LMFDB - Embedded newform 2183.1.d.c.2182.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2183,1,Mod(2182,2183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2183, 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("2183.2182");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2183 = 37 \cdot 59 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2183.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:	no
Analytic conductor:	$1.08945892258$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.2183.1
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.10403062487.3

Embedding invariants

Embedding label			2182.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	2183.2182
Dual form			2183.1.d.c.2182.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.41421i q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{2} -1.41421i q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +1.41421i q^{10} -1.41421i q^{11} +1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} -1.41421i q^{17} +1.00000 q^{18} +1.41421i q^{19} +1.41421i q^{22} -1.00000 q^{25} -1.00000 q^{26} -1.41421i q^{29} +1.00000 q^{31} +1.41421i q^{34} +1.41421i q^{35} -1.00000 q^{37} -1.41421i q^{38} -1.41421i q^{40} -1.00000 q^{41} +1.41421i q^{45} +1.41421i q^{47} +1.00000 q^{50} -1.00000 q^{53} -2.00000 q^{55} -1.00000 q^{56} +1.41421i q^{58} +1.00000 q^{59} -1.00000 q^{61} -1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.41421i q^{65} -1.41421i q^{67} -1.41421i q^{70} -1.00000 q^{71} -1.00000 q^{72} +1.41421i q^{73} +1.00000 q^{74} +1.41421i q^{77} +1.41421i q^{80} +1.00000 q^{81} +1.00000 q^{82} -2.00000 q^{85} -1.41421i q^{88} -1.00000 q^{89} -1.41421i q^{90} -1.00000 q^{91} -1.41421i q^{94} +2.00000 q^{95} -1.00000 q^{97} +1.41421i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{7} + 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^7 + 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{7} + 2 q^{8} - 2 q^{9} + 2 q^{13} + 2 q^{14} - 2 q^{16} + 2 q^{18} - 2 q^{25} - 2 q^{26} + 2 q^{31} - 2 q^{37} - 2 q^{41} + 2 q^{50} - 2 q^{53} - 4 q^{55} - 2 q^{56} + 2 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{63} + 2 q^{64} - 2 q^{71} - 2 q^{72} + 2 q^{74} + 2 q^{81} + 2 q^{82} - 4 q^{85} - 2 q^{89} - 2 q^{91} + 4 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^7 + 2 * q^8 - 2 * q^9 + 2 * q^13 + 2 * q^14 - 2 * q^16 + 2 * q^18 - 2 * q^25 - 2 * q^26 + 2 * q^31 - 2 * q^37 - 2 * q^41 + 2 * q^50 - 2 * q^53 - 4 * q^55 - 2 * q^56 + 2 * q^59 - 2 * q^61 - 2 * q^62 + 2 * q^63 + 2 * q^64 - 2 * q^71 - 2 * q^72 + 2 * q^74 + 2 * q^81 + 2 * q^82 - 4 * q^85 - 2 * q^89 - 2 * q^91 + 4 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2183\mathbb{Z}\right)^\times$.

$n$	$297$	$1889$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$5$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$6$	0	0
$7$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$7$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$8$	1.00000	1.00000
$9$	−1.00000	−1.00000
$10$	1.41421i	1.41421i
$11$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$11$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$12$	0	0
$13$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$13$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$14$	1.00000	1.00000
$15$	0	0
$16$	−1.00000	−1.00000
$17$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$17$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$18$	1.00000	1.00000
$19$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$19$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$20$	0	0
$21$	0	0
$22$	1.41421i	1.41421i
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−1.00000
$26$	−1.00000	−1.00000
$27$	0	0
$28$	0	0
$29$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$29$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$30$	0	0
$31$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$31$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$32$	0	0
$33$	0	0
$34$	1.41421i	1.41421i
$35$	1.41421i	1.41421i
$36$	0	0
$37$	−1.00000	−1.00000
$37$	−1.00000	−1.00000
$38$	− 1.41421i	− 1.41421i
$39$	0	0
$40$	− 1.41421i	− 1.41421i
$41$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$41$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.41421i	1.41421i
$46$	0	0
$47$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$47$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$48$	0	0
$49$	0	0
$50$	1.00000	1.00000
$51$	0	0
$52$	0	0
$53$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$53$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$54$	0	0
$55$	−2.00000	−2.00000
$56$	−1.00000	−1.00000
$57$	0	0
$58$	1.41421i	1.41421i
$59$	1.00000	1.00000
$59$	1.00000	1.00000
$60$	0	0
$61$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$62$	−1.00000	−1.00000
$63$	1.00000	1.00000
$64$	1.00000	1.00000
$65$	− 1.41421i	− 1.41421i
$66$	0	0
$67$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$67$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$68$	0	0
$69$	0	0
$70$	− 1.41421i	− 1.41421i
$71$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$72$	−1.00000	−1.00000
$73$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$73$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$74$	1.00000	1.00000
$75$	0	0
$76$	0	0
$77$	1.41421i	1.41421i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.41421i	1.41421i
$81$	1.00000	1.00000
$82$	1.00000	1.00000
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−2.00000	−2.00000
$86$	0	0
$87$	0	0
$88$	− 1.41421i	− 1.41421i
$89$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$90$	− 1.41421i	− 1.41421i
$91$	−1.00000	−1.00000
$92$	0	0
$93$	0	0
$94$	− 1.41421i	− 1.41421i
$95$	2.00000	2.00000
$96$	0	0
$97$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$97$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$98$	0	0
$99$	1.41421i	1.41421i
$100$	0	0
$101$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$101$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$102$	0	0
$103$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$103$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$104$	1.00000	1.00000
$105$	0	0
$106$	1.00000	1.00000
$107$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$107$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	2.00000	2.00000
$111$	0	0
$112$	1.00000	1.00000
$113$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$113$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−1.00000
$118$	−1.00000	−1.00000
$119$	1.41421i	1.41421i
$120$	0	0
$121$	−1.00000	−1.00000
$122$	1.00000	1.00000
$123$	0	0
$124$	0	0
$125$	0	0
$126$	−1.00000	−1.00000
$127$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$127$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	1.41421i	1.41421i
$131$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$131$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$132$	0	0
$133$	− 1.41421i	− 1.41421i
$134$	1.41421i	1.41421i
$135$	0	0
$136$	− 1.41421i	− 1.41421i
$137$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$137$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$138$	0	0
$139$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$140$	0	0
$141$	0	0
$142$	1.00000	1.00000
$143$	− 1.41421i	− 1.41421i
$144$	1.00000	1.00000
$145$	−2.00000	−2.00000
$146$	− 1.41421i	− 1.41421i
$147$	0	0
$148$	0	0
$149$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$149$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	1.41421i	1.41421i
$153$	1.41421i	1.41421i
$154$	− 1.41421i	− 1.41421i
$155$	− 1.41421i	− 1.41421i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−1.00000	−1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$167$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$168$	0	0
$169$	0	0
$170$	2.00000	2.00000
$171$	− 1.41421i	− 1.41421i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	1.00000	1.00000
$176$	1.41421i	1.41421i
$177$	0	0
$178$	1.00000	1.00000
$179$	−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
$181$	−2.00000	−2.00000	−1.00000			$\pi$
$181$	−2.00000	−2.00000	−1.00000			$\pi$
$182$	1.00000	1.00000
$183$	0	0
$184$	0	0
$185$	1.41421i	1.41421i
$186$	0	0
$187$	−2.00000	−2.00000
$188$	0	0
$189$	0	0
$190$	−2.00000	−2.00000
$191$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$191$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	1.00000	1.00000
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	− 1.41421i	− 1.41421i
$199$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$199$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$200$	−1.00000	−1.00000
$201$	0	0
$202$	− 1.41421i	− 1.41421i
$203$	1.41421i	1.41421i
$204$	0	0
$205$	1.41421i	1.41421i
$206$	−1.00000	−1.00000
$207$	0	0
$208$	−1.00000	−1.00000
$209$	2.00000	2.00000
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	1.00000	1.00000
$215$	0	0
$216$	0	0
$217$	−1.00000	−1.00000
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.41421i	− 1.41421i
$222$	0	0
$223$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$223$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$224$	0	0
$225$	1.00000	1.00000
$226$	−1.00000	−1.00000
$227$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$227$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$228$	0	0
$229$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$229$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$230$	0	0
$231$	0	0
$232$	− 1.41421i	− 1.41421i
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	1.00000	1.00000
$235$	2.00000	2.00000
$236$	0	0
$237$	0	0
$238$	− 1.41421i	− 1.41421i
$239$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$239$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$240$	0	0
$241$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$241$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$242$	1.00000	1.00000
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.41421i	1.41421i
$248$	1.00000	1.00000
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	−1.00000	−1.00000
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	1.00000	1.00000
$260$	0	0
$261$	1.41421i	1.41421i
$262$	1.00000	1.00000
$263$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$263$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$264$	0	0
$265$	1.41421i	1.41421i
$266$	1.41421i	1.41421i
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	1.41421i	1.41421i
$273$	0	0
$274$	−1.00000	−1.00000
$275$	1.41421i	1.41421i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.00000	1.00000
$279$	−1.00000	−1.00000
$280$	1.41421i	1.41421i
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$283$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$284$	0	0
$285$	0	0
$286$	1.41421i	1.41421i
$287$	1.00000	1.00000
$288$	0	0
$289$	−1.00000	−1.00000
$290$	2.00000	2.00000
$291$	0	0
$292$	0	0
$293$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$293$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$294$	0	0
$295$	− 1.41421i	− 1.41421i
$296$	−1.00000	−1.00000
$297$	0	0
$298$	1.41421i	1.41421i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	− 1.41421i	− 1.41421i
$305$	1.41421i	1.41421i
$306$	− 1.41421i	− 1.41421i
$307$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$307$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$308$	0	0
$309$	0	0
$310$	1.41421i	1.41421i
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$313$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$314$	0	0
$315$	− 1.41421i	− 1.41421i
$316$	0	0
$317$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$317$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$318$	0	0
$319$	−2.00000	−2.00000
$320$	− 1.41421i	− 1.41421i
$321$	0	0
$322$	0	0
$323$	2.00000	2.00000
$324$	0	0
$325$	−1.00000	−1.00000
$326$	0	0
$327$	0	0
$328$	−1.00000	−1.00000
$329$	− 1.41421i	− 1.41421i
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	1.00000	1.00000
$334$	1.41421i	1.41421i
$335$	−2.00000	−2.00000
$336$	0	0
$337$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$337$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.41421i	− 1.41421i
$342$	1.41421i	1.41421i
$343$	1.00000	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$347$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$348$	0	0
$349$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$349$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$350$	−1.00000	−1.00000
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	1.41421i	1.41421i
$356$	0	0
$357$	0	0
$358$	1.00000	1.00000
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	1.41421i	1.41421i
$361$	−1.00000	−1.00000
$362$	2.00000	2.00000
$363$	0	0
$364$	0	0
$365$	2.00000	2.00000
$366$	0	0
$367$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$367$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$368$	0	0
$369$	1.00000	1.00000
$370$	− 1.41421i	− 1.41421i
$371$	1.00000	1.00000
$372$	0	0
$373$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$373$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$374$	2.00000	2.00000
$375$	0	0
$376$	1.41421i	1.41421i
$377$	− 1.41421i	− 1.41421i
$378$	0	0
$379$	−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
$382$	−1.00000	−1.00000
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	2.00000	2.00000
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$389$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$397$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$398$	1.41421i	1.41421i
$399$	0	0
$400$	1.00000	1.00000
$401$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$402$	0	0
$403$	1.00000	1.00000
$404$	0	0
$405$	− 1.41421i	− 1.41421i
$406$	− 1.41421i	− 1.41421i
$407$	1.41421i	1.41421i
$408$	0	0
$409$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$409$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$410$	− 1.41421i	− 1.41421i
$411$	0	0
$412$	0	0
$413$	−1.00000	−1.00000
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−2.00000	−2.00000
$419$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$419$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$420$	0	0
$421$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$421$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$422$	0	0
$423$	− 1.41421i	− 1.41421i
$424$	−1.00000	−1.00000
$425$	1.41421i	1.41421i
$426$	0	0
$427$	1.00000	1.00000
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$434$	1.00000	1.00000
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$439$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$440$	−2.00000	−2.00000
$441$	0	0
$442$	1.41421i	1.41421i
$443$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$443$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$444$	0	0
$445$	1.41421i	1.41421i
$446$	−1.00000	−1.00000
$447$	0	0
$448$	−1.00000	−1.00000
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−1.00000	−1.00000
$451$	1.41421i	1.41421i
$452$	0	0
$453$	0	0
$454$	−1.00000	−1.00000
$455$	1.41421i	1.41421i
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	1.41421i	1.41421i
$459$	0	0
$460$	0	0
$461$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$461$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$462$	0	0
$463$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$463$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$464$	1.41421i	1.41421i
$465$	0	0
$466$	0	0
$467$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$467$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$468$	0	0
$469$	1.41421i	1.41421i
$470$	−2.00000	−2.00000
$471$	0	0
$472$	1.00000	1.00000
$473$	0	0
$474$	0	0
$475$	− 1.41421i	− 1.41421i
$476$	0	0
$477$	1.00000	1.00000
$478$	− 1.41421i	− 1.41421i
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	−1.00000	−1.00000
$482$	− 1.41421i	− 1.41421i
$483$	0	0
$484$	0	0
$485$	1.41421i	1.41421i
$486$	0	0
$487$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$488$	−1.00000	−1.00000
$489$	0	0
$490$	0	0
$491$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$492$	0	0
$493$	−2.00000	−2.00000
$494$	− 1.41421i	− 1.41421i
$495$	2.00000	2.00000
$496$	−1.00000	−1.00000
$497$	1.00000	1.00000
$498$	0	0
$499$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$499$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$503$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$504$	1.00000	1.00000
$505$	2.00000	2.00000
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$509$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$510$	0	0
$511$	− 1.41421i	− 1.41421i
$512$	1.00000	1.00000
$513$	0	0
$514$	0	0
$515$	− 1.41421i	− 1.41421i
$516$	0	0
$517$	2.00000	2.00000
$518$	−1.00000	−1.00000
$519$	0	0
$520$	− 1.41421i	− 1.41421i
$521$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$521$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$522$	− 1.41421i	− 1.41421i
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	1.00000	1.00000
$527$	− 1.41421i	− 1.41421i
$528$	0	0
$529$	−1.00000	−1.00000
$530$	− 1.41421i	− 1.41421i
$531$	−1.00000	−1.00000
$532$	0	0
$533$	−1.00000	−1.00000
$534$	0	0
$535$	1.41421i	1.41421i
$536$	− 1.41421i	− 1.41421i
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$547$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$548$	0	0
$549$	1.00000	1.00000
$550$	− 1.41421i	− 1.41421i
$551$	2.00000	2.00000
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1.00000	1.00000
$559$	0	0
$560$	− 1.41421i	− 1.41421i
$561$	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	0
$565$	− 1.41421i	− 1.41421i
$566$	−1.00000	−1.00000
$567$	−1.00000	−1.00000
$568$	−1.00000	−1.00000
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$571$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$572$	0	0
$573$	0	0
$574$	−1.00000	−1.00000
$575$	0	0
$576$	−1.00000	−1.00000
$577$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$577$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$578$	1.00000	1.00000
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.41421i	1.41421i
$584$	1.41421i	1.41421i
$585$	1.41421i	1.41421i
$586$	−1.00000	−1.00000
$587$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$587$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$588$	0	0
$589$	1.41421i	1.41421i
$590$	1.41421i	1.41421i
$591$	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
$595$	2.00000	2.00000
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$599$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$600$	0	0
$601$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$601$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$602$	0	0
$603$	1.41421i	1.41421i
$604$	0	0
$605$	1.41421i	1.41421i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	− 1.41421i	− 1.41421i
$611$	1.41421i	1.41421i
$612$	0	0
$613$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$613$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$614$	−1.00000	−1.00000
$615$	0	0
$616$	1.41421i	1.41421i
$617$	−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
$619$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.00000	1.00000
$624$	0	0
$625$	−1.00000	−1.00000
$626$	−1.00000	−1.00000
$627$	0	0
$628$	0	0
$629$	1.41421i	1.41421i
$630$	1.41421i	1.41421i
$631$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$631$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$632$	0	0
$633$	0	0
$634$	−1.00000	−1.00000
$635$	− 1.41421i	− 1.41421i
$636$	0	0
$637$	0	0
$638$	2.00000	2.00000
$639$	1.00000	1.00000
$640$	1.41421i	1.41421i
$641$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	−2.00000	−2.00000
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	1.00000	1.00000
$649$	− 1.41421i	− 1.41421i
$650$	1.00000	1.00000
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	1.41421i	1.41421i
$656$	1.00000	1.00000
$657$	− 1.41421i	− 1.41421i
$658$	1.41421i	1.41421i
$659$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$659$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.00000	−2.00000
$666$	−1.00000	−1.00000
$667$	0	0
$668$	0	0
$669$	0	0
$670$	2.00000	2.00000
$671$	1.41421i	1.41421i
$672$	0	0
$673$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$674$	1.41421i	1.41421i
$675$	0	0
$676$	0	0
$677$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$677$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$678$	0	0
$679$	1.00000	1.00000
$680$	−2.00000	−2.00000
$681$	0	0
$682$	1.41421i	1.41421i
$683$	−2.00000	−2.00000	−1.00000			$\pi$
$683$	−2.00000	−2.00000	−1.00000			$\pi$
$684$	0	0
$685$	− 1.41421i	− 1.41421i
$686$	−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			$\pi$
$692$	0	0
$693$	− 1.41421i	− 1.41421i
$694$	−1.00000	−1.00000
$695$	1.41421i	1.41421i
$696$	0	0
$697$	1.41421i	1.41421i
$698$	1.41421i	1.41421i
$699$	0	0
$700$	0	0
$701$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$701$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$702$	0	0
$703$	− 1.41421i	− 1.41421i
$704$	− 1.41421i	− 1.41421i
$705$	0	0
$706$	0	0
$707$	− 1.41421i	− 1.41421i
$708$	0	0
$709$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$709$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$710$	− 1.41421i	− 1.41421i
$711$	0	0
$712$	−1.00000	−1.00000
$713$	0	0
$714$	0	0
$715$	−2.00000	−2.00000
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$719$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$720$	− 1.41421i	− 1.41421i
$721$	−1.00000	−1.00000
$722$	1.00000	1.00000
$723$	0	0
$724$	0	0
$725$	1.41421i	1.41421i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	−1.00000	−1.00000
$729$	−1.00000	−1.00000
$730$	−2.00000	−2.00000
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	− 1.41421i	− 1.41421i
$735$	0	0
$736$	0	0
$737$	−2.00000	−2.00000
$738$	−1.00000	−1.00000
$739$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$739$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$740$	0	0
$741$	0	0
$742$	−1.00000	−1.00000
$743$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$744$	0	0
$745$	−2.00000	−2.00000
$746$	−1.00000	−1.00000
$747$	0	0
$748$	0	0
$749$	1.00000	1.00000
$750$	0	0
$751$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$751$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$752$	− 1.41421i	− 1.41421i
$753$	0	0
$754$	1.41421i	1.41421i
$755$	0	0
$756$	0	0
$757$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$757$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$758$	1.00000	1.00000
$759$	0	0
$760$	2.00000	2.00000
$761$	2.00000	2.00000	1.00000			$0$
$761$	2.00000	2.00000	1.00000			$0$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.00000	2.00000
$766$	0	0
$767$	1.00000	1.00000
$768$	0	0
$769$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$769$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$770$	−2.00000	−2.00000
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−1.00000	−1.00000
$776$	−1.00000	−1.00000
$777$	0	0
$778$	1.41421i	1.41421i
$779$	− 1.41421i	− 1.41421i
$780$	0	0
$781$	1.41421i	1.41421i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.00000	−1.00000
$792$	1.41421i	1.41421i
$793$	−1.00000	−1.00000
$794$	1.41421i	1.41421i
$795$	0	0
$796$	0	0
$797$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$797$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$798$	0	0
$799$	2.00000	2.00000
$800$	0	0
$801$	1.00000	1.00000
$802$	1.00000	1.00000
$803$	2.00000	2.00000
$804$	0	0
$805$	0	0
$806$	−1.00000	−1.00000
$807$	0	0
$808$	1.41421i	1.41421i
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	1.41421i	1.41421i
$811$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$811$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$812$	0	0
$813$	0	0
$814$	− 1.41421i	− 1.41421i
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−1.00000	−1.00000
$819$	1.00000	1.00000
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	1.00000	1.00000
$825$	0	0
$826$	1.00000	1.00000
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	1.00000
$833$	0	0
$834$	0	0
$835$	−2.00000	−2.00000
$836$	0	0
$837$	0	0
$838$	1.41421i	1.41421i
$839$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$839$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$840$	0	0
$841$	−1.00000	−1.00000
$842$	−1.00000	−1.00000
$843$	0	0
$844$	0	0
$845$	0	0
$846$	1.41421i	1.41421i
$847$	1.00000	1.00000
$848$	1.00000	1.00000
$849$	0	0
$850$	− 1.41421i	− 1.41421i
$851$	0	0
$852$	0	0
$853$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$853$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$854$	−1.00000	−1.00000
$855$	−2.00000	−2.00000
$856$	−1.00000	−1.00000
$857$	−2.00000	−2.00000	−1.00000			$\pi$
$857$	−2.00000	−2.00000	−1.00000			$\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	1.00000	1.00000
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 1.41421i	− 1.41421i
$872$	0	0
$873$	1.00000	1.00000
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$877$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$878$	− 1.41421i	− 1.41421i
$879$	0	0
$880$	2.00000	2.00000
$881$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$881$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	− 1.41421i	− 1.41421i
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−1.00000	−1.00000
$890$	− 1.41421i	− 1.41421i
$891$	− 1.41421i	− 1.41421i
$892$	0	0
$893$	−2.00000	−2.00000
$894$	0	0
$895$	1.41421i	1.41421i
$896$	1.00000	1.00000
$897$	0	0
$898$	0	0
$899$	− 1.41421i	− 1.41421i
$900$	0	0
$901$	1.41421i	1.41421i
$902$	− 1.41421i	− 1.41421i
$903$	0	0
$904$	1.00000	1.00000
$905$	2.82843i	2.82843i
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	− 1.41421i	− 1.41421i
$910$	− 1.41421i	− 1.41421i
$911$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$911$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.00000	1.00000
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	1.41421i	1.41421i
$923$	−1.00000	−1.00000
$924$	0	0
$925$	1.00000	1.00000
$926$	−1.00000	−1.00000
$927$	−1.00000	−1.00000
$928$	0	0
$929$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$929$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	1.00000	1.00000
$935$	2.82843i	2.82843i
$936$	−1.00000	−1.00000
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	− 1.41421i	− 1.41421i
$939$	0	0
$940$	0	0
$941$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$941$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$942$	0	0
$943$	0	0
$944$	−1.00000	−1.00000
$945$	0	0
$946$	0	0
$947$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$948$	0	0
$949$	1.41421i	1.41421i
$950$	1.41421i	1.41421i
$951$	0	0
$952$	1.41421i	1.41421i
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	−1.00000	−1.00000
$955$	− 1.41421i	− 1.41421i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.00000	−1.00000
$960$	0	0
$961$	0	0
$962$	1.00000	1.00000
$963$	1.00000	1.00000
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$967$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$968$	−1.00000	−1.00000
$969$	0	0
$970$	− 1.41421i	− 1.41421i
$971$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$971$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$972$	0	0
$973$	1.00000	1.00000
$974$	1.41421i	1.41421i
$975$	0	0
$976$	1.00000	1.00000
$977$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$977$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$978$	0	0
$979$	1.41421i	1.41421i
$980$	0	0
$981$	0	0
$982$	1.00000	1.00000
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	2.00000	2.00000
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−2.00000	−2.00000
$991$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$991$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$992$	0	0
$993$	0	0
$994$	−1.00000	−1.00000
$995$	−2.00000	−2.00000
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.41421i	1.41421i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2183.1.d.c.2182.1	✓	2
37.36	even	2		2183.1.d.d.2182.2	yes	2
59.58	odd	2		2183.1.d.d.2182.1	yes	2
2183.2182	odd	2	inner	2183.1.d.c.2182.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2183.1.d.c.2182.1	✓	2	1.1	even	1	trivial
2183.1.d.c.2182.2	yes	2	2183.2182	odd	2	inner
2183.1.d.d.2182.1	yes	2	59.58	odd	2
2183.1.d.d.2182.2	yes	2	37.36	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2183 = 37 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2183.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.41421i q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.41421i q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +1.41421i q^{10} -1.41421i q^{11} +1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} -1.41421i q^{17} +1.00000 q^{18} +1.41421i q^{19} +1.41421i q^{22} -1.00000 q^{25} -1.00000 q^{26} -1.41421i q^{29} +1.00000 q^{31} +1.41421i q^{34} +1.41421i q^{35} -1.00000 q^{37} -1.41421i q^{38} -1.41421i q^{40} -1.00000 q^{41} +1.41421i q^{45} +1.41421i q^{47} +1.00000 q^{50} -1.00000 q^{53} -2.00000 q^{55} -1.00000 q^{56} +1.41421i q^{58} +1.00000 q^{59} -1.00000 q^{61} -1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.41421i q^{65} -1.41421i q^{67} -1.41421i q^{70} -1.00000 q^{71} -1.00000 q^{72} +1.41421i q^{73} +1.00000 q^{74} +1.41421i q^{77} +1.41421i q^{80} +1.00000 q^{81} +1.00000 q^{82} -2.00000 q^{85} -1.41421i q^{88} -1.00000 q^{89} -1.41421i q^{90} -1.00000 q^{91} -1.41421i q^{94} +2.00000 q^{95} -1.00000 q^{97} +1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{7} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^7 + 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{7} + 2 q^{8} - 2 q^{9} + 2 q^{13} + 2 q^{14} - 2 q^{16} + 2 q^{18} - 2 q^{25} - 2 q^{26} + 2 q^{31} - 2 q^{37} - 2 q^{41} + 2 q^{50} - 2 q^{53} - 4 q^{55} - 2 q^{56} + 2 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{63} + 2 q^{64} - 2 q^{71} - 2 q^{72} + 2 q^{74} + 2 q^{81} + 2 q^{82} - 4 q^{85} - 2 q^{89} - 2 q^{91} + 4 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^7 + 2 * q^8 - 2 * q^9 + 2 * q^13 + 2 * q^14 - 2 * q^16 + 2 * q^18 - 2 * q^25 - 2 * q^26 + 2 * q^31 - 2 * q^37 - 2 * q^41 + 2 * q^50 - 2 * q^53 - 4 * q^55 - 2 * q^56 + 2 * q^59 - 2 * q^61 - 2 * q^62 + 2 * q^63 + 2 * q^64 - 2 * q^71 - 2 * q^72 + 2 * q^74 + 2 * q^81 + 2 * q^82 - 4 * q^85 - 2 * q^89 - 2 * q^91 + 4 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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