LMFDB - Embedded newform 3800.1.cv.f.1651.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(251,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 0, 2]))

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

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

chi := DirichletCharacter("3800.251");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3800.cv (of order $18$, degree $6$, 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:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.43477921384960000.1

Embedding invariants

Embedding label			1651.2
Root			$-0.984808 - 0.173648i$ of defining polynomial
Character	$\chi$	$=$	3800.1651
Dual form			3800.1.cv.f.2251.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.984808 + 0.173648i) q^{2} +(0.939693 + 0.342020i) q^{4} +(-1.62760 + 0.939693i) q^{7} +(0.866025 + 0.500000i) q^{8} +(-0.173648 - 0.984808i) q^{9} +(-0.173648 + 0.300767i) q^{11} +(0.642788 + 0.766044i) q^{13} +(-1.76604 + 0.642788i) q^{14} +(0.766044 + 0.642788i) q^{16} -1.00000i q^{18} +(-0.173648 + 0.984808i) q^{19} +(-0.223238 + 0.266044i) q^{22} +(-0.524005 + 1.43969i) q^{23} +(0.500000 + 0.866025i) q^{26} +(-1.85083 + 0.326352i) q^{28} +(0.642788 + 0.766044i) q^{32} +(0.173648 - 0.984808i) q^{36} +1.53209i q^{37} +(-0.342020 + 0.939693i) q^{38} +(0.266044 + 0.223238i) q^{41} +(-0.266044 + 0.223238i) q^{44} +(-0.766044 + 1.32683i) q^{46} +(0.984808 - 0.173648i) q^{47} +(1.26604 - 2.19285i) q^{49} +(0.342020 + 0.939693i) q^{52} +(0.642788 - 1.76604i) q^{53} -1.87939 q^{56} +(0.173648 - 0.984808i) q^{59} +(1.20805 + 1.43969i) q^{63} +(0.500000 + 0.866025i) q^{64} +(0.342020 - 0.939693i) q^{72} +(-0.266044 + 1.50881i) q^{74} +(-0.500000 + 0.866025i) q^{76} -0.652704i q^{77} +(-0.939693 + 0.342020i) q^{81} +(0.223238 + 0.266044i) q^{82} +(-0.300767 + 0.173648i) q^{88} +(-1.17365 + 0.984808i) q^{89} +(-1.76604 - 0.642788i) q^{91} +(-0.984808 + 1.17365i) q^{92} +1.00000 q^{94} +(1.62760 - 1.93969i) q^{98} +(0.326352 + 0.118782i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{14} + 6 q^{26} - 6 q^{41} + 6 q^{44} + 6 q^{49} + 6 q^{64} + 6 q^{74} - 6 q^{76} - 12 q^{89} - 12 q^{91} + 12 q^{94} + 6 q^{99}+O(q^{100}) $ 12 * q - 12 * q^14 + 6 * q^26 - 6 * q^41 + 6 * q^44 + 6 * q^49 + 6 * q^64 + 6 * q^74 - 6 * q^76 - 12 * q^89 - 12 * q^91 + 12 * q^94 + 6 * q^99

Character values

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

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$e\left(\frac{5}{9}\right)$	$-1$	$-1$	$1$

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 $

$2$	0.984808	+	0.173648i	0.984808	+	0.173648i
$2$	0.984808	+	0.173648i	0.984808	+	0.173648i
$3$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$3$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$4$	0.939693	+	0.342020i	0.939693	+	0.342020i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.642788	+	0.766044i	$0.722222\pi$
$7$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.984808	+	0.173648i	$0.944444\pi$
$8$	0.866025	+	0.500000i	0.866025	+	0.500000i
$9$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$10$		0			0
$11$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$11$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$12$		0			0
$13$	0.642788	+	0.766044i	0.642788	+	0.766044i	0.984808	−	0.173648i	$-0.0555556\pi$
$13$	0.642788	+	0.766044i	0.642788	+	0.766044i	−0.342020	+	0.939693i	$0.611111\pi$
$14$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$15$		0			0
$16$	0.766044	+	0.642788i	0.766044	+	0.642788i
$17$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$17$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$18$		−	1.00000i		−	1.00000i
$19$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$19$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$20$		0			0
$21$		0			0
$22$	−0.223238	+	0.266044i	−0.223238	+	0.266044i
$23$	−0.524005	+	1.43969i	−0.524005	+	1.43969i	0.342020	+	0.939693i	$0.388889\pi$
$23$	−0.524005	+	1.43969i	−0.524005	+	1.43969i	−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$		0			0
$26$	0.500000	+	0.866025i	0.500000	+	0.866025i
$27$		0			0
$28$	−1.85083	+	0.326352i	−1.85083	+	0.326352i
$29$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$29$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	0.642788	+	0.766044i	0.642788	+	0.766044i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.173648	−	0.984808i	0.173648	−	0.984808i
$37$			1.53209i			1.53209i	0.642788	+	0.766044i	$0.277778\pi$
$37$			1.53209i			1.53209i	−0.642788	+	0.766044i	$0.722222\pi$
$38$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$39$		0			0
$40$		0			0
$41$	0.266044	+	0.223238i	0.266044	+	0.223238i	0.766044	−	0.642788i	$-0.222222\pi$
$41$	0.266044	+	0.223238i	0.266044	+	0.223238i	−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$43$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$44$	−0.266044	+	0.223238i	−0.266044	+	0.223238i
$45$		0			0
$46$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$47$	0.984808	−	0.173648i	0.984808	−	0.173648i	0.342020	−	0.939693i	$-0.388889\pi$
$47$	0.984808	−	0.173648i	0.984808	−	0.173648i	0.642788	+	0.766044i	$0.277778\pi$
$48$		0			0
$49$	1.26604	−	2.19285i	1.26604	−	2.19285i
$50$		0			0
$51$		0			0
$52$	0.342020	+	0.939693i	0.342020	+	0.939693i
$53$	0.642788	−	1.76604i	0.642788	−	1.76604i		−	1.00000i	$-0.5\pi$
$53$	0.642788	−	1.76604i	0.642788	−	1.76604i	0.642788	−	0.766044i	$-0.277778\pi$
$54$		0			0
$55$		0			0
$56$	−1.87939			−1.87939
$57$		0			0
$58$		0			0
$59$	0.173648	−	0.984808i	0.173648	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$59$	0.173648	−	0.984808i	0.173648	−	0.984808i	0.939693	−	0.342020i	$-0.111111\pi$
$60$		0			0
$61$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$61$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$62$		0			0
$63$	1.20805	+	1.43969i	1.20805	+	1.43969i
$64$	0.500000	+	0.866025i	0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$67$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$71$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$72$	0.342020	−	0.939693i	0.342020	−	0.939693i
$73$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$73$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$74$	−0.266044	+	1.50881i	−0.266044	+	1.50881i
$75$		0			0
$76$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$77$		−	0.652704i		−	0.652704i
$78$		0			0
$79$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$79$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$80$		0			0
$81$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$82$	0.223238	+	0.266044i	0.223238	+	0.266044i
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.300767	+	0.173648i	−0.300767	+	0.173648i
$89$	−1.17365	+	0.984808i	−1.17365	+	0.984808i	−0.173648	+	0.984808i	$0.555556\pi$
$89$	−1.17365	+	0.984808i	−1.17365	+	0.984808i	−1.00000			$\pi$
$90$		0			0
$91$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$92$	−0.984808	+	1.17365i	−0.984808	+	1.17365i
$93$		0			0
$94$	1.00000			1.00000
$95$		0			0
$96$		0			0
$97$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$97$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$98$	1.62760	−	1.93969i	1.62760	−	1.93969i
$99$	0.326352	+	0.118782i	0.326352	+	0.118782i
$100$		0			0
$101$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$101$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$102$		0			0
$103$	−1.62760	−	0.939693i	−1.62760	−	0.939693i	−0.984808	−	0.173648i	$-0.944444\pi$
$103$	−1.62760	−	0.939693i	−1.62760	−	0.939693i	−0.642788	−	0.766044i	$-0.722222\pi$
$104$	0.173648	+	0.984808i	0.173648	+	0.984808i
$105$		0			0
$106$	0.939693	−	1.62760i	0.939693	−	1.62760i
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$109$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$110$		0			0
$111$		0			0
$112$	−1.85083	−	0.326352i	−1.85083	−	0.326352i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.642788	−	0.766044i	0.642788	−	0.766044i
$118$	0.342020	−	0.939693i	0.342020	−	0.939693i
$119$		0			0
$120$		0			0
$121$	0.439693	+	0.761570i	0.439693	+	0.761570i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.939693	+	1.62760i	0.939693	+	1.62760i
$127$	0.984808	+	1.17365i	0.984808	+	1.17365i	0.984808	+	0.173648i	$0.0555556\pi$
$127$	0.984808	+	1.17365i	0.984808	+	1.17365i			1.00000i	$0.5\pi$
$128$	0.342020	+	0.939693i	0.342020	+	0.939693i
$129$		0			0
$130$		0			0
$131$	0.266044	−	1.50881i	0.266044	−	1.50881i	−0.500000	−	0.866025i	$-0.666667\pi$
$131$	0.266044	−	1.50881i	0.266044	−	1.50881i	0.766044	−	0.642788i	$-0.222222\pi$
$132$		0			0
$133$	−0.642788	−	1.76604i	−0.642788	−	1.76604i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$137$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$138$		0			0
$139$	0.766044	−	0.642788i	0.766044	−	0.642788i	−0.173648	−	0.984808i	$-0.555556\pi$
$139$	0.766044	−	0.642788i	0.766044	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.342020	+	0.0603074i	−0.342020	+	0.0603074i
$144$	0.500000	−	0.866025i	0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.524005	+	1.43969i	−0.524005	+	1.43969i
$149$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$149$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$153$		0			0
$154$	0.113341	−	0.642788i	0.113341	−	0.642788i
$155$		0			0
$156$		0			0
$157$	−0.524005	−	1.43969i	−0.524005	−	1.43969i	−0.866025	−	0.500000i	$-0.833333\pi$
$157$	−0.524005	−	1.43969i	−0.524005	−	1.43969i	0.342020	−	0.939693i	$-0.388889\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.500000	−	2.83564i	−0.500000	−	2.83564i
$162$	−0.984808	+	0.173648i	−0.984808	+	0.173648i
$163$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$163$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$164$	0.173648	+	0.300767i	0.173648	+	0.300767i
$165$		0			0
$166$		0			0
$167$	0.118782	−	0.326352i	0.118782	−	0.326352i	−0.866025	−	0.500000i	$-0.833333\pi$
$167$	0.118782	−	0.326352i	0.118782	−	0.326352i	0.984808	+	0.173648i	$0.0555556\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	1.00000			1.00000
$172$		0			0
$173$	−0.342020	−	0.0603074i	−0.342020	−	0.0603074i		−	1.00000i	$-0.5\pi$
$173$	−0.342020	−	0.0603074i	−0.342020	−	0.0603074i	−0.342020	+	0.939693i	$0.611111\pi$
$174$		0			0
$175$		0			0
$176$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$177$		0			0
$178$	−1.32683	+	0.766044i	−1.32683	+	0.766044i
$179$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$179$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$180$		0			0
$181$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$181$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$182$	−1.62760	−	0.939693i	−1.62760	−	0.939693i
$183$		0			0
$184$	−1.17365	+	0.984808i	−1.17365	+	0.984808i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.984808	+	0.173648i	0.984808	+	0.173648i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$193$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$194$		0			0
$195$		0			0
$196$	1.93969	−	1.62760i	1.93969	−	1.62760i
$197$	0.300767	−	0.173648i	0.300767	−	0.173648i	−0.342020	−	0.939693i	$-0.611111\pi$
$197$	0.300767	−	0.173648i	0.300767	−	0.173648i	0.642788	+	0.766044i	$0.277778\pi$
$198$	0.300767	+	0.173648i	0.300767	+	0.173648i
$199$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$199$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$207$	1.50881	+	0.266044i	1.50881	+	0.266044i
$208$			1.00000i			1.00000i
$209$	−0.266044	−	0.223238i	−0.266044	−	0.223238i
$210$		0			0
$211$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	−0.939693	−	0.342020i	$-0.888889\pi$
$211$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	1.00000			$0$
$212$	1.20805	−	1.43969i	1.20805	−	1.43969i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.118782	+	0.326352i	0.118782	+	0.326352i	0.984808	−	0.173648i	$-0.0555556\pi$
$223$	0.118782	+	0.326352i	0.118782	+	0.326352i	−0.866025	+	0.500000i	$0.833333\pi$
$224$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$233$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$234$	0.766044	−	0.642788i	0.766044	−	0.642788i
$235$		0			0
$236$	0.500000	−	0.866025i	0.500000	−	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$241$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$242$	0.300767	+	0.826352i	0.300767	+	0.826352i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$251$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$252$	0.642788	+	1.76604i	0.642788	+	1.76604i
$253$	−0.342020	−	0.407604i	−0.342020	−	0.407604i
$254$	0.766044	+	1.32683i	0.766044	+	1.32683i
$255$		0			0
$256$	0.173648	+	0.984808i	0.173648	+	0.984808i
$257$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$257$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$258$		0			0
$259$	−1.43969	−	2.49362i	−1.43969	−	2.49362i
$260$		0			0
$261$		0			0
$262$	0.524005	−	1.43969i	0.524005	−	1.43969i
$263$	0.223238	−	0.266044i	0.223238	−	0.266044i	−0.642788	−	0.766044i	$-0.722222\pi$
$263$	0.223238	−	0.266044i	0.223238	−	0.266044i	0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$267$		0			0
$268$		0			0
$269$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$269$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$270$		0			0
$271$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$271$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$277$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$278$	0.866025	−	0.500000i	0.866025	−	0.500000i
$279$		0			0
$280$		0			0
$281$	1.76604	+	0.642788i	1.76604	+	0.642788i	1.00000			$0$
$281$	1.76604	+	0.642788i	1.76604	+	0.642788i	0.766044	+	0.642788i	$0.222222\pi$
$282$		0			0
$283$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$283$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$284$		0			0
$285$		0			0
$286$	−0.347296			−0.347296
$287$	−0.642788	−	0.113341i	−0.642788	−	0.113341i
$288$	0.642788	−	0.766044i	0.642788	−	0.766044i
$289$	0.939693	+	0.342020i	0.939693	+	0.342020i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.62760	−	0.939693i	−1.62760	−	0.939693i	−0.984808	−	0.173648i	$-0.944444\pi$
$293$	−1.62760	−	0.939693i	−1.62760	−	0.939693i	−0.642788	−	0.766044i	$-0.722222\pi$
$294$		0			0
$295$		0			0
$296$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$297$		0			0
$298$		0			0
$299$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$305$		0			0
$306$		0			0
$307$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$307$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$308$	0.223238	−	0.613341i	0.223238	−	0.613341i
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$313$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$314$	−0.266044	−	1.50881i	−0.266044	−	1.50881i
$315$		0			0
$316$		0			0
$317$	0.223238	+	0.266044i	0.223238	+	0.266044i	0.866025	−	0.500000i	$-0.166667\pi$
$317$	0.223238	+	0.266044i	0.223238	+	0.266044i	−0.642788	+	0.766044i	$0.722222\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		−	2.87939i		−	2.87939i
$323$		0			0
$324$	−1.00000			−1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$	0.118782	+	0.326352i	0.118782	+	0.326352i
$329$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$330$		0			0
$331$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$331$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$332$		0			0
$333$	1.50881	−	0.266044i	1.50881	−	0.266044i
$334$	0.173648	−	0.300767i	0.173648	−	0.300767i
$335$		0			0
$336$		0			0
$337$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$337$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$	0.984808	+	0.173648i	0.984808	+	0.173648i
$343$			2.87939i			2.87939i
$344$		0			0
$345$		0			0
$346$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$347$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$347$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$	−0.342020	+	0.0603074i	−0.342020	+	0.0603074i
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$357$		0			0
$358$	0.984808	−	1.17365i	0.984808	−	1.17365i
$359$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$359$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$360$		0			0
$361$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$362$		0			0
$363$		0			0
$364$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$365$		0			0
$366$		0			0
$367$	−0.642788	−	0.766044i	−0.642788	−	0.766044i	0.342020	−	0.939693i	$-0.388889\pi$
$367$	−0.642788	−	0.766044i	−0.642788	−	0.766044i	−0.984808	+	0.173648i	$0.944444\pi$
$368$	−1.32683	+	0.766044i	−1.32683	+	0.766044i
$369$	0.173648	−	0.300767i	0.173648	−	0.300767i
$370$		0			0
$371$	0.613341	+	3.47843i	0.613341	+	3.47843i
$372$		0			0
$373$	1.62760	−	0.939693i	1.62760	−	0.939693i	0.642788	−	0.766044i	$-0.277778\pi$
$373$	1.62760	−	0.939693i	1.62760	−	0.939693i	0.984808	−	0.173648i	$-0.0555556\pi$
$374$		0			0
$375$		0			0
$376$	0.939693	+	0.342020i	0.939693	+	0.342020i
$377$		0			0
$378$		0			0
$379$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$379$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.642788	+	0.766044i	−0.642788	+	0.766044i	−0.984808	−	0.173648i	$-0.944444\pi$
$383$	−0.642788	+	0.766044i	−0.642788	+	0.766044i	0.342020	+	0.939693i	$0.388889\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$389$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$390$		0			0
$391$		0			0
$392$	2.19285	−	1.26604i	2.19285	−	1.26604i
$393$		0			0
$394$	0.326352	−	0.118782i	0.326352	−	0.118782i
$395$		0			0
$396$	0.266044	+	0.223238i	0.266044	+	0.223238i
$397$	0.342020	+	0.0603074i	0.342020	+	0.0603074i	0.342020	−	0.939693i	$-0.388889\pi$
$397$	0.342020	+	0.0603074i	0.342020	+	0.0603074i			1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$401$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.460802	−	0.266044i	−0.460802	−	0.266044i
$408$		0			0
$409$	0.326352	+	1.85083i	0.326352	+	1.85083i	0.500000	+	0.866025i	$0.333333\pi$
$409$	0.326352	+	1.85083i	0.326352	+	1.85083i	−0.173648	+	0.984808i	$0.555556\pi$
$410$		0			0
$411$		0			0
$412$	−1.20805	−	1.43969i	−1.20805	−	1.43969i
$413$	0.642788	+	1.76604i	0.642788	+	1.76604i
$414$	1.43969	+	0.524005i	1.43969	+	0.524005i
$415$		0			0
$416$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$417$		0			0
$418$	−0.223238	−	0.266044i	−0.223238	−	0.266044i
$419$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$419$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$420$		0			0
$421$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$421$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$422$	0.118782	−	0.326352i	0.118782	−	0.326352i
$423$	−0.342020	−	0.939693i	−0.342020	−	0.939693i
$424$	1.43969	−	1.20805i	1.43969	−	1.20805i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$431$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$432$		0			0
$433$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$433$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.32683	−	0.766044i	−1.32683	−	0.766044i
$438$		0			0
$439$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$439$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$440$		0			0
$441$	−2.37939	−	0.866025i	−2.37939	−	0.866025i
$442$		0			0
$443$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$443$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$444$		0			0
$445$		0			0
$446$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$447$		0			0
$448$	−1.62760	−	0.939693i	−1.62760	−	0.939693i
$449$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$449$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$450$		0			0
$451$	−0.113341	+	0.0412527i	−0.113341	+	0.0412527i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$461$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$462$		0			0
$463$	−0.300767	+	0.173648i	−0.300767	+	0.173648i	−0.642788	−	0.766044i	$-0.722222\pi$
$463$	−0.300767	+	0.173648i	−0.300767	+	0.173648i	0.342020	+	0.939693i	$0.388889\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$	0.866025	−	0.500000i	0.866025	−	0.500000i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.642788	−	0.766044i	0.642788	−	0.766044i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.85083	−	0.326352i	−1.85083	−	0.326352i
$478$		0			0
$479$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$479$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$480$		0			0
$481$	−1.17365	+	0.984808i	−1.17365	+	0.984808i
$482$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$483$		0			0
$484$	0.152704	+	0.866025i	0.152704	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.642788	+	0.766044i	$0.722222\pi$
$487$	−1.62760	+	0.939693i	−1.62760	+	0.939693i	−0.984808	+	0.173648i	$0.944444\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.17365	+	0.984808i	1.17365	+	0.984808i	1.00000			$0$
$491$	1.17365	+	0.984808i	1.17365	+	0.984808i	0.173648	+	0.984808i	$0.444444\pi$
$492$		0			0
$493$		0			0
$494$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−0.766044	+	0.642788i	$0.777778\pi$
$499$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−1.00000			$1.00000\pi$
$500$		0			0
$501$		0			0
$502$	0.866025	+	0.500000i	0.866025	+	0.500000i
$503$	0.342020	−	0.0603074i	0.342020	−	0.0603074i		−	1.00000i	$-0.5\pi$
$503$	0.342020	−	0.0603074i	0.342020	−	0.0603074i	0.342020	+	0.939693i	$0.388889\pi$
$504$	0.326352	+	1.85083i	0.326352	+	1.85083i
$505$		0			0
$506$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$507$		0			0
$508$	0.524005	+	1.43969i	0.524005	+	1.43969i
$509$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$509$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.118782	+	0.326352i	−0.118782	+	0.326352i
$518$	−0.984808	−	2.70574i	−0.984808	−	2.70574i
$519$		0			0
$520$		0			0
$521$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$521$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$523$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$524$	0.766044	−	1.32683i	0.766044	−	1.32683i
$525$		0			0
$526$	0.266044	−	0.223238i	0.266044	−	0.223238i
$527$		0			0
$528$		0			0
$529$	−1.03209	−	0.866025i	−1.03209	−	0.866025i
$530$		0			0
$531$	−1.00000			−1.00000
$532$		−	1.87939i		−	1.87939i
$533$			0.347296i			0.347296i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.439693	+	0.761570i	0.439693	+	0.761570i
$540$		0			0
$541$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$541$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$547$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.766044	+	0.642788i	0.766044	+	0.642788i
$555$		0			0
$556$	0.939693	−	0.342020i	0.939693	−	0.342020i
$557$	−1.20805	−	1.43969i	−1.20805	−	1.43969i	−0.866025	−	0.500000i	$-0.833333\pi$
$557$	−1.20805	−	1.43969i	−1.20805	−	1.43969i	−0.342020	−	0.939693i	$-0.611111\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.62760	+	0.939693i	1.62760	+	0.939693i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.20805	−	1.43969i	1.20805	−	1.43969i
$568$		0			0
$569$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$569$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$570$		0			0
$571$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$572$	−0.342020	−	0.0603074i	−0.342020	−	0.0603074i
$573$		0			0
$574$	−0.613341	−	0.223238i	−0.613341	−	0.223238i
$575$		0			0
$576$	0.766044	−	0.642788i	0.766044	−	0.642788i
$577$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$577$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$578$	0.866025	+	0.500000i	0.866025	+	0.500000i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.419550	+	0.500000i	0.419550	+	0.500000i
$584$		0			0
$585$		0			0
$586$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$587$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$587$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.984808	+	1.17365i	−0.984808	+	1.17365i
$593$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$593$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−1.50881	+	0.266044i	−1.50881	+	0.266044i
$599$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$599$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$600$		0			0
$601$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$601$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	1.87939i		−	1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$607$		−	1.87939i		−	1.87939i	0.342020	−	0.939693i	$-0.388889\pi$
$608$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$609$		0			0
$610$		0			0
$611$	0.766044	+	0.642788i	0.766044	+	0.642788i
$612$		0			0
$613$	0.118782	+	0.326352i	0.118782	+	0.326352i	0.984808	−	0.173648i	$-0.0555556\pi$
$613$	0.118782	+	0.326352i	0.118782	+	0.326352i	−0.866025	+	0.500000i	$0.833333\pi$
$614$		0			0
$615$		0			0
$616$	0.326352	−	0.565258i	0.326352	−	0.565258i
$617$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$617$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$618$		0			0
$619$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$619$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.984808	−	2.70574i	0.984808	−	2.70574i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		−	1.53209i		−	1.53209i
$629$		0			0
$630$		0			0
$631$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$631$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$632$		0			0
$633$		0			0
$634$	0.173648	+	0.300767i	0.173648	+	0.300767i
$635$		0			0
$636$		0			0
$637$	2.49362	−	0.439693i	2.49362	−	0.439693i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$641$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$642$		0			0
$643$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$643$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$644$	0.500000	−	2.83564i	0.500000	−	2.83564i
$645$		0			0
$646$		0			0
$647$			1.53209i			1.53209i	0.642788	+	0.766044i	$0.277778\pi$
$647$			1.53209i			1.53209i	−0.642788	+	0.766044i	$0.722222\pi$
$648$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$649$	0.266044	+	0.223238i	0.266044	+	0.223238i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.62760	−	0.939693i	1.62760	−	0.939693i	0.642788	−	0.766044i	$-0.277778\pi$
$653$	1.62760	−	0.939693i	1.62760	−	0.939693i	0.984808	−	0.173648i	$-0.0555556\pi$
$654$		0			0
$655$		0			0
$656$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$657$		0			0
$658$	−1.62760	+	0.939693i	−1.62760	+	0.939693i
$659$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	−0.766044	−	0.642788i	$-0.777778\pi$
$659$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$661$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$662$	1.20805	−	1.43969i	1.20805	−	1.43969i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.53209			1.53209
$667$		0			0
$668$	0.223238	−	0.266044i	0.223238	−	0.266044i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$673$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.32683	−	0.766044i	1.32683	−	0.766044i	0.342020	−	0.939693i	$-0.388889\pi$
$677$	1.32683	−	0.766044i	1.32683	−	0.766044i	0.984808	+	0.173648i	$0.0555556\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	0.939693	+	0.342020i	0.939693	+	0.342020i
$685$		0			0
$686$	−0.500000	+	2.83564i	−0.500000	+	2.83564i
$687$		0			0
$688$		0			0
$689$	1.76604	−	0.642788i	1.76604	−	0.642788i
$690$		0			0
$691$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$691$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$692$	−0.300767	−	0.173648i	−0.300767	−	0.173648i
$693$	−0.642788	+	0.113341i	−0.642788	+	0.113341i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$701$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$702$		0			0
$703$	−1.50881	−	0.266044i	−1.50881	−	0.266044i
$704$	−0.347296			−0.347296
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$709$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$710$		0			0
$711$		0			0
$712$	−1.50881	+	0.266044i	−1.50881	+	0.266044i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.17365	−	0.984808i	1.17365	−	0.984808i
$717$		0			0
$718$		0			0
$719$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$719$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$720$		0			0
$721$	3.53209			3.53209
$722$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.684040	−	1.87939i	−0.684040	−	1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$727$	−0.684040	−	1.87939i	−0.684040	−	1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$728$	−1.20805	−	1.43969i	−1.20805	−	1.43969i
$729$	0.500000	+	0.866025i	0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.62760	−	0.939693i	−1.62760	−	0.939693i	−0.984808	−	0.173648i	$-0.944444\pi$
$733$	−1.62760	−	0.939693i	−1.62760	−	0.939693i	−0.642788	−	0.766044i	$-0.722222\pi$
$734$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$735$		0			0
$736$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$737$		0			0
$738$	0.223238	−	0.266044i	0.223238	−	0.266044i
$739$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	0.500000	+	0.866025i	$0.333333\pi$
$739$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	−0.766044	+	0.642788i	$0.777778\pi$
$740$		0			0
$741$		0			0
$742$			3.53209i			3.53209i
$743$	1.85083	+	0.326352i	1.85083	+	0.326352i	0.984808	−	0.173648i	$-0.0555556\pi$
$743$	1.85083	+	0.326352i	1.85083	+	0.326352i	0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$	1.76604	−	0.642788i	1.76604	−	0.642788i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$751$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$752$	0.866025	+	0.500000i	0.866025	+	0.500000i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.223238	+	0.266044i	−0.223238	+	0.266044i	−0.866025	−	0.500000i	$-0.833333\pi$
$757$	−0.223238	+	0.266044i	−0.223238	+	0.266044i	0.642788	+	0.766044i	$0.277778\pi$
$758$	0.984808	+	0.173648i	0.984808	+	0.173648i
$759$		0			0
$760$		0			0
$761$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$761$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$767$	0.866025	−	0.500000i	0.866025	−	0.500000i
$768$		0			0
$769$	0.173648	+	0.984808i	0.173648	+	0.984808i	0.939693	+	0.342020i	$0.111111\pi$
$769$	0.173648	+	0.984808i	0.173648	+	0.984808i	−0.766044	+	0.642788i	$0.777778\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.984808	−	1.17365i	−0.984808	−	1.17365i	−0.984808	−	0.173648i	$-0.944444\pi$
$773$	−0.984808	−	1.17365i	−0.984808	−	1.17365i		−	1.00000i	$-0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.266044	+	0.223238i	−0.266044	+	0.223238i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	2.37939	−	0.866025i	2.37939	−	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$	0.342020	−	0.0603074i	0.342020	−	0.0603074i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.223238	+	0.266044i	0.223238	+	0.266044i
$793$		0			0
$794$	0.326352	+	0.118782i	0.326352	+	0.118782i
$795$		0			0
$796$		0			0
$797$			0.347296i			0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$797$			0.347296i			0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	1.17365	+	0.984808i	1.17365	+	0.984808i
$802$	0.684040	−	1.87939i	0.684040	−	1.87939i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$809$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$810$		0			0
$811$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.500000	+	0.866025i	$0.666667\pi$
$811$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.939693	+	0.342020i	$0.888889\pi$
$812$		0			0
$813$		0			0
$814$	−0.407604	−	0.342020i	−0.407604	−	0.342020i
$815$		0			0
$816$		0			0
$817$		0			0
$818$			1.87939i			1.87939i
$819$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$820$		0			0
$821$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$821$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$822$		0			0
$823$	−0.984808	−	1.17365i	−0.984808	−	1.17365i	−0.984808	−	0.173648i	$-0.944444\pi$
$823$	−0.984808	−	1.17365i	−0.984808	−	1.17365i		−	1.00000i	$-0.5\pi$
$824$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$825$		0			0
$826$	0.326352	+	1.85083i	0.326352	+	1.85083i
$827$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$827$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$828$	1.32683	+	0.766044i	1.32683	+	0.766044i
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$837$		0			0
$838$	1.85083	+	0.326352i	1.85083	+	0.326352i
$839$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$839$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$840$		0			0
$841$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$842$		0			0
$843$		0			0
$844$	0.173648	−	0.300767i	0.173648	−	0.300767i
$845$		0			0
$846$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$847$	−1.43128	−	0.826352i	−1.43128	−	0.826352i
$848$	1.62760	−	0.939693i	1.62760	−	0.939693i
$849$		0			0
$850$		0			0
$851$	−2.20574	−	0.802823i	−2.20574	−	0.802823i
$852$		0			0
$853$	0.984808	+	0.173648i	0.984808	+	0.173648i	0.642788	−	0.766044i	$-0.277778\pi$
$853$	0.984808	+	0.173648i	0.984808	+	0.173648i	0.342020	+	0.939693i	$0.388889\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$857$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$858$		0			0
$859$	−1.76604	−	0.642788i	−1.76604	−	0.642788i	−0.766044	−	0.642788i	$-0.777778\pi$
$859$	−1.76604	−	0.642788i	−1.76604	−	0.642788i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.300767	+	0.173648i	0.300767	+	0.173648i	0.642788	−	0.766044i	$-0.277778\pi$
$863$	0.300767	+	0.173648i	0.300767	+	0.173648i	−0.342020	+	0.939693i	$0.611111\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−1.17365	−	0.984808i	−1.17365	−	0.984808i
$875$		0			0
$876$		0			0
$877$	1.20805	−	1.43969i	1.20805	−	1.43969i	0.342020	−	0.939693i	$-0.388889\pi$
$877$	1.20805	−	1.43969i	1.20805	−	1.43969i	0.866025	−	0.500000i	$-0.166667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$881$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$882$	−2.19285	−	1.26604i	−2.19285	−	1.26604i
$883$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$883$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.642788	−	0.766044i	−0.642788	−	0.766044i	0.342020	−	0.939693i	$-0.388889\pi$
$887$	−0.642788	−	0.766044i	−0.642788	−	0.766044i	−0.984808	+	0.173648i	$0.944444\pi$
$888$		0			0
$889$	−2.70574	−	0.984808i	−2.70574	−	0.984808i
$890$		0			0
$891$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$892$			0.347296i			0.347296i
$893$			1.00000i			1.00000i
$894$		0			0
$895$		0			0
$896$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$897$		0			0
$898$	0.118782	+	0.326352i	0.118782	+	0.326352i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−0.118782	+	0.0209445i	−0.118782	+	0.0209445i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$907$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.984808	+	2.70574i	0.984808	+	2.70574i
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$927$	−0.642788	+	1.76604i	−0.642788	+	1.76604i
$928$		0			0
$929$	0.326352	−	1.85083i	0.326352	−	1.85083i	−0.173648	−	0.984808i	$-0.555556\pi$
$929$	0.326352	−	1.85083i	0.326352	−	1.85083i	0.500000	−	0.866025i	$-0.333333\pi$
$930$		0			0
$931$	1.93969	+	1.62760i	1.93969	+	1.62760i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.939693	−	0.342020i	0.939693	−	0.342020i
$937$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$937$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$941$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$942$		0			0
$943$	−0.460802	+	0.266044i	−0.460802	+	0.266044i
$944$	0.766044	−	0.642788i	0.766044	−	0.642788i
$945$		0			0
$946$		0			0
$947$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$947$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$953$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$954$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$	−1.32683	+	0.766044i	−1.32683	+	0.766044i
$963$		0			0
$964$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$965$		0			0
$966$		0			0
$967$	1.96962	+	0.347296i	1.96962	+	0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$967$	1.96962	+	0.347296i	1.96962	+	0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$968$			0.879385i			0.879385i
$969$		0			0
$970$		0			0
$971$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$971$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$972$		0			0
$973$	−0.642788	+	1.76604i	−0.642788	+	1.76604i
$974$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$975$		0			0
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$	−0.0923963	−	0.524005i	−0.0923963	−	0.524005i
$980$		0			0
$981$		0			0
$982$	0.984808	+	1.17365i	0.984808	+	1.17365i
$983$	0.524005	+	1.43969i	0.524005	+	1.43969i	0.866025	+	0.500000i	$0.166667\pi$
$983$	0.524005	+	1.43969i	0.524005	+	1.43969i	−0.342020	+	0.939693i	$0.611111\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.984808	+	0.173648i	−0.984808	+	0.173648i
$989$		0			0
$990$		0			0
$991$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$991$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.50881	+	0.266044i	−1.50881	+	0.266044i	−0.866025	−	0.500000i	$-0.833333\pi$
$997$	−1.50881	+	0.266044i	−1.50881	+	0.266044i	−0.642788	+	0.766044i	$0.722222\pi$
$998$	−1.85083	+	0.326352i	−1.85083	+	0.326352i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.1.cv.f.1651.2		12
5.2	odd	4		760.1.bz.b.739.1	yes	6
5.3	odd	4		760.1.bz.a.739.1	yes	6
5.4	even	2	inner	3800.1.cv.f.1651.1		12
8.3	odd	2	inner	3800.1.cv.f.1651.1		12
19.9	even	9	inner	3800.1.cv.f.2251.1		12
20.3	even	4		3040.1.dv.b.2639.1		6
20.7	even	4		3040.1.dv.a.2639.1		6
40.3	even	4		760.1.bz.b.739.1	yes	6
40.13	odd	4		3040.1.dv.a.2639.1		6
40.19	odd	2	CM	3800.1.cv.f.1651.2		12
40.27	even	4		760.1.bz.a.739.1	yes	6
40.37	odd	4		3040.1.dv.b.2639.1		6
95.9	even	18	inner	3800.1.cv.f.2251.2		12
95.28	odd	36		760.1.bz.a.579.1	✓	6
95.47	odd	36		760.1.bz.b.579.1	yes	6
152.123	odd	18	inner	3800.1.cv.f.2251.2		12
380.47	even	36		3040.1.dv.a.2479.1		6
380.123	even	36		3040.1.dv.b.2479.1		6
760.123	even	36		760.1.bz.b.579.1	yes	6
760.237	odd	36		3040.1.dv.b.2479.1		6
760.427	even	36		760.1.bz.a.579.1	✓	6
760.579	odd	18	inner	3800.1.cv.f.2251.1		12
760.693	odd	36		3040.1.dv.a.2479.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.1.bz.a.579.1	✓	6	95.28	odd	36
760.1.bz.a.579.1	✓	6	760.427	even	36
760.1.bz.a.739.1	yes	6	5.3	odd	4
760.1.bz.a.739.1	yes	6	40.27	even	4
760.1.bz.b.579.1	yes	6	95.47	odd	36
760.1.bz.b.579.1	yes	6	760.123	even	36
760.1.bz.b.739.1	yes	6	5.2	odd	4
760.1.bz.b.739.1	yes	6	40.3	even	4
3040.1.dv.a.2479.1		6	380.47	even	36
3040.1.dv.a.2479.1		6	760.693	odd	36
3040.1.dv.a.2639.1		6	20.7	even	4
3040.1.dv.a.2639.1		6	40.13	odd	4
3040.1.dv.b.2479.1		6	380.123	even	36
3040.1.dv.b.2479.1		6	760.237	odd	36
3040.1.dv.b.2639.1		6	20.3	even	4
3040.1.dv.b.2639.1		6	40.37	odd	4
3800.1.cv.f.1651.1		12	5.4	even	2	inner
3800.1.cv.f.1651.1		12	8.3	odd	2	inner
3800.1.cv.f.1651.2		12	1.1	even	1	trivial
3800.1.cv.f.1651.2		12	40.19	odd	2	CM
3800.1.cv.f.2251.1		12	19.9	even	9	inner
3800.1.cv.f.2251.1		12	760.579	odd	18	inner
3800.1.cv.f.2251.2		12	95.9	even	18	inner
3800.1.cv.f.2251.2		12	152.123	odd	18	inner

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.cv (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(0.984808 + 0.173648i) q^{2} +(0.939693 + 0.342020i) q^{4} +(-1.62760 + 0.939693i) q^{7} +(0.866025 + 0.500000i) q^{8} +(-0.173648 - 0.984808i) q^{9} +(-0.173648 + 0.300767i) q^{11} +(0.642788 + 0.766044i) q^{13} +(-1.76604 + 0.642788i) q^{14} +(0.766044 + 0.642788i) q^{16} -1.00000i q^{18} +(-0.173648 + 0.984808i) q^{19} +(-0.223238 + 0.266044i) q^{22} +(-0.524005 + 1.43969i) q^{23} +(0.500000 + 0.866025i) q^{26} +(-1.85083 + 0.326352i) q^{28} +(0.642788 + 0.766044i) q^{32} +(0.173648 - 0.984808i) q^{36} +1.53209i q^{37} +(-0.342020 + 0.939693i) q^{38} +(0.266044 + 0.223238i) q^{41} +(-0.266044 + 0.223238i) q^{44} +(-0.766044 + 1.32683i) q^{46} +(0.984808 - 0.173648i) q^{47} +(1.26604 - 2.19285i) q^{49} +(0.342020 + 0.939693i) q^{52} +(0.642788 - 1.76604i) q^{53} -1.87939 q^{56} +(0.173648 - 0.984808i) q^{59} +(1.20805 + 1.43969i) q^{63} +(0.500000 + 0.866025i) q^{64} +(0.342020 - 0.939693i) q^{72} +(-0.266044 + 1.50881i) q^{74} +(-0.500000 + 0.866025i) q^{76} -0.652704i q^{77} +(-0.939693 + 0.342020i) q^{81} +(0.223238 + 0.266044i) q^{82} +(-0.300767 + 0.173648i) q^{88} +(-1.17365 + 0.984808i) q^{89} +(-1.76604 - 0.642788i) q^{91} +(-0.984808 + 1.17365i) q^{92} +1.00000 q^{94} +(1.62760 - 1.93969i) q^{98} +(0.326352 + 0.118782i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{14} + 6 q^{26} - 6 q^{41} + 6 q^{44} + 6 q^{49} + 6 q^{64} + 6 q^{74} - 6 q^{76} - 12 q^{89} - 12 q^{91} + 12 q^{94} + 6 q^{99}+O(q^{100}) \) 12 * q - 12 * q^14 + 6 * q^26 - 6 * q^41 + 6 * q^44 + 6 * q^49 + 6 * q^64 + 6 * q^74 - 6 * q^76 - 12 * q^89 - 12 * q^91 + 12 * q^94 + 6 * q^99

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