385
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	2.34730	2.34730
$494$	1.53209	1.53209
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$503$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$504$	1.65270	1.65270
$505$	0	0
$506$	0	0
$507$	0.467911	0.467911
$508$	0	0
$509$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$510$	0	0
$511$	−0.652704	−0.652704
$512$	1.00000	1.00000
$513$	−0.652704	−0.652704
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	1.87939	1.87939
$519$	−0.347296	−0.347296
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1.34730	−1.34730
$523$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$523$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$524$	0	0
$525$	0	0
$526$	2.00000	2.00000
$527$	0	0
$528$	0	0
$529$	−0.879385	−0.879385
$530$	0	0
$531$	−0.305407	−0.305407
$532$	−1.87939	−1.87939
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−1.87939	−1.87939
$537$	−0.347296	−0.347296
$538$	−1.00000	−1.00000
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0.347296	0.347296
$543$	−0.347296	−0.347296
$544$	1.53209	1.53209
$545$	0	0
$546$	−1.00000	−1.00000
$547$	2.00000	2.00000	1.00000			$0$
$547$	2.00000	2.00000	1.00000			$0$
$548$	−1.87939	−1.87939
$549$	0	0
$550$	0	0
$551$	1.53209	1.53209
$552$	0.120615	0.120615
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$563$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$564$	−0.347296	−0.347296
$565$	0	0
$566$	0	0
$567$	−1.22668	−1.22668
$568$	0	0
$569$	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
$573$	0.120615	0.120615
$574$	0	0
$575$	0	0
$576$	−0.879385	−0.879385
$577$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$577$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$578$	1.34730	1.34730
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0.347296	0.347296
$585$	0	0
$586$	1.53209	1.53209
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.879385	0.879385
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−1.00000	−1.00000
$593$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$593$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.652704	−0.652704
$598$	0.532089	0.532089
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	1.65270	1.65270
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.00000	1.00000
$609$	−1.00000	−1.00000
$610$	0	0
$611$	−1.53209	−1.53209
$612$	−1.34730	−1.34730
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−1.00000	−1.00000
$615$	0	0
$616$	0	0
$617$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−0.226682	−0.226682
$622$	−1.87939	−1.87939
$623$	0	0
$624$	0.532089	0.532089
$625$	0	0
$626$	1.53209	1.53209
$627$	0	0
$628$	0	0
$629$	−1.53209	−1.53209
$630$	0	0
$631$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$632$	0	0
$633$	0.120615	0.120615
$634$	0.347296	0.347296
$635$	0	0
$636$	−0.652704	−0.652704
$637$	3.87939	3.87939
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0.532089	0.532089
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−0.652704	−0.652704
$645$	0	0
$646$	1.53209	1.53209
$647$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$647$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$648$	0.652704	0.652704
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−0.652704	−0.652704
$655$	0	0
$656$	0	0
$657$	−0.305407	−0.305407
$658$	1.87939	1.87939
$659$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$659$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$660$	0	0
$661$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$661$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$662$	1.53209	1.53209
$663$	0.815207	0.815207
$664$	0	0
$665$	0	0
$666$	0.879385	0.879385
$667$	0.532089	0.532089
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	−0.652704	−0.652704
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	1.34730	1.34730
$677$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$677$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.652704	−0.652704
$682$	0	0
$683$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$684$	−0.879385	−0.879385
$685$	0	0
$686$	−2.87939	−2.87939
$687$	0	0
$688$	0	0
$689$	−2.87939	−2.87939
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−1.00000	−1.00000
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0.532089	0.532089
$697$	0	0
$698$	0	0
$699$	−0.347296	−0.347296
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−1.00000	−1.00000
$703$	−1.00000	−1.00000
$704$	0	0
$705$	0	0
$706$	−1.87939	−1.87939
$707$	0	0
$708$	0.120615	0.120615
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	−1.00000	−1.00000
$715$	0	0
$716$	−1.00000	−1.00000
$717$	0.120615	0.120615
$718$	1.53209	1.53209
$719$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$719$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$720$	0	0
$721$	0	0
$722$	1.00000	1.00000
$723$	0	0
$724$	−1.00000	−1.00000
$725$	0	0
$726$	0.347296	0.347296
$727$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$727$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$728$	−2.87939	−2.87939
$729$	−0.347296	−0.347296
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	2.00000	2.00000
$735$	0	0
$736$	0.347296	0.347296
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0.532089	0.532089
$742$	3.53209	3.53209
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0.347296	0.347296
$747$	0	0
$748$	0	0
$749$	−2.87939	−2.87939
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−1.00000	−1.00000
$753$	0	0
$754$	2.34730	2.34730
$755$	0	0
$756$	1.22668	1.22668
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−1.87939	−1.87939
$759$	0	0
$760$	0	0
$761$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$761$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$762$	0	0
$763$	3.53209	3.53209
$764$	0.347296	0.347296
$765$	0	0
$766$	0	0
$767$	0.532089	0.532089
$768$	0.347296	0.347296
$769$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$769$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$773$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.652704	0.652704
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0.532089	0.532089
$783$	−1.00000	−1.00000
$784$	2.53209	2.53209
$785$	0	0
$786$	0	0
$787$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$787$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$788$	0	0
$789$	0.694593	0.694593
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−1.87939	−1.87939
$797$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$797$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$798$	−0.652704	−0.652704
$799$	−1.53209	−1.53209
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−0.652704	−0.652704
$805$	0	0
$806$	0	0
$807$	−0.347296	−0.347296
$808$	0	0
$809$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$809$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$810$	0	0
$811$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$811$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$812$	−2.87939	−2.87939
$813$	0.120615	0.120615
$814$	0	0
$815$	0	0
$816$	0.532089	0.532089
$817$	0	0
$818$	0	0
$819$	2.53209	2.53209
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−0.652704	−0.652704
$823$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$823$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$824$	0	0
$825$	0	0
$826$	−0.652704	−0.652704
$827$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$827$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$828$	−0.305407	−0.305407
$829$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$829$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$830$	0	0
$831$	0	0
$832$	1.53209	1.53209
$833$	3.87939	3.87939
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.34730	1.34730
$842$	0.347296	0.347296
$843$	0	0
$844$	0.347296	0.347296
$845$	0	0
$846$	0.879385	0.879385
$847$	−1.87939	−1.87939
$848$	−1.87939	−1.87939
$849$	0	0
$850$	0	0
$851$	−0.347296	−0.347296
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	1.53209	1.53209
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−0.652704	−0.652704
$865$	0	0
$866$	0	0
$867$	0.467911	0.467911
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−2.87939	−2.87939
$872$	−1.87939	−1.87939
$873$	0	0
$874$	0.347296	0.347296
$875$	0	0
$876$	0.120615	0.120615
$877$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$877$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$878$	0	0
$879$	0.532089	0.532089
$880$	0	0
$881$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$882$	−2.22668	−2.22668
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	2.34730	2.34730
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−0.347296	−0.347296
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.00000	−1.00000
$894$	0	0
$895$	0	0
$896$	−1.87939	−1.87939
$897$	0.184793	0.184793
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−2.87939	−2.87939
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$907$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$908$	−1.87939	−1.87939
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.347296	0.347296
$913$	0	0
$914$	1.53209	1.53209
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−1.00000	−1.00000
$919$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$919$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$920$	0	0
$921$	−0.347296	−0.347296
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−1.00000	−1.00000
$927$	0	0
$928$	1.53209	1.53209
$929$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$929$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$930$	0	0
$931$	2.53209	2.53209
$932$	−1.00000	−1.00000
$933$	−0.652704	−0.652704
$934$	0	0
$935$	0	0
$936$	−1.34730	−1.34730
$937$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$937$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$938$	3.53209	3.53209
$939$	0.532089	0.532089
$940$	0	0
$941$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$941$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$942$	0	0
$943$	0	0
$944$	0.347296	0.347296
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0.532089	0.532089
$950$	0	0
$951$	0.120615	0.120615
$952$	−2.87939	−2.87939
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	1.65270	1.65270
$955$	0	0
$956$	0.347296	0.347296
$957$	0	0
$958$	−1.00000	−1.00000
$959$	3.53209	3.53209
$960$	0	0
$961$	1.00000	1.00000
$962$	−1.53209	−1.53209
$963$	−1.34730	−1.34730
$964$	0	0
$965$	0	0
$966$	−0.226682	−0.226682
$967$	2.00000	2.00000	1.00000			$0$
$967$	2.00000	2.00000	1.00000			$0$
$968$	1.00000	1.00000
$969$	0.532089	0.532089
$970$	0	0
$971$	2.00000	2.00000	1.00000			$0$
$971$	2.00000	2.00000	1.00000			$0$
$972$	0.879385	0.879385
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1.65270	1.65270
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	2.34730	2.34730
$987$	0.652704	0.652704
$988$	1.53209	1.53209
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0.532089	0.532089
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0.652704	0.652704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.1.o.f.1101.2	yes	3
5.2	odd	4		3800.1.b.c.949.5		6
5.3	odd	4		3800.1.b.c.949.2		6
5.4	even	2		3800.1.o.d.1101.2	yes	3
8.5	even	2		3800.1.o.c.1101.2	✓	3
19.18	odd	2		3800.1.o.c.1101.2	✓	3
40.13	odd	4		3800.1.b.d.949.5		6
40.29	even	2		3800.1.o.e.1101.2	yes	3
40.37	odd	4		3800.1.b.d.949.2		6
95.18	even	4		3800.1.b.d.949.5		6
95.37	even	4		3800.1.b.d.949.2		6
95.94	odd	2		3800.1.o.e.1101.2	yes	3
152.37	odd	2	CM	3800.1.o.f.1101.2	yes	3
760.37	even	4		3800.1.b.c.949.5		6
760.189	odd	2		3800.1.o.d.1101.2	yes	3
760.493	even	4		3800.1.b.c.949.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.1.b.c.949.2		6	5.3	odd	4
3800.1.b.c.949.2		6	760.493	even	4
3800.1.b.c.949.5		6	5.2	odd	4
3800.1.b.c.949.5		6	760.37	even	4
3800.1.b.d.949.2		6	40.37	odd	4
3800.1.b.d.949.2		6	95.37	even	4
3800.1.b.d.949.5		6	40.13	odd	4
3800.1.b.d.949.5		6	95.18	even	4
3800.1.o.c.1101.2	✓	3	8.5	even	2
3800.1.o.c.1101.2	✓	3	19.18	odd	2
3800.1.o.d.1101.2	yes	3	5.4	even	2
3800.1.o.d.1101.2	yes	3	760.189	odd	2
3800.1.o.e.1101.2	yes	3	40.29	even	2
3800.1.o.e.1101.2	yes	3	95.94	odd	2
3800.1.o.f.1101.2	yes	3	1.1	even	1	trivial
3800.1.o.f.1101.2	yes	3	152.37	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3800.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} +0.347296 q^{6} -1.87939 q^{7} +1.00000 q^{8} -0.879385 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} +0.347296 q^{6} -1.87939 q^{7} +1.00000 q^{8} -0.879385 q^{9} +0.347296 q^{12} +1.53209 q^{13} -1.87939 q^{14} +1.00000 q^{16} +1.53209 q^{17} -0.879385 q^{18} +1.00000 q^{19} -0.652704 q^{21} +0.347296 q^{23} +0.347296 q^{24} +1.53209 q^{26} -0.652704 q^{27} -1.87939 q^{28} +1.53209 q^{29} +1.00000 q^{32} +1.53209 q^{34} -0.879385 q^{36} -1.00000 q^{37} +1.00000 q^{38} +0.532089 q^{39} -0.652704 q^{42} +0.347296 q^{46} -1.00000 q^{47} +0.347296 q^{48} +2.53209 q^{49} +0.532089 q^{51} +1.53209 q^{52} -1.87939 q^{53} -0.652704 q^{54} -1.87939 q^{56} +0.347296 q^{57} +1.53209 q^{58} +0.347296 q^{59} +1.65270 q^{63} +1.00000 q^{64} -1.87939 q^{67} +1.53209 q^{68} +0.120615 q^{69} -0.879385 q^{72} +0.347296 q^{73} -1.00000 q^{74} +1.00000 q^{76} +0.532089 q^{78} +0.652704 q^{81} -0.652704 q^{84} +0.532089 q^{87} -2.87939 q^{91} +0.347296 q^{92} -1.00000 q^{94} +0.347296 q^{96} +2.53209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9} + 3 q^{16} + 3 q^{18} + 3 q^{19} - 3 q^{21} - 3 q^{27} + 3 q^{32} + 3 q^{36} - 3 q^{37} + 3 q^{38} - 3 q^{39} - 3 q^{42} - 3 q^{47} + 3 q^{49} - 3 q^{51} - 3 q^{54} + 6 q^{63} + 3 q^{64} + 6 q^{69} + 3 q^{72} - 3 q^{74} + 3 q^{76} - 3 q^{78} + 3 q^{81} - 3 q^{84} - 3 q^{87} - 3 q^{91} - 3 q^{94} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^9 + 3 * q^16 + 3 * q^18 + 3 * q^19 - 3 * q^21 - 3 * q^27 + 3 * q^32 + 3 * q^36 - 3 * q^37 + 3 * q^38 - 3 * q^39 - 3 * q^42 - 3 * q^47 + 3 * q^49 - 3 * q^51 - 3 * q^54 + 6 * q^63 + 3 * q^64 + 6 * q^69 + 3 * q^72 - 3 * q^74 + 3 * q^76 - 3 * q^78 + 3 * q^81 - 3 * q^84 - 3 * q^87 - 3 * q^91 - 3 * q^94 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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