LMFDB - Embedded newform 1057.1.bq.a.160.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1057,1,Mod(20,1057)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1057, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 34]))

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

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

chi := DirichletCharacter("1057.20");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1057 = 7 \cdot 151 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1057.bq (of order $50$, degree $20$, 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:	$0.527511718353$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

Embedding invariants

Embedding label			160.1
Root			$-0.535827 - 0.844328i$ of defining polynomial
Character	$\chi$	$=$	1057.160
Dual form			1057.1.bq.a.839.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.450527 - 1.38658i) q^{2} +(-0.910614 - 0.661600i) q^{4} +(-0.425779 - 0.904827i) q^{7} +(-0.148122 + 0.107617i) q^{8} +(-0.992115 + 0.125333i) q^{9} +O(q^{10})$	\(q+(0.450527 - 1.38658i) q^{2} +(-0.910614 - 0.661600i) q^{4} +(-0.425779 - 0.904827i) q^{7} +(-0.148122 + 0.107617i) q^{8} +(-0.992115 + 0.125333i) q^{9} +(0.0388067 - 0.616814i) q^{11} +(-1.44644 + 0.182728i) q^{14} +(-0.265337 - 0.816623i) q^{16} +(-0.273190 + 1.43211i) q^{18} +(-0.837780 - 0.331700i) q^{22} +(-0.500000 - 0.363271i) q^{23} +(0.968583 - 0.248690i) q^{25} +(-0.210913 + 1.10564i) q^{28} +(1.18532 + 0.469303i) q^{29} -1.43494 q^{32} +(0.986354 + 0.542253i) q^{36} +(-0.393950 - 0.476203i) q^{37} +(-0.0534698 + 0.113629i) q^{43} +(-0.443422 + 0.536005i) q^{44} +(-0.728969 + 0.529627i) q^{46} +(-0.637424 + 0.770513i) q^{49} +(0.0915446 - 1.45506i) q^{50} +(1.03799 + 1.63560i) q^{53} +(0.160442 + 0.0882039i) q^{56} +(1.18475 - 1.43211i) q^{58} +(0.535827 + 0.844328i) q^{63} +(-0.381145 + 1.17304i) q^{64} +(0.371808 + 0.0469702i) q^{67} +(0.844844 + 1.79538i) q^{71} +(0.133466 - 0.125333i) q^{72} +(-0.837780 + 0.331700i) q^{74} +(-0.574633 + 0.227513i) q^{77} +(1.06279 - 0.998027i) q^{79} +(0.968583 - 0.248690i) q^{81} +(0.133466 + 0.125333i) q^{86} +(0.0606317 + 0.0955403i) q^{88} +(0.214967 + 0.661600i) q^{92} +(0.781202 + 1.23098i) q^{98} +(0.0388067 + 0.616814i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{4}+O(q^{10}) $ 20 * q - 5 * q^4	$ 20 q - 5 q^{4} - 5 q^{14} - 5 q^{16} - 5 q^{18} - 10 q^{23} - 5 q^{29} - 10 q^{32} - 10 q^{44} - 5 q^{53} - 5 q^{56} - 5 q^{58} - 5 q^{64} - 5 q^{67} - 5 q^{71} + 20 q^{72} + 20 q^{79} + 20 q^{86} + 15 q^{88} + 15 q^{92}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^14 - 5 * q^16 - 5 * q^18 - 10 * q^23 - 5 * q^29 - 10 * q^32 - 10 * q^44 - 5 * q^53 - 5 * q^56 - 5 * q^58 - 5 * q^64 - 5 * q^67 - 5 * q^71 + 20 * q^72 + 20 * q^79 + 20 * q^86 + 15 * q^88 + 15 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1057\mathbb{Z}\right)^\times$.

$n$	$605$	$610$
$\chi(n)$	$-1$	$e\left(\frac{2}{25}\right)$

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 $

$2$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$2$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$3$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$3$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$4$	−0.910614	−	0.661600i	−0.910614	−	0.661600i
$5$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$5$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$6$		0			0
$7$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$7$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$8$	−0.148122	+	0.107617i	−0.148122	+	0.107617i
$9$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$10$		0			0
$11$	0.0388067	−	0.616814i	0.0388067	−	0.616814i	−0.929776	−	0.368125i	$-0.880000\pi$
$11$	0.0388067	−	0.616814i	0.0388067	−	0.616814i	0.968583	−	0.248690i	$-0.0800000\pi$
$12$		0			0
$13$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$13$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$14$	−1.44644	+	0.182728i	−1.44644	+	0.182728i
$15$		0			0
$16$	−0.265337	−	0.816623i	−0.265337	−	0.816623i
$17$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$17$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$18$	−0.273190	+	1.43211i	−0.273190	+	1.43211i
$19$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$19$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$20$		0			0
$21$		0			0
$22$	−0.837780	−	0.331700i	−0.837780	−	0.331700i
$23$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$23$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$24$		0			0
$25$	0.968583	−	0.248690i	0.968583	−	0.248690i
$26$		0			0
$27$		0			0
$28$	−0.210913	+	1.10564i	−0.210913	+	1.10564i
$29$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.876307	−	0.481754i	$-0.160000\pi$
$29$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.309017	+	0.951057i	$0.400000\pi$
$30$		0			0
$31$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$31$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$32$	−1.43494			−1.43494
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.986354	+	0.542253i	0.986354	+	0.542253i
$37$	−0.393950	−	0.476203i	−0.393950	−	0.476203i	0.535827	−	0.844328i	$-0.320000\pi$
$37$	−0.393950	−	0.476203i	−0.393950	−	0.476203i	−0.929776	+	0.368125i	$0.880000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$41$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$42$		0			0
$43$	−0.0534698	+	0.113629i	−0.0534698	+	0.113629i	−0.929776	−	0.368125i	$-0.880000\pi$
$43$	−0.0534698	+	0.113629i	−0.0534698	+	0.113629i	0.876307	+	0.481754i	$0.160000\pi$
$44$	−0.443422	+	0.536005i	−0.443422	+	0.536005i
$45$		0			0
$46$	−0.728969	+	0.529627i	−0.728969	+	0.529627i
$47$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$47$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$48$		0			0
$49$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$50$	0.0915446	−	1.45506i	0.0915446	−	1.45506i
$51$		0			0
$52$		0			0
$53$	1.03799	+	1.63560i	1.03799	+	1.63560i	0.728969	+	0.684547i	$0.240000\pi$
$53$	1.03799	+	1.63560i	1.03799	+	1.63560i	0.309017	+	0.951057i	$0.400000\pi$
$54$		0			0
$55$		0			0
$56$	0.160442	+	0.0882039i	0.160442	+	0.0882039i
$57$		0			0
$58$	1.18475	−	1.43211i	1.18475	−	1.43211i
$59$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$59$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$61$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$62$		0			0
$63$	0.535827	+	0.844328i	0.535827	+	0.844328i
$64$	−0.381145	+	1.17304i	−0.381145	+	1.17304i
$65$		0			0
$66$		0			0
$67$	0.371808	+	0.0469702i	0.371808	+	0.0469702i	0.309017	−	0.951057i	$-0.400000\pi$
$67$	0.371808	+	0.0469702i	0.371808	+	0.0469702i	0.0627905	+	0.998027i	$0.480000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.844844	+	1.79538i	0.844844	+	1.79538i	0.535827	+	0.844328i	$0.320000\pi$
$71$	0.844844	+	1.79538i	0.844844	+	1.79538i	0.309017	+	0.951057i	$0.400000\pi$
$72$	0.133466	−	0.125333i	0.133466	−	0.125333i
$73$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$73$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$74$	−0.837780	+	0.331700i	−0.837780	+	0.331700i
$75$		0			0
$76$		0			0
$77$	−0.574633	+	0.227513i	−0.574633	+	0.227513i
$78$		0			0
$79$	1.06279	−	0.998027i	1.06279	−	0.998027i	0.0627905	−	0.998027i	$-0.480000\pi$
$79$	1.06279	−	0.998027i	1.06279	−	0.998027i	1.00000			$0$
$80$		0			0
$81$	0.968583	−	0.248690i	0.968583	−	0.248690i
$82$		0			0
$83$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$83$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$84$		0			0
$85$		0			0
$86$	0.133466	+	0.125333i	0.133466	+	0.125333i
$87$		0			0
$88$	0.0606317	+	0.0955403i	0.0606317	+	0.0955403i
$89$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$89$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$90$		0			0
$91$		0			0
$92$	0.214967	+	0.661600i	0.214967	+	0.661600i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$97$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$98$	0.781202	+	1.23098i	0.781202	+	1.23098i
$99$	0.0388067	+	0.616814i	0.0388067	+	0.616814i
$100$	−1.04654	−	0.414354i	−1.04654	−	0.414354i
$101$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$101$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$102$		0			0
$103$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$103$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$104$		0			0
$105$		0			0
$106$	2.73554	−	0.702367i	2.73554	−	0.702367i
$107$	−0.0800484	+	0.0967619i	−0.0800484	+	0.0967619i	−0.809017	−	0.587785i	$-0.800000\pi$
$107$	−0.0800484	+	0.0967619i	−0.0800484	+	0.0967619i	0.728969	+	0.684547i	$0.240000\pi$
$108$		0			0
$109$	0.0702235	+	0.368125i	0.0702235	+	0.368125i	1.00000			$0$
$109$	0.0702235	+	0.368125i	0.0702235	+	0.368125i	−0.929776	+	0.368125i	$0.880000\pi$
$110$		0			0
$111$		0			0
$112$	−0.625928	+	0.587785i	−0.625928	+	0.587785i
$113$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$113$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$114$		0			0
$115$		0			0
$116$	−0.768882	−	1.21156i	−0.768882	−	1.21156i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.613161	+	0.0774602i	0.613161	+	0.0774602i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1.41213	−	0.362574i	1.41213	−	0.362574i
$127$	−1.11716	−	1.35041i	−1.11716	−	1.35041i	−0.929776	−	0.368125i	$-0.880000\pi$
$127$	−1.11716	−	1.35041i	−1.11716	−	1.35041i	−0.187381	−	0.982287i	$-0.560000\pi$
$128$	0.293909	+	0.213537i	0.293909	+	0.213537i
$129$		0			0
$130$		0			0
$131$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$131$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$132$		0			0
$133$		0			0
$134$	0.232637	−	0.494380i	0.232637	−	0.494380i
$135$		0			0
$136$		0			0
$137$	−1.06320	+	0.134314i	−1.06320	+	0.134314i	−0.637424	−	0.770513i	$-0.720000\pi$
$137$	−1.06320	+	0.134314i	−1.06320	+	0.134314i	−0.425779	+	0.904827i	$0.640000\pi$
$138$		0			0
$139$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$139$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$140$		0			0
$141$		0			0
$142$	2.87007	−	0.362574i	2.87007	−	0.362574i
$143$		0			0
$144$	0.365595	+	0.776928i	0.365595	+	0.776928i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.0436801	+	0.694275i	0.0436801	+	0.694275i
$149$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$149$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$150$		0			0
$151$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$151$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$152$		0			0
$153$		0			0
$154$	0.0565777	+	0.899277i	0.0565777	+	0.899277i
$155$		0			0
$156$		0			0
$157$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$157$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$158$	−0.905028	−	1.92328i	−0.905028	−	1.92328i
$159$		0			0
$160$		0			0
$161$	−0.115808	+	0.607087i	−0.115808	+	0.607087i
$162$	0.0915446	−	1.45506i	0.0915446	−	1.45506i
$163$	1.03799	−	1.63560i	1.03799	−	1.63560i	0.309017	−	0.951057i	$-0.400000\pi$
$163$	1.03799	−	1.63560i	1.03799	−	1.63560i	0.728969	−	0.684547i	$-0.240000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$167$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$168$		0			0
$169$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$170$		0			0
$171$		0			0
$172$	0.123867	−	0.0680967i	0.123867	−	0.0680967i
$173$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$173$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$174$		0			0
$175$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$176$	−0.514002	+	0.131973i	−0.514002	+	0.131973i
$177$		0			0
$178$		0			0
$179$	0.348445	−	1.82662i	0.348445	−	1.82662i	−0.187381	−	0.982287i	$-0.560000\pi$
$179$	0.348445	−	1.82662i	0.348445	−	1.82662i	0.535827	−	0.844328i	$-0.320000\pi$
$180$		0			0
$181$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$181$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$182$		0			0
$183$		0			0
$184$	0.113155			0.113155
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.273190	+	0.256543i	−0.273190	+	0.256543i	−0.809017	−	0.587785i	$-0.800000\pi$
$191$	−0.273190	+	0.256543i	−0.273190	+	0.256543i	0.535827	+	0.844328i	$0.320000\pi$
$192$		0			0
$193$	−0.362989	−	1.90285i	−0.362989	−	1.90285i	−0.425779	−	0.904827i	$-0.640000\pi$
$193$	−0.362989	−	1.90285i	−0.362989	−	1.90285i	0.0627905	−	0.998027i	$-0.480000\pi$
$194$		0			0
$195$		0			0
$196$	1.09022	−	0.279921i	1.09022	−	0.279921i
$197$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$197$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$198$	0.872746	+	0.224083i	0.872746	+	0.224083i
$199$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$199$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$200$	−0.116705	+	0.141073i	−0.116705	+	0.141073i
$201$		0			0
$202$		0			0
$203$	−0.0800484	−	1.27233i	−0.0800484	−	1.27233i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.541587	+	0.297740i	0.541587	+	0.297740i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.159566	−	0.339095i	0.159566	−	0.339095i	−0.809017	−	0.587785i	$-0.800000\pi$
$211$	0.159566	−	0.339095i	0.159566	−	0.339095i	0.968583	+	0.248690i	$0.0800000\pi$
$212$	0.136911	−	2.17614i	0.136911	−	2.17614i
$213$		0			0
$214$	0.0981041	+	0.154587i	0.0981041	+	0.154587i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.542072	+	0.0684796i	0.542072	+	0.0684796i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$223$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$224$	0.610970	+	1.29838i	0.610970	+	1.29838i
$225$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$226$	−0.383650	−	1.18075i	−0.383650	−	1.18075i
$227$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$227$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$228$		0			0
$229$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$229$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$230$		0			0
$231$		0			0
$232$	−0.226078	+	0.0580470i	−0.226078	+	0.0580470i
$233$	−0.996398	+	1.57007i	−0.996398	+	1.57007i	−0.187381	+	0.982287i	$0.560000\pi$
$233$	−0.996398	+	1.57007i	−0.996398	+	1.57007i	−0.809017	+	0.587785i	$0.800000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.456288	−	0.718995i	−0.456288	−	0.718995i	0.535827	−	0.844328i	$-0.320000\pi$
$239$	−0.456288	−	0.718995i	−0.456288	−	0.718995i	−0.992115	+	0.125333i	$0.960000\pi$
$240$		0			0
$241$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$241$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$242$	0.383650	−	0.815299i	0.383650	−	0.815299i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$251$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$252$	0.0706758	−	1.12336i	0.0706758	−	1.12336i
$253$	−0.243474	+	0.294310i	−0.243474	+	0.294310i
$254$	−2.37577	+	0.940632i	−2.37577	+	0.940632i
$255$		0			0
$256$	−0.569350	+	0.413657i	−0.569350	+	0.413657i
$257$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$257$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$258$		0			0
$259$	−0.263146	+	0.559214i	−0.263146	+	0.559214i
$260$		0			0
$261$	−1.23480	−	0.317042i	−1.23480	−	0.317042i
$262$		0			0
$263$	−0.620759	+	0.582932i	−0.620759	+	0.582932i	−0.929776	−	0.368125i	$-0.880000\pi$
$263$	−0.620759	+	0.582932i	−0.620759	+	0.582932i	0.309017	+	0.951057i	$0.400000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.307498	−	0.288760i	−0.307498	−	0.288760i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$271$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$272$		0			0
$273$		0			0
$274$	−0.292765	+	1.53473i	−0.292765	+	1.53473i
$275$	−0.115808	−	0.607087i	−0.115808	−	0.607087i
$276$		0			0
$277$	1.69755	−	0.435857i	1.69755	−	0.435857i	0.728969	−	0.684547i	$-0.240000\pi$
$277$	1.69755	−	0.435857i	1.69755	−	0.435857i	0.968583	+	0.248690i	$0.0800000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.41789	+	0.779494i	−1.41789	+	0.779494i	−0.992115	−	0.125333i	$-0.960000\pi$
$281$	−1.41789	+	0.779494i	−1.41789	+	0.779494i	−0.425779	+	0.904827i	$0.640000\pi$
$282$		0			0
$283$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$283$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$284$	0.418499	−	2.19385i	0.418499	−	2.19385i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.42363	−	0.179846i	1.42363	−	0.179846i
$289$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$293$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$294$		0			0
$295$		0			0
$296$	0.109600	+	0.0281406i	0.109600	+	0.0281406i
$297$		0			0
$298$	2.34039	+	1.70039i	2.34039	+	1.70039i
$299$		0			0
$300$		0			0
$301$	0.125581			0.125581
$302$	1.27760	+	0.702367i	1.27760	+	0.702367i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$307$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$308$	0.673792	+	0.173000i	0.673792	+	0.173000i
$309$		0			0
$310$		0			0
$311$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$311$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$312$		0			0
$313$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$313$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$314$		0			0
$315$		0			0
$316$	−1.62809	+	0.205675i	−1.62809	+	0.205675i
$317$	0.238883	+	1.25227i	0.238883	+	1.25227i	0.876307	+	0.481754i	$0.160000\pi$
$317$	0.238883	+	1.25227i	0.238883	+	1.25227i	−0.637424	+	0.770513i	$0.720000\pi$
$318$		0			0
$319$	0.335471	−	0.712913i	0.335471	−	0.712913i
$320$		0			0
$321$		0			0
$322$	0.789600	+	0.434086i	0.789600	+	0.434086i
$323$		0			0
$324$	−1.04654	−	0.414354i	−1.04654	−	0.414354i
$325$		0			0
$326$	−1.80026	−	2.17614i	−1.80026	−	2.17614i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.968583	+	0.248690i	$0.0800000\pi$
$331$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.876307	+	0.481754i	$0.160000\pi$
$332$		0			0
$333$	0.450527	+	0.423073i	0.450527	+	0.423073i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.574221	+	0.904827i	0.574221	+	0.904827i	1.00000			$0$
$337$	0.574221	+	0.904827i	0.574221	+	0.904827i	−0.425779	+	0.904827i	$0.640000\pi$
$338$	1.27760	+	0.702367i	1.27760	+	0.702367i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.968583	+	0.248690i	0.968583	+	0.248690i
$344$	−0.00430837	−	0.0225853i	−0.00430837	−	0.0225853i
$345$		0			0
$346$		0			0
$347$	−0.362989	+	0.0931997i	−0.362989	+	0.0931997i	−0.425779	−	0.904827i	$-0.640000\pi$
$347$	−0.362989	+	0.0931997i	−0.362989	+	0.0931997i	0.0627905	+	0.998027i	$0.480000\pi$
$348$		0			0
$349$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$349$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$350$	−1.35556	+	0.536702i	−1.35556	+	0.536702i
$351$		0			0
$352$	−0.0556854	+	0.885095i	−0.0556854	+	0.885095i
$353$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$353$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−2.37577	−	1.30609i	−2.37577	−	1.30609i
$359$	−1.06320	+	1.67534i	−1.06320	+	1.67534i	−0.425779	+	0.904827i	$0.640000\pi$
$359$	−1.06320	+	1.67534i	−1.06320	+	1.67534i	−0.637424	+	0.770513i	$0.720000\pi$
$360$		0			0
$361$	0.309017	+	0.951057i	0.309017	+	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$367$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$368$	−0.163987	+	0.504701i	−0.163987	+	0.504701i
$369$		0			0
$370$		0			0
$371$	1.03799	−	1.63560i	1.03799	−	1.63560i
$372$		0			0
$373$	−0.456288	−	0.969661i	−0.456288	−	0.969661i	−0.992115	−	0.125333i	$-0.960000\pi$
$373$	−0.456288	−	0.969661i	−0.456288	−	0.969661i	0.535827	−	0.844328i	$-0.320000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.62954	+	0.645180i	−1.62954	+	0.645180i	−0.992115	−	0.125333i	$-0.960000\pi$
$379$	−1.62954	+	0.645180i	−1.62954	+	0.645180i	−0.637424	+	0.770513i	$0.720000\pi$
$380$		0			0
$381$		0			0
$382$	0.232637	+	0.494380i	0.232637	+	0.494380i
$383$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$383$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$384$		0			0
$385$		0			0
$386$	−2.80200	−	0.353975i	−2.80200	−	0.353975i
$387$	0.0388067	−	0.119435i	0.0388067	−	0.119435i
$388$		0			0
$389$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$389$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$390$		0			0
$391$		0			0
$392$	0.0114963	−	0.182728i	0.0114963	−	0.182728i
$393$		0			0
$394$	0.656841	+	2.02155i	0.656841	+	2.02155i
$395$		0			0
$396$	0.372746	−	0.587354i	0.372746	−	0.587354i
$397$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$397$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$398$		0			0
$399$		0			0
$400$	−0.460087	−	0.724981i	−0.460087	−	0.724981i
$401$	0.125581	+	1.99605i	0.125581	+	1.99605i	0.0627905	+	0.998027i	$0.480000\pi$
$401$	0.125581	+	1.99605i	0.125581	+	1.99605i	0.0627905	+	0.998027i	$0.480000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−1.80026	−	0.462227i	−1.80026	−	0.462227i
$407$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$408$		0			0
$409$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$409$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.656841	−	0.616814i	0.656841	−	0.616814i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$419$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$420$		0			0
$421$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$421$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$422$	−0.398294	−	0.374023i	−0.398294	−	0.374023i
$423$		0			0
$424$	−0.329768	−	0.130564i	−0.329768	−	0.130564i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.136911	−	0.0351527i	0.136911	−	0.0351527i
$429$		0			0
$430$		0			0
$431$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.187381	−	0.982287i	$-0.560000\pi$
$431$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.809017	+	0.587785i	$0.800000\pi$
$432$		0			0
$433$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$433$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$434$		0			0
$435$		0			0
$436$	0.179605	−	0.381679i	0.179605	−	0.381679i
$437$		0			0
$438$		0			0
$439$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$439$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$440$		0			0
$441$	0.535827	−	0.844328i	0.535827	−	0.844328i
$442$		0			0
$443$	0.303189	−	1.58937i	0.303189	−	1.58937i	−0.425779	−	0.904827i	$-0.640000\pi$
$443$	0.303189	−	1.58937i	0.303189	−	1.58937i	0.728969	−	0.684547i	$-0.240000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.22369	−	0.154587i	1.22369	−	0.154587i
$449$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$449$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$450$	0.0915446	+	1.45506i	0.0915446	+	1.45506i
$451$		0			0
$452$	−0.958498			−0.958498
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$457$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$461$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$462$		0			0
$463$	−0.273190	+	1.43211i	−0.273190	+	1.43211i	0.535827	+	0.844328i	$0.320000\pi$
$463$	−0.273190	+	1.43211i	−0.273190	+	1.43211i	−0.809017	+	0.587785i	$0.800000\pi$
$464$	0.0687334	−	1.09249i	0.0687334	−	1.09249i
$465$		0			0
$466$	1.72813	+	2.08895i	1.72813	+	2.08895i
$467$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$467$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$468$		0			0
$469$	−0.115808	−	0.356420i	−0.115808	−	0.356420i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.0680131	+	0.0373905i	0.0680131	+	0.0373905i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.23480	−	1.49261i	−1.23480	−	1.49261i
$478$	−1.20251	+	0.308753i	−1.20251	+	0.308753i
$479$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$479$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.507105	−	0.476203i	−0.507105	−	0.476203i
$485$		0			0
$486$		0			0
$487$	−0.929324	−	0.872693i	−0.929324	−	0.872693i	0.0627905	−	0.998027i	$-0.480000\pi$
$487$	−0.929324	−	0.872693i	−0.929324	−	0.872693i	−0.992115	+	0.125333i	$0.960000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$491$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.26480	−	1.52888i	1.26480	−	1.52888i
$498$		0			0
$499$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$499$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$503$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$504$	−0.170232	−	0.0673997i	−0.170232	−	0.0673997i
$505$		0			0
$506$	0.298393	+	0.470191i	0.298393	+	0.470191i
$507$		0			0
$508$	0.123867	+	1.96882i	0.123867	+	1.96882i
$509$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$509$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$510$		0			0
$511$		0			0
$512$	0.429324	+	1.32132i	0.429324	+	1.32132i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	0.656841	+	0.616814i	0.656841	+	0.616814i
$519$		0			0
$520$		0			0
$521$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$521$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$522$	−0.995914	+	1.56931i	−0.995914	+	1.56931i
$523$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$523$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$524$		0			0
$525$		0			0
$526$	0.528613	+	1.12336i	0.528613	+	1.12336i
$527$		0			0
$528$		0			0
$529$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.0601278	+	0.0330555i	−0.0601278	+	0.0330555i
$537$		0			0
$538$		0			0
$539$	0.450527	+	0.423073i	0.450527	+	0.423073i
$540$		0			0
$541$	−0.456288	−	0.718995i	−0.456288	−	0.718995i	0.535827	−	0.844328i	$-0.320000\pi$
$541$	−0.456288	−	0.718995i	−0.456288	−	0.718995i	−0.992115	+	0.125333i	$0.960000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.781202	−	1.23098i	0.781202	−	1.23098i	−0.187381	−	0.982287i	$-0.560000\pi$
$547$	0.781202	−	1.23098i	0.781202	−	1.23098i	0.968583	−	0.248690i	$-0.0800000\pi$
$548$	1.05703	+	0.581107i	1.05703	+	0.581107i
$549$		0			0
$550$	−0.893950	−	0.112932i	−0.893950	−	0.112932i
$551$		0			0
$552$		0			0
$553$	−1.35556	−	0.536702i	−1.35556	−	0.536702i
$554$	0.160442	−	2.55016i	0.160442	−	2.55016i
$555$		0			0
$556$		0			0
$557$	1.87631	+	0.481754i	1.87631	+	0.481754i	1.00000			$0$
$557$	1.87631	+	0.481754i	1.87631	+	0.481754i	0.876307	+	0.481754i	$0.160000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.442031	+	2.31721i	0.442031	+	2.31721i
$563$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$563$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$568$	−0.318354	−	0.175017i	−0.318354	−	0.175017i
$569$	−0.996398	−	1.57007i	−0.996398	−	1.57007i	−0.809017	−	0.587785i	$-0.800000\pi$
$569$	−0.996398	−	1.57007i	−0.996398	−	1.57007i	−0.187381	−	0.982287i	$-0.560000\pi$
$570$		0			0
$571$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$571$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.574633	−	0.227513i	−0.574633	−	0.227513i
$576$	0.231118	−	1.21156i	0.231118	−	1.21156i
$577$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$577$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$578$	−1.35556	+	0.536702i	−1.35556	+	0.536702i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.04914	−	0.576772i	1.04914	−	0.576772i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$587$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.284349	+	0.448063i	−0.284349	+	0.448063i
$593$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$593$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$594$		0			0
$595$		0			0
$596$	1.80687	−	1.31277i	1.80687	−	1.31277i
$597$		0			0
$598$		0			0
$599$	1.26480	−	0.159781i	1.26480	−	0.159781i	0.535827	−	0.844328i	$-0.320000\pi$
$599$	1.26480	−	0.159781i	1.26480	−	0.159781i	0.728969	+	0.684547i	$0.240000\pi$
$600$		0			0
$601$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$601$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$602$	0.0565777	−	0.174128i	0.0565777	−	0.174128i
$603$	−0.374763			−0.374763
$604$	0.820513	−	0.770513i	0.820513	−	0.770513i
$605$		0			0
$606$		0			0
$607$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$607$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.06320	+	0.134314i	−1.06320	+	0.134314i	−0.637424	−	0.770513i	$-0.720000\pi$
$613$	−1.06320	+	0.134314i	−1.06320	+	0.134314i	−0.425779	+	0.904827i	$0.640000\pi$
$614$		0			0
$615$		0			0
$616$	0.0606317	−	0.0955403i	0.0606317	−	0.0955403i
$617$	−0.393950	−	0.476203i	−0.393950	−	0.476203i	0.535827	−	0.844328i	$-0.320000\pi$
$617$	−0.393950	−	0.476203i	−0.393950	−	0.476203i	−0.929776	+	0.368125i	$0.880000\pi$
$618$		0			0
$619$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$619$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.876307	−	0.481754i	0.876307	−	0.481754i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.303189	+	1.58937i	0.303189	+	1.58937i	0.728969	+	0.684547i	$0.240000\pi$
$631$	0.303189	+	1.58937i	0.303189	+	1.58937i	−0.425779	+	0.904827i	$0.640000\pi$
$632$	−0.0500182	+	0.262205i	−0.0500182	+	0.262205i
$633$		0			0
$634$	1.84399	+	0.232950i	1.84399	+	0.232950i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.837372	−	0.786345i	−0.837372	−	0.786345i
$639$	−1.06320	−	1.67534i	−1.06320	−	1.67534i
$640$		0			0
$641$	0.238883	+	0.288760i	0.238883	+	0.288760i	0.876307	−	0.481754i	$-0.160000\pi$
$641$	0.238883	+	0.288760i	0.238883	+	0.288760i	−0.637424	+	0.770513i	$0.720000\pi$
$642$		0			0
$643$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$643$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$644$	0.507105	−	0.476203i	0.507105	−	0.476203i
$645$		0			0
$646$		0			0
$647$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$647$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$648$	−0.116705	+	0.141073i	−0.116705	+	0.141073i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.02732	+	0.802673i	−2.02732	+	0.802673i
$653$	0.812619	−	0.982287i	0.812619	−	0.982287i	−0.187381	−	0.982287i	$-0.560000\pi$
$653$	0.812619	−	0.982287i	0.812619	−	0.982287i	1.00000			$0$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.116762	−	1.85588i	−0.116762	−	1.85588i	−0.425779	−	0.904827i	$-0.640000\pi$
$659$	−0.116762	−	1.85588i	−0.116762	−	1.85588i	0.309017	−	0.951057i	$-0.400000\pi$
$660$		0			0
$661$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$661$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$662$	1.84399	−	2.22900i	1.84399	−	2.22900i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.789600	−	0.434086i	0.789600	−	0.434086i
$667$	−0.422178	−	0.665245i	−0.422178	−	0.665245i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.939097	−	1.47978i	0.939097	−	1.47978i	0.0627905	−	0.998027i	$-0.480000\pi$
$673$	0.939097	−	1.47978i	0.939097	−	1.47978i	0.876307	−	0.481754i	$-0.160000\pi$
$674$	1.51332	−	0.388554i	1.51332	−	0.388554i
$675$		0			0
$676$	0.820513	−	0.770513i	0.820513	−	0.770513i
$677$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$677$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.27760	−	1.19975i	1.27760	−	1.19975i	0.309017	−	0.951057i	$-0.400000\pi$
$683$	1.27760	−	1.19975i	1.27760	−	1.19975i	0.968583	−	0.248690i	$-0.0800000\pi$
$684$		0			0
$685$		0			0
$686$	0.781202	−	1.23098i	0.781202	−	1.23098i
$687$		0			0
$688$	0.106980	+	0.0135147i	0.106980	+	0.0135147i
$689$		0			0
$690$		0			0
$691$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$691$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$692$		0			0
$693$	0.541587	−	0.297740i	0.541587	−	0.297740i
$694$	−0.0343075	+	0.545302i	−0.0343075	+	0.545302i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.0706758	+	1.12336i	0.0706758	+	1.12336i
$701$	1.96858	+	0.248690i	1.96858	+	0.248690i	1.00000			$0$
$701$	1.96858	+	0.248690i	1.96858	+	0.248690i	0.968583	+	0.248690i	$0.0800000\pi$
$702$		0			0
$703$		0			0
$704$	0.708759	+	0.280618i	0.708759	+	0.280618i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$709$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$710$		0			0
$711$	−0.929324	+	1.12336i	−0.929324	+	1.12336i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.52579	+	1.43281i	−1.52579	+	1.43281i
$717$		0			0
$718$	1.84399	+	2.22900i	1.84399	+	2.22900i
$719$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$719$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$720$		0			0
$721$		0			0
$722$	1.45794			1.45794
$723$		0			0
$724$		0			0
$725$	1.26480	+	0.159781i	1.26480	+	0.159781i
$726$		0			0
$727$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$727$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$728$		0			0
$729$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$733$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$734$		0			0
$735$		0			0
$736$	0.717472	+	0.521274i	0.717472	+	0.521274i
$737$	0.0434005	−	0.227513i	0.0434005	−	0.227513i
$738$		0			0
$739$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$739$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$740$		0			0
$741$		0			0
$742$	−1.80026	−	2.17614i	−1.80026	−	2.17614i
$743$	0.0672897	−	0.106032i	0.0672897	−	0.106032i	−0.809017	−	0.587785i	$-0.800000\pi$
$743$	0.0672897	−	0.106032i	0.0672897	−	0.106032i	0.876307	+	0.481754i	$0.160000\pi$
$744$		0			0
$745$		0			0
$746$	−1.55008	+	0.195821i	−1.55008	+	0.195821i
$747$		0			0
$748$		0			0
$749$	0.121636	+	0.0312307i	0.121636	+	0.0312307i
$750$		0			0
$751$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$751$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$757$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$758$	0.160442	+	2.55016i	0.160442	+	2.55016i
$759$		0			0
$760$		0			0
$761$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$761$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$762$		0			0
$763$	0.303189	−	0.220280i	0.303189	−	0.220280i
$764$	0.418499	−	0.0528688i	0.418499	−	0.0528688i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$769$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$770$		0			0
$771$		0			0
$772$	−0.928385	+	1.97292i	−0.928385	+	1.97292i
$773$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$773$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$774$	−0.148122	−	0.107617i	−0.148122	−	0.107617i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.26401	−	0.918358i	−1.26401	−	0.918358i
$779$		0			0
$780$		0			0
$781$	1.14020	−	0.451439i	1.14020	−	0.451439i
$782$		0			0
$783$		0			0
$784$	0.798351	+	0.316090i	0.798351	+	0.316090i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	1.64103			1.64103
$789$		0			0
$790$		0			0
$791$	−0.746226	−	0.410241i	−0.746226	−	0.410241i
$792$	−0.0721280	−	0.0871877i	−0.0721280	−	0.0871877i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$797$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$798$		0			0
$799$		0			0
$800$	−1.38986	+	0.356856i	−1.38986	+	0.356856i
$801$		0			0
$802$	2.82427	+	0.725149i	2.82427	+	0.725149i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.728969	−	0.684547i	$-0.240000\pi$
$809$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.535827	+	0.844328i	$0.320000\pi$
$810$		0			0
$811$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$811$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$812$	−0.768882	+	1.21156i	−0.768882	+	1.21156i
$813$		0			0
$814$	0.172086	+	0.529627i	0.172086	+	0.529627i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$821$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$822$		0			0
$823$	−1.73879	+	0.955910i	−1.73879	+	0.955910i	−0.809017	+	0.587785i	$0.800000\pi$
$823$	−1.73879	+	0.955910i	−1.73879	+	0.955910i	−0.929776	+	0.368125i	$0.880000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.41213	−	1.32608i	1.41213	−	1.32608i	0.535827	−	0.844328i	$-0.320000\pi$
$827$	1.41213	−	1.32608i	1.41213	−	1.32608i	0.876307	−	0.481754i	$-0.160000\pi$
$828$	−0.296192	−	0.629441i	−0.296192	−	0.629441i
$829$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$829$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$839$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$840$		0			0
$841$	0.455778	+	0.428004i	0.455778	+	0.428004i
$842$	−0.728969	+	2.24353i	−0.728969	+	2.24353i
$843$		0			0
$844$	−0.369649	+	0.203216i	−0.369649	+	0.203216i
$845$		0			0
$846$		0			0
$847$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$848$	1.06026	−	1.28163i	1.06026	−	1.28163i
$849$		0			0
$850$		0			0
$851$	0.0239838	+	0.381212i	0.0239838	+	0.381212i
$852$		0			0
$853$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$853$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$854$		0			0
$855$		0			0
$856$	0.00144371	−	0.0229472i	0.00144371	−	0.0229472i
$857$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$857$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$858$		0			0
$859$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$859$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$860$		0			0
$861$		0			0
$862$	−0.995914	+	1.20385i	−0.995914	+	1.20385i
$863$	−0.620759	+	1.31918i	−0.620759	+	1.31918i	0.309017	+	0.951057i	$0.400000\pi$
$863$	−0.620759	+	1.31918i	−0.620759	+	1.31918i	−0.929776	+	0.368125i	$0.880000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.574354	−	0.694275i	−0.574354	−	0.694275i
$870$		0			0
$871$		0			0
$872$	−0.0500182	−	0.0469702i	−0.0500182	−	0.0469702i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.968583	+	0.248690i	$0.0800000\pi$
$877$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.876307	+	0.481754i	$0.160000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$881$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$882$	−0.929324	−	1.12336i	−0.929324	−	1.12336i
$883$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$883$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$884$		0			0
$885$		0			0
$886$	−2.06720	−	1.13645i	−2.06720	−	1.13645i
$887$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$887$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$888$		0			0
$889$	−0.746226	+	1.58581i	−0.746226	+	1.58581i
$890$		0			0
$891$	−0.115808	−	0.607087i	−0.115808	−	0.607087i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.0680740	−	0.356856i	0.0680740	−	0.356856i
$897$		0			0
$898$	2.19334	−	1.59355i	2.19334	−	1.59355i
$899$		0			0
$900$	1.09022	+	0.279921i	1.09022	+	0.279921i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.0481792	+	0.148280i	−0.0481792	+	0.148280i
$905$		0			0
$906$		0			0
$907$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$907$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.92189	+	0.242791i	−1.92189	+	0.242791i	−0.992115	−	0.125333i	$-0.960000\pi$
$911$	−1.92189	+	0.242791i	−1.92189	+	0.242791i	−0.929776	+	0.368125i	$0.880000\pi$
$912$		0			0
$913$		0			0
$914$	−1.71963	+	1.24939i	−1.71963	+	1.24939i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.27485	−	1.54103i	−1.27485	−	1.54103i	−0.637424	−	0.770513i	$-0.720000\pi$
$919$	−1.27485	−	1.54103i	−1.27485	−	1.54103i	−0.637424	−	0.770513i	$-0.720000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$926$	1.86266	+	1.02401i	1.86266	+	1.02401i
$927$		0			0
$928$	−1.70087	−	0.673424i	−1.70087	−	0.673424i
$929$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$929$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$930$		0			0
$931$		0			0
$932$	1.94609	−	0.770513i	1.94609	−	0.770513i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$937$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$938$	−0.546380			−0.546380
$939$		0			0
$940$		0			0
$941$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$941$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.0824867	−	0.0774602i	0.0824867	−	0.0774602i
$947$	1.69755	+	0.435857i	1.69755	+	0.435857i	0.968583	−	0.248690i	$-0.0800000\pi$
$947$	1.69755	+	0.435857i	1.69755	+	0.435857i	0.728969	+	0.684547i	$0.240000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.03799	+	0.266509i	1.03799	+	0.266509i	0.728969	−	0.684547i	$-0.240000\pi$
$953$	1.03799	+	0.266509i	1.03799	+	0.266509i	0.309017	+	0.951057i	$0.400000\pi$
$954$	−2.62594	+	1.03968i	−2.62594	+	1.03968i
$955$		0			0
$956$	−0.0601846	+	0.956607i	−0.0601846	+	0.956607i
$957$		0			0
$958$		0			0
$959$	0.574221	+	0.904827i	0.574221	+	0.904827i
$960$		0			0
$961$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$962$		0			0
$963$	0.0672897	−	0.106032i	0.0672897	−	0.106032i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.116762	+	1.85588i	−0.116762	+	1.85588i	0.309017	+	0.951057i	$0.400000\pi$
$967$	−0.116762	+	1.85588i	−0.116762	+	1.85588i	−0.425779	+	0.904827i	$0.640000\pi$
$968$	−0.0991588	+	0.0545130i	−0.0991588	+	0.0545130i
$969$		0			0
$970$		0			0
$971$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$971$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$972$		0			0
$973$		0			0
$974$	−1.62875	+	0.895411i	−1.62875	+	0.895411i
$975$		0			0
$976$		0			0
$977$	0.362576	+	0.770513i	0.362576	+	0.770513i	1.00000			$0$
$977$	0.362576	+	0.770513i	0.362576	+	0.770513i	−0.637424	+	0.770513i	$0.720000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.115808	−	0.356420i	−0.115808	−	0.356420i
$982$	0.482809	+	1.48593i	0.482809	+	1.48593i
$983$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$983$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.0680131	−	0.0373905i	0.0680131	−	0.0373905i
$990$		0			0
$991$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$991$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$992$		0			0
$993$		0			0
$994$	−1.55008	−	2.44254i	−1.55008	−	2.44254i
$995$		0			0
$996$		0			0
$997$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$997$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$998$	0.872746	+	2.68604i	0.872746	+	2.68604i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1057.1.bq.a.160.1	✓	20
7.6	odd	2	CM	1057.1.bq.a.160.1	✓	20
151.84	even	25	inner	1057.1.bq.a.839.1	yes	20
1057.839	odd	50	inner	1057.1.bq.a.839.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1057.1.bq.a.160.1	✓	20	1.1	even	1	trivial
1057.1.bq.a.160.1	✓	20	7.6	odd	2	CM
1057.1.bq.a.839.1	yes	20	151.84	even	25	inner
1057.1.bq.a.839.1	yes	20	1057.839	odd	50	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:	\( N \)	\(=\)	\( 1057 = 7 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1057.bq (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.450527 - 1.38658i) q^{2} +(-0.910614 - 0.661600i) q^{4} +(-0.425779 - 0.904827i) q^{7} +(-0.148122 + 0.107617i) q^{8} +(-0.992115 + 0.125333i) q^{9} +O(q^{10})\)	\(q+(0.450527 - 1.38658i) q^{2} +(-0.910614 - 0.661600i) q^{4} +(-0.425779 - 0.904827i) q^{7} +(-0.148122 + 0.107617i) q^{8} +(-0.992115 + 0.125333i) q^{9} +(0.0388067 - 0.616814i) q^{11} +(-1.44644 + 0.182728i) q^{14} +(-0.265337 - 0.816623i) q^{16} +(-0.273190 + 1.43211i) q^{18} +(-0.837780 - 0.331700i) q^{22} +(-0.500000 - 0.363271i) q^{23} +(0.968583 - 0.248690i) q^{25} +(-0.210913 + 1.10564i) q^{28} +(1.18532 + 0.469303i) q^{29} -1.43494 q^{32} +(0.986354 + 0.542253i) q^{36} +(-0.393950 - 0.476203i) q^{37} +(-0.0534698 + 0.113629i) q^{43} +(-0.443422 + 0.536005i) q^{44} +(-0.728969 + 0.529627i) q^{46} +(-0.637424 + 0.770513i) q^{49} +(0.0915446 - 1.45506i) q^{50} +(1.03799 + 1.63560i) q^{53} +(0.160442 + 0.0882039i) q^{56} +(1.18475 - 1.43211i) q^{58} +(0.535827 + 0.844328i) q^{63} +(-0.381145 + 1.17304i) q^{64} +(0.371808 + 0.0469702i) q^{67} +(0.844844 + 1.79538i) q^{71} +(0.133466 - 0.125333i) q^{72} +(-0.837780 + 0.331700i) q^{74} +(-0.574633 + 0.227513i) q^{77} +(1.06279 - 0.998027i) q^{79} +(0.968583 - 0.248690i) q^{81} +(0.133466 + 0.125333i) q^{86} +(0.0606317 + 0.0955403i) q^{88} +(0.214967 + 0.661600i) q^{92} +(0.781202 + 1.23098i) q^{98} +(0.0388067 + 0.616814i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4}+O(q^{10}) \) 20 * q - 5 * q^4	\( 20 q - 5 q^{4} - 5 q^{14} - 5 q^{16} - 5 q^{18} - 10 q^{23} - 5 q^{29} - 10 q^{32} - 10 q^{44} - 5 q^{53} - 5 q^{56} - 5 q^{58} - 5 q^{64} - 5 q^{67} - 5 q^{71} + 20 q^{72} + 20 q^{79} + 20 q^{86} + 15 q^{88} + 15 q^{92}+O(q^{100}) \) 20 * q - 5 * q^4 - 5 * q^14 - 5 * q^16 - 5 * q^18 - 10 * q^23 - 5 * q^29 - 10 * q^32 - 10 * q^44 - 5 * q^53 - 5 * q^56 - 5 * q^58 - 5 * q^64 - 5 * q^67 - 5 * q^71 + 20 * q^72 + 20 * q^79 + 20 * q^86 + 15 * q^88 + 15 * q^92

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