LMFDB - Embedded newform 3800.1.b.a.949.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(949,3800)]

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, 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("3800.949");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3800.b (of order $2$, degree $1$, not 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:	no
Analytic conductor:	$1.89644704801$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.152.1
Artin image:	$C_4\times D_6$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{24} - \cdots)$

Embedding invariants

Embedding label			949.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3800.949
Dual form			3800.1.b.a.949.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000 q^{19} -1.00000 q^{21} -1.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} -1.00000 q^{29} +1.00000i q^{32} -1.00000 q^{34} +2.00000i q^{37} +1.00000i q^{38} -1.00000 q^{39} -1.00000i q^{42} +1.00000 q^{46} -2.00000i q^{47} +1.00000i q^{48} -1.00000 q^{51} -1.00000i q^{52} +1.00000i q^{53} -1.00000 q^{54} +1.00000 q^{56} +1.00000i q^{57} -1.00000i q^{58} -1.00000 q^{59} -1.00000 q^{64} -1.00000i q^{67} -1.00000i q^{68} +1.00000 q^{69} -1.00000i q^{73} -2.00000 q^{74} -1.00000 q^{76} -1.00000i q^{78} -1.00000 q^{81} +1.00000 q^{84} -1.00000i q^{87} -1.00000 q^{91} +1.00000i q^{92} +2.00000 q^{94} -1.00000 q^{96} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{14} + 2 q^{16} + 2 q^{19} - 2 q^{21} + 2 q^{24} - 2 q^{26} - 2 q^{29} - 2 q^{34} - 2 q^{39} + 2 q^{46} - 2 q^{51} - 2 q^{54} + 2 q^{56} - 2 q^{59} - 2 q^{64} + 2 q^{69} - 4 q^{74} - 2 q^{76} - 2 q^{81} + 2 q^{84} - 2 q^{91} + 4 q^{94} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^14 + 2 * q^16 + 2 * q^19 - 2 * q^21 + 2 * q^24 - 2 * q^26 - 2 * q^29 - 2 * q^34 - 2 * q^39 + 2 * q^46 - 2 * q^51 - 2 * q^54 + 2 * q^56 - 2 * q^59 - 2 * q^64 + 2 * q^69 - 4 * q^74 - 2 * q^76 - 2 * q^81 + 2 * q^84 - 2 * q^91 + 4 * q^94 - 2 * q^96

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 $

$2$	1.00000i	1.00000i
$2$	1.00000i	1.00000i
$3$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$3$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$4$	−1.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	−1.00000	−1.00000
$7$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$7$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$8$	− 1.00000i	− 1.00000i
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 1.00000i	− 1.00000i
$13$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$13$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$14$	−1.00000	−1.00000
$15$	0	0
$16$	1.00000	1.00000
$17$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$17$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$18$	0	0
$19$	1.00000	1.00000
$19$	1.00000	1.00000
$20$	0	0
$21$	−1.00000	−1.00000
$22$	0	0
$23$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$23$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$24$	1.00000	1.00000
$25$	0	0
$26$	−1.00000	−1.00000
$27$	1.00000i	1.00000i
$28$	− 1.00000i	− 1.00000i
$29$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$29$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.00000i	1.00000i
$33$	0	0
$34$	−1.00000	−1.00000
$35$	0	0
$36$	0	0
$37$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$37$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$38$	1.00000i	1.00000i
$39$	−1.00000	−1.00000
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	− 1.00000i	− 1.00000i
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	1.00000	1.00000
$47$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$47$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$48$	1.00000i	1.00000i
$49$	0	0
$50$	0	0
$51$	−1.00000	−1.00000
$52$	− 1.00000i	− 1.00000i
$53$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$53$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$54$	−1.00000	−1.00000
$55$	0	0
$56$	1.00000	1.00000
$57$	1.00000i	1.00000i
$58$	− 1.00000i	− 1.00000i
$59$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$67$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$68$	− 1.00000i	− 1.00000i
$69$	1.00000	1.00000
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$73$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$74$	−2.00000	−2.00000
$75$	0	0
$76$	−1.00000	−1.00000
$77$	0	0
$78$	− 1.00000i	− 1.00000i
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−1.00000	−1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	1.00000	1.00000
$85$	0	0
$86$	0	0
$87$	− 1.00000i	− 1.00000i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−1.00000	−1.00000
$92$	1.00000i	1.00000i
$93$	0	0
$94$	2.00000	2.00000
$95$	0	0
$96$	−1.00000	−1.00000
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	− 1.00000i	− 1.00000i
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	1.00000	1.00000
$105$	0	0
$106$	−1.00000	−1.00000
$107$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$107$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$108$	− 1.00000i	− 1.00000i
$109$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$110$	0	0
$111$	−2.00000	−2.00000
$112$	1.00000i	1.00000i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−1.00000	−1.00000
$115$	0	0
$116$	1.00000	1.00000
$117$	0	0
$118$	− 1.00000i	− 1.00000i
$119$	−1.00000	−1.00000
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	− 1.00000i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	1.00000i	1.00000i
$134$	1.00000	1.00000
$135$	0	0
$136$	1.00000	1.00000
$137$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$137$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$138$	1.00000i	1.00000i
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	2.00000	2.00000
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	1.00000	1.00000
$147$	0	0
$148$	− 2.00000i	− 2.00000i
$149$	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			$\pi$
$152$	− 1.00000i	− 1.00000i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	1.00000	1.00000
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−1.00000	−1.00000
$160$	0	0
$161$	1.00000	1.00000
$162$	− 1.00000i	− 1.00000i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	1.00000i	1.00000i
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$173$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$174$	1.00000	1.00000
$175$	0	0
$176$	0	0
$177$	− 1.00000i	− 1.00000i
$178$	0	0
$179$	2.00000	2.00000	1.00000			$0$
$179$	2.00000	2.00000	1.00000			$0$
$180$	0	0
$181$	−2.00000	−2.00000	−1.00000			$\pi$
$181$	−2.00000	−2.00000	−1.00000			$\pi$
$182$	− 1.00000i	− 1.00000i
$183$	0	0
$184$	−1.00000	−1.00000
$185$	0	0
$186$	0	0
$187$	0	0
$188$	2.00000i	2.00000i
$189$	−1.00000	−1.00000
$190$	0	0
$191$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$192$	− 1.00000i	− 1.00000i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$199$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$200$	0	0
$201$	1.00000	1.00000
$202$	0	0
$203$	− 1.00000i	− 1.00000i
$204$	1.00000	1.00000
$205$	0	0
$206$	0	0
$207$	0	0
$208$	1.00000i	1.00000i
$209$	0	0
$210$	0	0
$211$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$211$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$212$	− 1.00000i	− 1.00000i
$213$	0	0
$214$	1.00000	1.00000
$215$	0	0
$216$	1.00000	1.00000
$217$	0	0
$218$	− 1.00000i	− 1.00000i
$219$	1.00000	1.00000
$220$	0	0
$221$	−1.00000	−1.00000
$222$	− 2.00000i	− 2.00000i
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−1.00000	−1.00000
$225$	0	0
$226$	0	0
$227$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$227$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$228$	− 1.00000i	− 1.00000i
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	1.00000i	1.00000i
$233$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$233$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	1.00000	1.00000
$237$	0	0
$238$	− 1.00000i	− 1.00000i
$239$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$239$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.00000i	1.00000i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.00000i	1.00000i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−2.00000	−2.00000
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$263$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	−1.00000	−1.00000
$267$	0	0
$268$	1.00000i	1.00000i
$269$	2.00000	2.00000	1.00000			$0$
$269$	2.00000	2.00000	1.00000			$0$
$270$	0	0
$271$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$271$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$272$	1.00000i	1.00000i
$273$	− 1.00000i	− 1.00000i
$274$	−1.00000	−1.00000
$275$	0	0
$276$	−1.00000	−1.00000
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	2.00000i	2.00000i
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0	0
$290$	0	0
$291$	0	0
$292$	1.00000i	1.00000i
$293$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$293$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	2.00000
$297$	0	0
$298$	0	0
$299$	1.00000	1.00000
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	1.00000	1.00000
$305$	0	0
$306$	0	0
$307$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$307$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$312$	1.00000i	1.00000i
$313$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$313$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$317$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$318$	− 1.00000i	− 1.00000i
$319$	0	0
$320$	0	0
$321$	1.00000	1.00000
$322$	1.00000i	1.00000i
$323$	1.00000i	1.00000i
$324$	1.00000	1.00000
$325$	0	0
$326$	0	0
$327$	− 1.00000i	− 1.00000i
$328$	0	0
$329$	2.00000	2.00000
$330$	0	0
$331$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$331$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−1.00000	−1.00000
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	1.00000i
$344$	0	0
$345$	0	0
$346$	2.00000	2.00000
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	1.00000i	1.00000i
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	−1.00000	−1.00000
$352$	0	0
$353$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$353$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$354$	1.00000	1.00000
$355$	0	0
$356$	0	0
$357$	− 1.00000i	− 1.00000i
$358$	2.00000i	2.00000i
$359$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$359$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	− 2.00000i	− 2.00000i
$363$	1.00000i	1.00000i
$364$	1.00000	1.00000
$365$	0	0
$366$	0	0
$367$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$367$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$368$	− 1.00000i	− 1.00000i
$369$	0	0
$370$	0	0
$371$	−1.00000	−1.00000
$372$	0	0
$373$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$373$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$374$	0	0
$375$	0	0
$376$	−2.00000	−2.00000
$377$	− 1.00000i	− 1.00000i
$378$	− 1.00000i	− 1.00000i
$379$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$380$	0	0
$381$	0	0
$382$	− 1.00000i	− 1.00000i
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	1.00000
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	1.00000	1.00000
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	1.00000i	1.00000i
$399$	−1.00000	−1.00000
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	1.00000i	1.00000i
$403$	0	0
$404$	0	0
$405$	0	0
$406$	1.00000	1.00000
$407$	0	0
$408$	1.00000i	1.00000i
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−1.00000	−1.00000
$412$	0	0
$413$	− 1.00000i	− 1.00000i
$414$	0	0
$415$	0	0
$416$	−1.00000	−1.00000
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$421$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$422$	1.00000i	1.00000i
$423$	0	0
$424$	1.00000	1.00000
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.00000i	1.00000i
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.00000i	1.00000i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	1.00000	1.00000
$437$	− 1.00000i	− 1.00000i
$438$	1.00000i	1.00000i
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	− 1.00000i	− 1.00000i
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	2.00000	2.00000
$445$	0	0
$446$	0	0
$447$	0	0
$448$	− 1.00000i	− 1.00000i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	1.00000	1.00000
$455$	0	0
$456$	1.00000	1.00000
$457$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$457$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$458$	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
$463$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$463$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$464$	−1.00000	−1.00000
$465$	0	0
$466$	−2.00000	−2.00000
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	1.00000	1.00000
$470$	0	0
$471$	0	0
$472$	1.00000i	1.00000i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	1.00000	1.00000
$477$	0	0
$478$	1.00000i	1.00000i
$479$	−2.00000	−2.00000	−1.00000			$\pi$
$479$	−2.00000	−2.00000	−1.00000			$\pi$
$480$	0	0
$481$	−2.00000	−2.00000
$482$	0	0
$483$	1.00000i	1.00000i
$484$	−1.00000	−1.00000
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	− 1.00000i	− 1.00000i
$494$	−1.00000	−1.00000
$495$	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
$503$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$503$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.00000	2.00000	1.00000			$0$
$509$	2.00000	2.00000	1.00000			$0$
$510$	0	0
$511$	1.00000	1.00000
$512$	1.00000i	1.00000i
$513$	1.00000i	1.00000i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 2.00000i	− 2.00000i
$519$	2.00000	2.00000
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$523$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$524$	0	0
$525$	0	0
$526$	−2.00000	−2.00000
$527$	0	0
$528$	0	0
$529$	0	0
$530$	0	0
$531$	0	0
$532$	− 1.00000i	− 1.00000i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−1.00000	−1.00000
$537$	2.00000i	2.00000i
$538$	2.00000i	2.00000i
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	− 1.00000i	− 1.00000i
$543$	− 2.00000i	− 2.00000i
$544$	−1.00000	−1.00000
$545$	0	0
$546$	1.00000	1.00000
$547$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$547$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$548$	− 1.00000i	− 1.00000i
$549$	0	0
$550$	0	0
$551$	−1.00000	−1.00000
$552$	− 1.00000i	− 1.00000i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$563$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$564$	−2.00000	−2.00000
$565$	0	0
$566$	0	0
$567$	− 1.00000i	− 1.00000i
$568$	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
$573$	− 1.00000i	− 1.00000i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$577$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−1.00000	−1.00000
$585$	0	0
$586$	−1.00000	−1.00000
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	2.00000i	2.00000i
$593$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$593$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.00000i	1.00000i
$598$	1.00000i	1.00000i
$599$	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
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	1.00000i	1.00000i
$609$	1.00000	1.00000
$610$	0	0
$611$	2.00000	2.00000
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−2.00000	−2.00000
$615$	0	0
$616$	0	0
$617$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$617$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	1.00000	1.00000
$622$	− 1.00000i	− 1.00000i
$623$	0	0
$624$	−1.00000	−1.00000
$625$	0	0
$626$	1.00000	1.00000
$627$	0	0
$628$	0	0
$629$	−2.00000	−2.00000
$630$	0	0
$631$	2.00000	2.00000	1.00000			$0$
$631$	2.00000	2.00000	1.00000			$0$
$632$	0	0
$633$	1.00000i	1.00000i
$634$	1.00000	1.00000
$635$	0	0
$636$	1.00000	1.00000
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	1.00000i	1.00000i
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	−1.00000	−1.00000
$645$	0	0
$646$	−1.00000	−1.00000
$647$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$647$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$648$	1.00000i	1.00000i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	1.00000	1.00000
$655$	0	0
$656$	0	0
$657$	0	0
$658$	2.00000i	2.00000i
$659$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$659$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$660$	0	0
$661$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$661$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$662$	1.00000i	1.00000i
$663$	− 1.00000i	− 1.00000i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.00000i	1.00000i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	− 1.00000i	− 1.00000i
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$677$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.00000	1.00000
$682$	0	0
$683$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$683$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−1.00000
$687$	0	0
$688$	0	0
$689$	−1.00000	−1.00000
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	2.00000i	2.00000i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	−1.00000	−1.00000
$697$	0	0
$698$	0	0
$699$	−2.00000	−2.00000
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	− 1.00000i	− 1.00000i
$703$	2.00000i	2.00000i
$704$	0	0
$705$	0	0
$706$	1.00000	1.00000
$707$	0	0
$708$	1.00000i	1.00000i
$709$	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
$716$	−2.00000	−2.00000
$717$	1.00000i	1.00000i
$718$	1.00000i	1.00000i
$719$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$719$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$720$	0	0
$721$	0	0
$722$	1.00000i	1.00000i
$723$	0	0
$724$	2.00000	2.00000
$725$	0	0
$726$	−1.00000	−1.00000
$727$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$727$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$728$	1.00000i	1.00000i
$729$	−1.00000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	2.00000	2.00000
$735$	0	0
$736$	1.00000	1.00000
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	−1.00000	−1.00000
$742$	− 1.00000i	− 1.00000i
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	−1.00000	−1.00000
$747$	0	0
$748$	0	0
$749$	1.00000	1.00000
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	− 2.00000i	− 2.00000i
$753$	0	0
$754$	1.00000	1.00000
$755$	0	0
$756$	1.00000	1.00000
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	− 1.00000i	− 1.00000i
$759$	0	0
$760$	0	0
$761$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$762$	0	0
$763$	− 1.00000i	− 1.00000i
$764$	1.00000	1.00000
$765$	0	0
$766$	0	0
$767$	− 1.00000i	− 1.00000i
$768$	1.00000i	1.00000i
$769$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$769$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$773$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.00000i	− 2.00000i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	1.00000i	1.00000i
$783$	− 1.00000i	− 1.00000i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$787$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$788$	0	0
$789$	−2.00000	−2.00000
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−1.00000	−1.00000
$797$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$797$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$798$	− 1.00000i	− 1.00000i
$799$	2.00000	2.00000
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−1.00000	−1.00000
$805$	0	0
$806$	0	0
$807$	2.00000i	2.00000i
$808$	0	0
$809$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$809$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$810$	0	0
$811$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$811$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$812$	1.00000i	1.00000i
$813$	− 1.00000i	− 1.00000i
$814$	0	0
$815$	0	0
$816$	−1.00000	−1.00000
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	− 1.00000i	− 1.00000i
$823$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$823$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$824$	0	0
$825$	0	0
$826$	1.00000	1.00000
$827$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$827$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$828$	0	0
$829$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 1.00000i
$833$	0	0
$834$	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
$841$	0	0
$842$	1.00000i	1.00000i
$843$	0	0
$844$	−1.00000	−1.00000
$845$	0	0
$846$	0	0
$847$	1.00000i	1.00000i
$848$	1.00000i	1.00000i
$849$	0	0
$850$	0	0
$851$	2.00000	2.00000
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−1.00000	−1.00000
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	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		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−1.00000	−1.00000
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	1.00000	1.00000
$872$	1.00000i	1.00000i
$873$	0	0
$874$	1.00000	1.00000
$875$	0	0
$876$	−1.00000	−1.00000
$877$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$877$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$878$	0	0
$879$	−1.00000	−1.00000
$880$	0	0
$881$	2.00000	2.00000	1.00000			$0$
$881$	2.00000	2.00000	1.00000			$0$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	1.00000	1.00000
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	2.00000i	2.00000i
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2.00000i	− 2.00000i
$894$	0	0
$895$	0	0
$896$	1.00000	1.00000
$897$	1.00000i	1.00000i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−1.00000	−1.00000
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$907$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$908$	1.00000i	1.00000i
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	1.00000i	1.00000i
$913$	0	0
$914$	−1.00000	−1.00000
$915$	0	0
$916$	0	0
$917$	0	0
$918$	− 1.00000i	− 1.00000i
$919$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$919$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$920$	0	0
$921$	−2.00000	−2.00000
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−2.00000	−2.00000
$927$	0	0
$928$	− 1.00000i	− 1.00000i
$929$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$929$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$930$	0	0
$931$	0	0
$932$	− 2.00000i	− 2.00000i
$933$	− 1.00000i	− 1.00000i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$937$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$938$	1.00000i	1.00000i
$939$	1.00000	1.00000
$940$	0	0
$941$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$941$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$942$	0	0
$943$	0	0
$944$	−1.00000	−1.00000
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	1.00000	1.00000
$950$	0	0
$951$	1.00000	1.00000
$952$	1.00000i	1.00000i
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	−1.00000	−1.00000
$957$	0	0
$958$	− 2.00000i	− 2.00000i
$959$	−1.00000	−1.00000
$960$	0	0
$961$	1.00000	1.00000
$962$	− 2.00000i	− 2.00000i
$963$	0	0
$964$	0	0
$965$	0	0
$966$	−1.00000	−1.00000
$967$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$967$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$968$	− 1.00000i	− 1.00000i
$969$	−1.00000	−1.00000
$970$	0	0
$971$	−2.00000	−2.00000	−1.00000			$\pi$
$971$	−2.00000	−2.00000	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	1.00000	1.00000
$987$	2.00000i	2.00000i
$988$	− 1.00000i	− 1.00000i
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	1.00000i	1.00000i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	−2.00000	−2.00000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.1.b.a.949.2		2
5.2	odd	4		152.1.g.a.37.1	✓	1
5.3	odd	4		3800.1.o.b.1101.1		1
5.4	even	2	inner	3800.1.b.a.949.1		2
8.5	even	2		3800.1.b.b.949.1		2
15.2	even	4		1368.1.i.b.37.1		1
19.18	odd	2		3800.1.b.b.949.1		2
20.7	even	4		608.1.g.a.113.1		1
40.13	odd	4		3800.1.o.a.1101.1		1
40.27	even	4		608.1.g.b.113.1		1
40.29	even	2		3800.1.b.b.949.2		2
40.37	odd	4		152.1.g.b.37.1	yes	1
95.2	even	36		2888.1.s.a.2789.1		6
95.7	odd	12		2888.1.l.b.293.1		2
95.12	even	12		2888.1.l.a.293.1		2
95.17	odd	36		2888.1.s.b.2789.1		6
95.18	even	4		3800.1.o.a.1101.1		1
95.22	even	36		2888.1.s.a.333.1		6
95.27	even	12		2888.1.l.a.69.1		2
95.32	even	36		2888.1.s.a.477.1		6
95.37	even	4		152.1.g.b.37.1	yes	1
95.42	odd	36		2888.1.s.b.1029.1		6
95.47	odd	36		2888.1.s.b.1021.1		6
95.52	even	36		2888.1.s.a.2293.1		6
95.62	odd	36		2888.1.s.b.2293.1		6
95.67	even	36		2888.1.s.a.1021.1		6
95.72	even	36		2888.1.s.a.1029.1		6
95.82	odd	36		2888.1.s.b.477.1		6
95.87	odd	12		2888.1.l.b.69.1		2
95.92	odd	36		2888.1.s.b.333.1		6
95.94	odd	2		3800.1.b.b.949.2		2
120.77	even	4		1368.1.i.a.37.1		1
152.37	odd	2	CM	3800.1.b.a.949.2		2
285.227	odd	4		1368.1.i.a.37.1		1
380.227	odd	4		608.1.g.b.113.1		1
760.37	even	4		152.1.g.a.37.1	✓	1
760.117	even	36		2888.1.s.b.333.1		6
760.157	odd	36		2888.1.s.a.2293.1		6
760.189	odd	2	inner	3800.1.b.a.949.1		2
760.197	odd	12		2888.1.l.a.293.1		2
760.227	odd	4		608.1.g.a.113.1		1
760.237	odd	36		2888.1.s.a.1021.1		6
760.277	odd	12		2888.1.l.a.69.1		2
760.317	even	36		2888.1.s.b.477.1		6
760.357	even	36		2888.1.s.b.1029.1		6
760.397	odd	36		2888.1.s.a.2789.1		6
760.477	even	36		2888.1.s.b.2789.1		6
760.493	even	4		3800.1.o.b.1101.1		1
760.517	odd	36		2888.1.s.a.1029.1		6
760.557	odd	36		2888.1.s.a.477.1		6
760.597	even	12		2888.1.l.b.69.1		2
760.637	even	36		2888.1.s.b.1021.1		6
760.677	even	12		2888.1.l.b.293.1		2
760.717	even	36		2888.1.s.b.2293.1		6
760.757	odd	36		2888.1.s.a.333.1		6
2280.797	odd	4		1368.1.i.b.37.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.1.g.a.37.1	✓	1	5.2	odd	4
152.1.g.a.37.1	✓	1	760.37	even	4
152.1.g.b.37.1	yes	1	40.37	odd	4
152.1.g.b.37.1	yes	1	95.37	even	4
608.1.g.a.113.1		1	20.7	even	4
608.1.g.a.113.1		1	760.227	odd	4
608.1.g.b.113.1		1	40.27	even	4
608.1.g.b.113.1		1	380.227	odd	4
1368.1.i.a.37.1		1	120.77	even	4
1368.1.i.a.37.1		1	285.227	odd	4
1368.1.i.b.37.1		1	15.2	even	4
1368.1.i.b.37.1		1	2280.797	odd	4
2888.1.l.a.69.1		2	95.27	even	12
2888.1.l.a.69.1		2	760.277	odd	12
2888.1.l.a.293.1		2	95.12	even	12
2888.1.l.a.293.1		2	760.197	odd	12
2888.1.l.b.69.1		2	95.87	odd	12
2888.1.l.b.69.1		2	760.597	even	12
2888.1.l.b.293.1		2	95.7	odd	12
2888.1.l.b.293.1		2	760.677	even	12
2888.1.s.a.333.1		6	95.22	even	36
2888.1.s.a.333.1		6	760.757	odd	36
2888.1.s.a.477.1		6	95.32	even	36
2888.1.s.a.477.1		6	760.557	odd	36
2888.1.s.a.1021.1		6	95.67	even	36
2888.1.s.a.1021.1		6	760.237	odd	36
2888.1.s.a.1029.1		6	95.72	even	36
2888.1.s.a.1029.1		6	760.517	odd	36
2888.1.s.a.2293.1		6	95.52	even	36
2888.1.s.a.2293.1		6	760.157	odd	36
2888.1.s.a.2789.1		6	95.2	even	36
2888.1.s.a.2789.1		6	760.397	odd	36
2888.1.s.b.333.1		6	95.92	odd	36
2888.1.s.b.333.1		6	760.117	even	36
2888.1.s.b.477.1		6	95.82	odd	36
2888.1.s.b.477.1		6	760.317	even	36
2888.1.s.b.1021.1		6	95.47	odd	36
2888.1.s.b.1021.1		6	760.637	even	36
2888.1.s.b.1029.1		6	95.42	odd	36
2888.1.s.b.1029.1		6	760.357	even	36
2888.1.s.b.2293.1		6	95.62	odd	36
2888.1.s.b.2293.1		6	760.717	even	36
2888.1.s.b.2789.1		6	95.17	odd	36
2888.1.s.b.2789.1		6	760.477	even	36
3800.1.b.a.949.1		2	5.4	even	2	inner
3800.1.b.a.949.1		2	760.189	odd	2	inner
3800.1.b.a.949.2		2	1.1	even	1	trivial
3800.1.b.a.949.2		2	152.37	odd	2	CM
3800.1.b.b.949.1		2	8.5	even	2
3800.1.b.b.949.1		2	19.18	odd	2
3800.1.b.b.949.2		2	40.29	even	2
3800.1.b.b.949.2		2	95.94	odd	2
3800.1.o.a.1101.1		1	40.13	odd	4
3800.1.o.a.1101.1		1	95.18	even	4
3800.1.o.b.1101.1		1	5.3	odd	4
3800.1.o.b.1101.1		1	760.493	even	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000 q^{19} -1.00000 q^{21} -1.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} -1.00000 q^{29} +1.00000i q^{32} -1.00000 q^{34} +2.00000i q^{37} +1.00000i q^{38} -1.00000 q^{39} -1.00000i q^{42} +1.00000 q^{46} -2.00000i q^{47} +1.00000i q^{48} -1.00000 q^{51} -1.00000i q^{52} +1.00000i q^{53} -1.00000 q^{54} +1.00000 q^{56} +1.00000i q^{57} -1.00000i q^{58} -1.00000 q^{59} -1.00000 q^{64} -1.00000i q^{67} -1.00000i q^{68} +1.00000 q^{69} -1.00000i q^{73} -2.00000 q^{74} -1.00000 q^{76} -1.00000i q^{78} -1.00000 q^{81} +1.00000 q^{84} -1.00000i q^{87} -1.00000 q^{91} +1.00000i q^{92} +2.00000 q^{94} -1.00000 q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{14} + 2 q^{16} + 2 q^{19} - 2 q^{21} + 2 q^{24} - 2 q^{26} - 2 q^{29} - 2 q^{34} - 2 q^{39} + 2 q^{46} - 2 q^{51} - 2 q^{54} + 2 q^{56} - 2 q^{59} - 2 q^{64} + 2 q^{69} - 4 q^{74} - 2 q^{76} - 2 q^{81} + 2 q^{84} - 2 q^{91} + 4 q^{94} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^14 + 2 * q^16 + 2 * q^19 - 2 * q^21 + 2 * q^24 - 2 * q^26 - 2 * q^29 - 2 * q^34 - 2 * q^39 + 2 * q^46 - 2 * q^51 - 2 * q^54 + 2 * q^56 - 2 * q^59 - 2 * q^64 + 2 * q^69 - 4 * q^74 - 2 * q^76 - 2 * q^81 + 2 * q^84 - 2 * q^91 + 4 * q^94 - 2 * q^96

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