LMFDB - Embedded newform 1944.1.bh.a.187.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1944,1,Mod(43,1944)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1944, base_ring=CyclotomicField(162))

chi = DirichletCharacter(H, H._module([81, 81, 130]))

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

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

chi := DirichletCharacter("1944.43");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1944 = 2^{3} \cdot 3^{5} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1944.bh (of order $162$, degree $54$, 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:	$0.970182384559$
Analytic rank:	$0$
Dimension:	$54$
Coefficient field:	$\Q(\zeta_{162})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{54} - x^{27} + 1 $ x^54 - x^27 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{81}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{81} + \cdots)$

Embedding invariants

Embedding label			187.1
Root			$-0.925724 - 0.378200i$ of defining polynomial
Character	$\chi$	$=$	1944.187
Dual form			1944.1.bh.a.499.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.987990 - 0.154519i) q^{2} +(0.249441 - 0.968390i) q^{3} +(0.952248 - 0.305326i) q^{4} +(0.0968109 - 0.995303i) q^{6} +(0.893633 - 0.448799i) q^{8} +(-0.875558 - 0.483113i) q^{9} +O(q^{10})$	\(q+(0.987990 - 0.154519i) q^{2} +(0.249441 - 0.968390i) q^{3} +(0.952248 - 0.305326i) q^{4} +(0.0968109 - 0.995303i) q^{6} +(0.893633 - 0.448799i) q^{8} +(-0.875558 - 0.483113i) q^{9} +(-0.422135 - 0.0328109i) q^{11} +(-0.0581448 - 0.998308i) q^{12} +(0.813552 - 0.581492i) q^{16} +(0.256449 + 0.594515i) q^{17} +(-0.939693 - 0.342020i) q^{18} +(-0.515212 - 0.692050i) q^{19} +(-0.422135 + 0.0328109i) q^{22} +(-0.211704 - 0.977334i) q^{24} +(0.657521 - 0.753436i) q^{25} +(-0.686242 + 0.727374i) q^{27} +(0.713930 - 0.700217i) q^{32} +(-0.137071 + 0.400606i) q^{33} +(0.345233 + 0.547749i) q^{34} +(-0.981255 - 0.192712i) q^{36} +(-0.615959 - 0.604128i) q^{38} +(0.693969 + 1.79743i) q^{41} +(-0.179686 + 0.697585i) q^{43} +(-0.411995 + 0.0976445i) q^{44} +(-0.360178 - 0.932884i) q^{48} +(-0.963371 - 0.268173i) q^{49} +(0.533204 - 0.845986i) q^{50} +(0.639692 - 0.100046i) q^{51} +(-0.565607 + 0.824675i) q^{54} +(-0.798689 + 0.326300i) q^{57} +(-0.701910 + 1.46792i) q^{59} +(0.597159 - 0.802123i) q^{64} +(-0.0735240 + 0.416975i) q^{66} +(-0.231732 + 0.139851i) q^{67} +(0.425724 + 0.487826i) q^{68} +(-0.999248 - 0.0387754i) q^{72} +(-0.0324024 - 0.0213114i) q^{73} +(-0.565607 - 0.824675i) q^{75} +(-0.701910 - 0.501695i) q^{76} +(0.533204 + 0.845986i) q^{81} +(0.963371 + 1.66861i) q^{82} +(0.206600 - 0.535108i) q^{83} +(-0.0697382 + 0.716972i) q^{86} +(-0.391959 + 0.160133i) q^{88} +(-0.0694434 + 1.19230i) q^{89} +(-0.500000 - 0.866025i) q^{96} +(1.81936 + 0.826999i) q^{97} +(-0.993238 - 0.116093i) q^{98} +(0.353752 + 0.232666i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q+O(q^{10}) $ 54 * q	$ 54 q - 27 q^{68} - 27 q^{96}+O(q^{100}) $ 54 * q - 27 * q^68 - 27 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times$.

$n$	$487$	$973$	$1217$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{77}{81}\right)$

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 $

$2$	0.987990	−	0.154519i	0.987990	−	0.154519i
$2$	0.987990	−	0.154519i	0.987990	−	0.154519i
$3$	0.249441	−	0.968390i	0.249441	−	0.968390i
$3$	0.249441	−	0.968390i	0.249441	−	0.968390i
$4$	0.952248	−	0.305326i	0.952248	−	0.305326i
$5$		0			0		0.910363	−	0.413811i	$-0.135802\pi$
$5$		0			0		−0.910363	+	0.413811i	$0.864198\pi$
$6$	0.0968109	−	0.995303i	0.0968109	−	0.995303i
$7$		0			0		0.135331	−	0.990800i	$-0.456790\pi$
$7$		0			0		−0.135331	+	0.990800i	$0.543210\pi$
$8$	0.893633	−	0.448799i	0.893633	−	0.448799i
$9$	−0.875558	−	0.483113i	−0.875558	−	0.483113i
$10$		0			0
$11$	−0.422135	−	0.0328109i	−0.422135	−	0.0328109i	−0.135331	−	0.990800i	$-0.543210\pi$
$11$	−0.422135	−	0.0328109i	−0.422135	−	0.0328109i	−0.286803	+	0.957990i	$0.592593\pi$
$12$	−0.0581448	−	0.998308i	−0.0581448	−	0.998308i
$13$		0			0		0.323734	−	0.946148i	$-0.395062\pi$
$13$		0			0		−0.323734	+	0.946148i	$0.604938\pi$
$14$		0			0
$15$		0			0
$16$	0.813552	−	0.581492i	0.813552	−	0.581492i
$17$	0.256449	+	0.594515i	0.256449	+	0.594515i	0.996993	−	0.0774924i	$-0.0246914\pi$
$17$	0.256449	+	0.594515i	0.256449	+	0.594515i	−0.740544	+	0.672008i	$0.765432\pi$
$18$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$19$	−0.515212	−	0.692050i	−0.515212	−	0.692050i	0.466044	−	0.884762i	$-0.345679\pi$
$19$	−0.515212	−	0.692050i	−0.515212	−	0.692050i	−0.981255	+	0.192712i	$0.938272\pi$
$20$		0			0
$21$		0			0
$22$	−0.422135	+	0.0328109i	−0.422135	+	0.0328109i
$23$		0			0		0.790393	−	0.612601i	$-0.209877\pi$
$23$		0			0		−0.790393	+	0.612601i	$0.790123\pi$
$24$	−0.211704	−	0.977334i	−0.211704	−	0.977334i
$25$	0.657521	−	0.753436i	0.657521	−	0.753436i
$26$		0			0
$27$	−0.686242	+	0.727374i	−0.686242	+	0.727374i
$28$		0			0
$29$		0			0		−0.856167	−	0.516699i	$-0.827160\pi$
$29$		0			0		0.856167	+	0.516699i	$0.172840\pi$
$30$		0			0
$31$		0			0		−0.999248	−	0.0387754i	$-0.987654\pi$
$31$		0			0		0.999248	+	0.0387754i	$0.0123457\pi$
$32$	0.713930	−	0.700217i	0.713930	−	0.700217i
$33$	−0.137071	+	0.400606i	−0.137071	+	0.400606i
$34$	0.345233	+	0.547749i	0.345233	+	0.547749i
$35$		0			0
$36$	−0.981255	−	0.192712i	−0.981255	−	0.192712i
$37$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$37$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$38$	−0.615959	−	0.604128i	−0.615959	−	0.604128i
$39$		0			0
$40$		0			0
$41$	0.693969	+	1.79743i	0.693969	+	1.79743i	0.597159	+	0.802123i	$0.296296\pi$
$41$	0.693969	+	1.79743i	0.693969	+	1.79743i	0.0968109	+	0.995303i	$0.469136\pi$
$42$		0			0
$43$	−0.179686	+	0.697585i	−0.179686	+	0.697585i	0.813552	+	0.581492i	$0.197531\pi$
$43$	−0.179686	+	0.697585i	−0.179686	+	0.697585i	−0.993238	+	0.116093i	$0.962963\pi$
$44$	−0.411995	+	0.0976445i	−0.411995	+	0.0976445i
$45$		0			0
$46$		0			0
$47$		0			0		0.999248	−	0.0387754i	$-0.0123457\pi$
$47$		0			0		−0.999248	+	0.0387754i	$0.987654\pi$
$48$	−0.360178	−	0.932884i	−0.360178	−	0.932884i
$49$	−0.963371	−	0.268173i	−0.963371	−	0.268173i
$50$	0.533204	−	0.845986i	0.533204	−	0.845986i
$51$	0.639692	−	0.100046i	0.639692	−	0.100046i
$52$		0			0
$53$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$53$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$54$	−0.565607	+	0.824675i	−0.565607	+	0.824675i
$55$		0			0
$56$		0			0
$57$	−0.798689	+	0.326300i	−0.798689	+	0.326300i
$58$		0			0
$59$	−0.701910	+	1.46792i	−0.701910	+	1.46792i	0.173648	+	0.984808i	$0.444444\pi$
$59$	−0.701910	+	1.46792i	−0.701910	+	1.46792i	−0.875558	+	0.483113i	$0.839506\pi$
$60$		0			0
$61$		0			0		−0.952248	−	0.305326i	$-0.901235\pi$
$61$		0			0		0.952248	+	0.305326i	$0.0987654\pi$
$62$		0			0
$63$		0			0
$64$	0.597159	−	0.802123i	0.597159	−	0.802123i
$65$		0			0
$66$	−0.0735240	+	0.416975i	−0.0735240	+	0.416975i
$67$	−0.231732	+	0.139851i	−0.231732	+	0.139851i	−0.627812	−	0.778365i	$-0.716049\pi$
$67$	−0.231732	+	0.139851i	−0.231732	+	0.139851i	0.396080	+	0.918216i	$0.370370\pi$
$68$	0.425724	+	0.487826i	0.425724	+	0.487826i
$69$		0			0
$70$		0			0
$71$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$71$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$72$	−0.999248	−	0.0387754i	−0.999248	−	0.0387754i
$73$	−0.0324024	−	0.0213114i	−0.0324024	−	0.0213114i	0.533204	−	0.845986i	$-0.320988\pi$
$73$	−0.0324024	−	0.0213114i	−0.0324024	−	0.0213114i	−0.565607	+	0.824675i	$0.691358\pi$
$74$		0			0
$75$	−0.565607	−	0.824675i	−0.565607	−	0.824675i
$76$	−0.701910	−	0.501695i	−0.701910	−	0.501695i
$77$		0			0
$78$		0			0
$79$		0			0		−0.627812	−	0.778365i	$-0.716049\pi$
$79$		0			0		0.627812	+	0.778365i	$0.283951\pi$
$80$		0			0
$81$	0.533204	+	0.845986i	0.533204	+	0.845986i
$82$	0.963371	+	1.66861i	0.963371	+	1.66861i
$83$	0.206600	−	0.535108i	0.206600	−	0.535108i	−0.790393	−	0.612601i	$-0.790123\pi$
$83$	0.206600	−	0.535108i	0.206600	−	0.535108i	0.996993	+	0.0774924i	$0.0246914\pi$
$84$		0			0
$85$		0			0
$86$	−0.0697382	+	0.716972i	−0.0697382	+	0.716972i
$87$		0			0
$88$	−0.391959	+	0.160133i	−0.391959	+	0.160133i
$89$	−0.0694434	+	1.19230i	−0.0694434	+	1.19230i	0.766044	+	0.642788i	$0.222222\pi$
$89$	−0.0694434	+	1.19230i	−0.0694434	+	1.19230i	−0.835488	+	0.549509i	$0.814815\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$97$	1.81936	+	0.826999i	1.81936	+	0.826999i	0.925724	+	0.378200i	$0.123457\pi$
$97$	1.81936	+	0.826999i	1.81936	+	0.826999i	0.893633	+	0.448799i	$0.148148\pi$
$98$	−0.993238	−	0.116093i	−0.993238	−	0.116093i
$99$	0.353752	+	0.232666i	0.353752	+	0.232666i
$100$	0.396080	−	0.918216i	0.396080	−	0.918216i
$101$		0			0		−0.211704	−	0.977334i	$-0.567901\pi$
$101$		0			0		0.211704	+	0.977334i	$0.432099\pi$
$102$	0.616550	−	0.197689i	0.616550	−	0.197689i
$103$		0			0		−0.565607	−	0.824675i	$-0.691358\pi$
$103$		0			0		0.565607	+	0.824675i	$0.308642\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.34143	−	1.12560i	−1.34143	−	1.12560i	−0.981255	−	0.192712i	$-0.938272\pi$
$107$	−1.34143	−	1.12560i	−1.34143	−	1.12560i	−0.360178	−	0.932884i	$-0.617284\pi$
$108$	−0.431386	+	0.902167i	−0.431386	+	0.902167i
$109$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$109$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.445817	+	1.73077i	0.445817	+	1.73077i	0.657521	+	0.753436i	$0.271605\pi$
$113$	0.445817	+	1.73077i	0.445817	+	1.73077i	−0.211704	+	0.977334i	$0.567901\pi$
$114$	−0.738677	+	0.445794i	−0.738677	+	0.445794i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.466659	+	1.55875i	−0.466659	+	1.55875i
$119$		0			0
$120$		0			0
$121$	−0.810869	−	0.126818i	−0.810869	−	0.126818i
$122$		0			0
$123$	1.91371	−	0.223681i	1.91371	−	0.223681i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$127$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$128$	0.466044	−	0.884762i	0.466044	−	0.884762i
$129$	0.630713	+	0.348013i	0.630713	+	0.348013i
$130$		0			0
$131$	−0.0541960	−	0.102889i	−0.0541960	−	0.102889i	0.856167	−	0.516699i	$-0.172840\pi$
$131$	−0.0541960	−	0.102889i	−0.0541960	−	0.102889i	−0.910363	+	0.413811i	$0.864198\pi$
$132$	−0.00821044	+	0.423328i	−0.00821044	+	0.423328i
$133$		0			0
$134$	−0.207340	+	0.173979i	−0.207340	+	0.173979i
$135$		0			0
$136$	0.495989	+	0.416184i	0.495989	+	0.416184i
$137$	−0.914615	−	0.179625i	−0.914615	−	0.179625i	−0.286803	−	0.957990i	$-0.592593\pi$
$137$	−0.914615	−	0.179625i	−0.914615	−	0.179625i	−0.627812	+	0.778365i	$0.716049\pi$
$138$		0			0
$139$	−0.144318	−	1.05660i	−0.144318	−	1.05660i	−0.910363	−	0.413811i	$-0.864198\pi$
$139$	−0.144318	−	1.05660i	−0.144318	−	1.05660i	0.766044	−	0.642788i	$-0.222222\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.993238	+	0.116093i	−0.993238	+	0.116093i
$145$		0			0
$146$	−0.0353063	−	0.0160487i	−0.0353063	−	0.0160487i
$147$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$148$		0			0
$149$		0			0		0.981255	−	0.192712i	$-0.0617284\pi$
$149$		0			0		−0.981255	+	0.192712i	$0.938272\pi$
$150$	−0.686242	−	0.727374i	−0.686242	−	0.727374i
$151$		0			0		0.565607	−	0.824675i	$-0.308642\pi$
$151$		0			0		−0.565607	+	0.824675i	$0.691358\pi$
$152$	−0.771001	−	0.387211i	−0.771001	−	0.387211i
$153$	0.0626819	−	0.644427i	0.0626819	−	0.644427i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.0968109	−	0.995303i	$-0.469136\pi$
$157$		0			0		−0.0968109	+	0.995303i	$0.530864\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.657521	+	0.753436i	0.657521	+	0.753436i
$163$	0.835488	−	1.44711i	0.835488	−	1.44711i	−0.0581448	−	0.998308i	$-0.518519\pi$
$163$	0.835488	−	1.44711i	0.835488	−	1.44711i	0.893633	−	0.448799i	$-0.148148\pi$
$164$	1.20963	+	1.49971i	1.20963	+	1.49971i
$165$		0			0
$166$	0.121435	−	0.560605i	0.121435	−	0.560605i
$167$		0			0		−0.813552	−	0.581492i	$-0.802469\pi$
$167$		0			0		0.813552	+	0.581492i	$0.197531\pi$
$168$		0			0
$169$	−0.790393	−	0.612601i	−0.790393	−	0.612601i
$170$		0			0
$171$	0.116760	+	0.854835i	0.116760	+	0.854835i
$172$	0.0418849	+	0.719137i	0.0418849	+	0.719137i
$173$		0			0		−0.431386	−	0.902167i	$-0.641975\pi$
$173$		0			0		0.431386	+	0.902167i	$0.358025\pi$
$174$		0			0
$175$		0			0
$176$	−0.362508	+	0.218775i	−0.362508	+	0.218775i
$177$	1.24643	+	1.04588i	1.24643	+	1.04588i
$178$	0.115623	+	1.18871i	0.115623	+	1.18871i
$179$	0.473045	−	0.635410i	0.473045	−	0.635410i	−0.500000	−	0.866025i	$-0.666667\pi$
$179$	0.473045	−	0.635410i	0.473045	−	0.635410i	0.973045	+	0.230616i	$0.0740741\pi$
$180$		0			0
$181$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$181$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.0887494	−	0.259380i	−0.0887494	−	0.259380i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.0193913	−	0.999812i	$-0.493827\pi$
$191$		0			0		−0.0193913	+	0.999812i	$0.506173\pi$
$192$	−0.627812	−	0.778365i	−0.627812	−	0.778365i
$193$	0.266006	−	0.422048i	0.266006	−	0.422048i	−0.686242	−	0.727374i	$-0.740741\pi$
$193$	0.266006	−	0.422048i	0.266006	−	0.422048i	0.952248	+	0.305326i	$0.0987654\pi$
$194$	1.92529	+	0.535942i	1.92529	+	0.535942i
$195$		0			0
$196$	−0.999248	+	0.0387754i	−0.999248	+	0.0387754i
$197$		0			0		0.686242	−	0.727374i	$-0.259259\pi$
$197$		0			0		−0.686242	+	0.727374i	$0.740741\pi$
$198$	0.385455	+	0.175211i	0.385455	+	0.175211i
$199$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$199$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$200$	0.249441	−	0.968390i	0.249441	−	0.968390i
$201$	0.0776269	+	0.259292i	0.0776269	+	0.259292i
$202$		0			0
$203$		0			0
$204$	0.578598	−	0.290583i	0.578598	−	0.290583i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.194782	+	0.309043i	0.194782	+	0.309043i
$210$		0			0
$211$	−0.979857	+	0.961037i	−0.979857	+	0.961037i	−0.999248	−	0.0387754i	$-0.987654\pi$
$211$	−0.979857	+	0.961037i	−0.979857	+	0.961037i	0.0193913	+	0.999812i	$0.493827\pi$
$212$		0			0
$213$		0			0
$214$	−1.49925	−	0.904801i	−1.49925	−	0.904801i
$215$		0			0
$216$	−0.286803	+	0.957990i	−0.286803	+	0.957990i
$217$		0			0
$218$		0			0
$219$	−0.0287203	+	0.0260623i	−0.0287203	+	0.0260623i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.740544	−	0.672008i	$-0.234568\pi$
$223$		0			0		−0.740544	+	0.672008i	$0.765432\pi$
$224$		0			0
$225$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$226$	0.707900	+	1.64110i	0.707900	+	1.64110i
$227$	−1.02152	+	0.730136i	−1.02152	+	0.730136i	−0.963371	−	0.268173i	$-0.913580\pi$
$227$	−1.02152	+	0.730136i	−1.02152	+	0.730136i	−0.0581448	+	0.998308i	$0.518519\pi$
$228$	−0.660922	+	0.554579i	−0.660922	+	0.554579i
$229$		0			0		−0.0193913	−	0.999812i	$-0.506173\pi$
$229$		0			0		0.0193913	+	0.999812i	$0.493827\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.161769	+	0.106397i	−0.161769	+	0.106397i	−0.627812	−	0.778365i	$-0.716049\pi$
$233$	−0.161769	+	0.106397i	−0.161769	+	0.106397i	0.466044	+	0.884762i	$0.345679\pi$
$234$		0			0
$235$		0			0
$236$	−0.220198	+	1.61214i	−0.220198	+	1.61214i
$237$		0			0
$238$		0			0
$239$		0			0		0.952248	−	0.305326i	$-0.0987654\pi$
$239$		0			0		−0.952248	+	0.305326i	$0.901235\pi$
$240$		0			0
$241$	1.82921	−	0.286084i	1.82921	−	0.286084i	0.856167	−	0.516699i	$-0.172840\pi$
$241$	1.82921	−	0.286084i	1.82921	−	0.286084i	0.973045	+	0.230616i	$0.0740741\pi$
$242$	−0.820726			−0.820726
$243$	0.952248	−	0.305326i	0.952248	−	0.305326i
$244$		0			0
$245$		0			0
$246$	1.85617	−	0.516699i	1.85617	−	0.516699i
$247$		0			0
$248$		0			0
$249$	−0.466659	−	0.333548i	−0.466659	−	0.333548i
$250$		0			0
$251$	1.76580	−	0.886818i	1.76580	−	0.886818i	0.813552	−	0.581492i	$-0.197531\pi$
$251$	1.76580	−	0.886818i	1.76580	−	0.886818i	0.952248	−	0.305326i	$-0.0987654\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.323734	−	0.946148i	0.323734	−	0.946148i
$257$	0.0359020	+	1.85110i	0.0359020	+	1.85110i	0.396080	+	0.918216i	$0.370370\pi$
$257$	0.0359020	+	1.85110i	0.0359020	+	1.85110i	−0.360178	+	0.932884i	$0.617284\pi$
$258$	0.676913	+	0.246376i	0.676913	+	0.246376i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.0694434	−	0.0932786i	−0.0694434	−	0.0932786i
$263$		0			0		0.740544	−	0.672008i	$-0.234568\pi$
$263$		0			0		−0.740544	+	0.672008i	$0.765432\pi$
$264$	0.0573003	+	0.419513i	0.0573003	+	0.419513i
$265$		0			0
$266$		0			0
$267$	1.13729	+	0.364656i	1.13729	+	0.364656i
$268$	−0.177966	+	0.203927i	−0.177966	+	0.203927i
$269$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$269$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$270$		0			0
$271$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$271$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$272$	0.554341	+	0.334546i	0.554341	+	0.334546i
$273$		0			0
$274$	−0.931386	−	0.0361420i	−0.931386	−	0.0361420i
$275$	−0.302283	+	0.296477i	−0.302283	+	0.296477i
$276$		0			0
$277$		0			0		−0.533204	−	0.845986i	$-0.679012\pi$
$277$		0			0		0.533204	+	0.845986i	$0.320988\pi$
$278$	−0.305850	−	1.02161i	−0.305850	−	1.02161i
$279$		0			0
$280$		0			0
$281$	−0.979857	−	0.961037i	−0.979857	−	0.961037i	0.0193913	−	0.999812i	$-0.493827\pi$
$281$	−0.979857	−	0.961037i	−0.979857	−	0.961037i	−0.999248	+	0.0387754i	$0.987654\pi$
$282$		0			0
$283$	−1.22178	+	1.51477i	−1.22178	+	1.51477i	−0.431386	+	0.902167i	$0.641975\pi$
$283$	−1.22178	+	1.51477i	−1.22178	+	1.51477i	−0.790393	+	0.612601i	$0.790123\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.963371	+	0.268173i	−0.963371	+	0.268173i
$289$	0.398559	−	0.422448i	0.398559	−	0.422448i
$290$		0			0
$291$	1.25468	−	1.55556i	1.25468	−	1.55556i
$292$	−0.0373621	−	0.0104004i	−0.0373621	−	0.0104004i
$293$		0			0		0.533204	−	0.845986i	$-0.320988\pi$
$293$		0			0		−0.533204	+	0.845986i	$0.679012\pi$
$294$	−0.360178	+	0.932884i	−0.360178	+	0.932884i
$295$		0			0
$296$		0			0
$297$	0.313552	−	0.284533i	0.313552	−	0.284533i
$298$		0			0
$299$		0			0
$300$	−0.790393	−	0.612601i	−0.790393	−	0.612601i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.821573	−	0.263427i	−0.821573	−	0.263427i
$305$		0			0
$306$	−0.0376469	−	0.646372i	−0.0376469	−	0.646372i
$307$	0.115623	−	0.155308i	0.115623	−	0.155308i	−0.740544	−	0.672008i	$-0.765432\pi$
$307$	0.115623	−	0.155308i	0.115623	−	0.155308i	0.856167	+	0.516699i	$0.172840\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.657521	−	0.753436i	$-0.728395\pi$
$311$		0			0		0.657521	+	0.753436i	$0.271605\pi$
$312$		0			0
$313$	−0.615959	−	1.28817i	−0.615959	−	1.28817i	−0.939693	−	0.342020i	$-0.888889\pi$
$313$	−0.615959	−	1.28817i	−0.615959	−	1.28817i	0.323734	−	0.946148i	$-0.395062\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.790393	−	0.612601i	$-0.790123\pi$
$317$		0			0		0.790393	+	0.612601i	$0.209877\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−1.42462	+	1.01826i	−1.42462	+	1.01826i
$322$		0			0
$323$	0.279309	−	0.483777i	0.279309	−	0.483777i
$324$	0.766044	+	0.642788i	0.766044	+	0.642788i
$325$		0			0
$326$	0.601848	−	1.55883i	0.601848	−	1.55883i
$327$		0			0
$328$	1.42684	+	1.29479i	1.42684	+	1.29479i
$329$		0			0
$330$		0			0
$331$	0.733321	−	0.299595i	0.733321	−	0.299595i	0.0193913	−	0.999812i	$-0.493827\pi$
$331$	0.733321	−	0.299595i	0.733321	−	0.299595i	0.713930	+	0.700217i	$0.246914\pi$
$332$	0.0333522	−	0.572636i	0.0333522	−	0.572636i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.93894	+	0.380796i	−1.93894	+	0.380796i	−0.939693	+	0.342020i	$0.888889\pi$
$337$	−1.93894	+	0.380796i	−1.93894	+	0.380796i	−0.999248	+	0.0387754i	$0.987654\pi$
$338$	−0.875558	−	0.483113i	−0.875558	−	0.483113i
$339$	1.78727			1.78727
$340$		0			0
$341$		0			0
$342$	0.247446	+	0.826527i	0.247446	+	0.826527i
$343$		0			0
$344$	0.152502	+	0.704028i	0.152502	+	0.704028i
$345$		0			0
$346$		0			0
$347$	−0.269849	−	1.97564i	−0.269849	−	1.97564i	−0.211704	−	0.977334i	$-0.567901\pi$
$347$	−0.269849	−	1.97564i	−0.269849	−	1.97564i	−0.0581448	−	0.998308i	$-0.518519\pi$
$348$		0			0
$349$		0			0		−0.981255	−	0.192712i	$-0.938272\pi$
$349$		0			0		0.981255	+	0.192712i	$0.0617284\pi$
$350$		0			0
$351$		0			0
$352$	−0.324349	+	0.272161i	−0.324349	+	0.272161i
$353$	−1.49925	+	0.827250i	−1.49925	+	0.827250i	−0.999248	−	0.0387754i	$-0.987654\pi$
$353$	−1.49925	+	0.827250i	−1.49925	+	0.827250i	−0.500000	+	0.866025i	$0.666667\pi$
$354$	1.39307	+	0.840724i	1.39307	+	0.840724i
$355$		0			0
$356$	0.297912	+	1.15656i	0.297912	+	1.15656i
$357$		0			0
$358$	0.369181	−	0.700872i	0.369181	−	0.700872i
$359$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$359$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$360$		0			0
$361$	0.0733139	−	0.244885i	0.0733139	−	0.244885i
$362$		0			0
$363$	−0.325073	+	0.753604i	−0.325073	+	0.753604i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.963371	−	0.268173i	$-0.0864198\pi$
$367$		0			0		−0.963371	+	0.268173i	$0.913580\pi$
$368$		0			0
$369$	0.260748	−	1.90902i	0.260748	−	1.90902i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.249441	−	0.968390i	$-0.580247\pi$
$373$		0			0		0.249441	+	0.968390i	$0.419753\pi$
$374$	−0.127763	−	0.242551i	−0.127763	−	0.242551i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.09380	+	0.917810i	1.09380	+	0.917810i	0.996993	−	0.0774924i	$-0.0246914\pi$
$379$	1.09380	+	0.917810i	1.09380	+	0.917810i	0.0968109	+	0.995303i	$0.469136\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.565607	−	0.824675i	$-0.691358\pi$
$383$		0			0		0.565607	+	0.824675i	$0.308642\pi$
$384$	−0.740544	−	0.672008i	−0.740544	−	0.672008i
$385$		0			0
$386$	0.197597	−	0.458082i	0.197597	−	0.458082i
$387$	0.494338	−	0.523968i	0.494338	−	0.523968i
$388$	1.98498	+	0.232011i	1.98498	+	0.232011i
$389$		0			0		−0.910363	−	0.413811i	$-0.864198\pi$
$389$		0			0		0.910363	+	0.413811i	$0.135802\pi$
$390$		0			0
$391$		0			0
$392$	−0.981255	+	0.192712i	−0.981255	+	0.192712i
$393$	−0.113155	+	0.0268182i	−0.113155	+	0.0268182i
$394$		0			0
$395$		0			0
$396$	0.407899	+	0.113546i	0.407899	+	0.113546i
$397$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$397$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$398$		0			0
$399$		0			0
$400$	0.0968109	−	0.995303i	0.0968109	−	0.995303i
$401$	0.837714	+	0.760185i	0.837714	+	0.760185i	0.973045	−	0.230616i	$-0.0740741\pi$
$401$	0.837714	+	0.760185i	0.837714	+	0.760185i	−0.135331	+	0.990800i	$0.543210\pi$
$402$	0.116760	+	0.244183i	0.116760	+	0.244183i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	0.526749	−	0.376497i	0.526749	−	0.376497i
$409$	0.415471	−	1.91803i	0.415471	−	1.91803i	0.0193913	−	0.999812i	$-0.493827\pi$
$409$	0.415471	−	1.91803i	0.415471	−	1.91803i	0.396080	−	0.918216i	$-0.370370\pi$
$410$		0			0
$411$	−0.402089	+	0.840899i	−0.402089	+	0.840899i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.05920	−	0.123803i	−1.05920	−	0.123803i
$418$	0.240195	+	0.275233i	0.240195	+	0.275233i
$419$	−1.43063	+	0.863392i	−1.43063	+	0.863392i	−0.999248	−	0.0387754i	$-0.987654\pi$
$419$	−1.43063	+	0.863392i	−1.43063	+	0.863392i	−0.431386	+	0.902167i	$0.641975\pi$
$420$		0			0
$421$		0			0		−0.0968109	−	0.995303i	$-0.530864\pi$
$421$		0			0		0.0968109	+	0.995303i	$0.469136\pi$
$422$	−0.819590	+	1.10090i	−0.819590	+	1.10090i
$423$		0			0
$424$		0			0
$425$	0.616550	+	0.197689i	0.616550	+	0.197689i
$426$		0			0
$427$		0			0
$428$	−1.62105	−	0.662272i	−1.62105	−	0.662272i
$429$		0			0
$430$		0			0
$431$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$431$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$432$	−0.135331	+	0.990800i	−0.135331	+	0.990800i
$433$	−1.23574	−	0.449771i	−1.23574	−	0.449771i	−0.360178	−	0.932884i	$-0.617284\pi$
$433$	−1.23574	−	0.449771i	−1.23574	−	0.449771i	−0.875558	+	0.483113i	$0.839506\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.0243482	+	0.0301871i	−0.0243482	+	0.0301871i
$439$		0			0		0.999248	−	0.0387754i	$-0.0123457\pi$
$439$		0			0		−0.999248	+	0.0387754i	$0.987654\pi$
$440$		0			0
$441$	0.713930	+	0.700217i	0.713930	+	0.700217i
$442$		0			0
$443$	0.124442	−	0.483113i	0.124442	−	0.483113i	−0.875558	−	0.483113i	$-0.839506\pi$
$443$	0.124442	−	0.483113i	0.124442	−	0.483113i	1.00000			$0$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.731814	−	0.775678i	−0.731814	−	0.775678i	0.249441	−	0.968390i	$-0.419753\pi$
$449$	−0.731814	−	0.775678i	−0.731814	−	0.775678i	−0.981255	+	0.192712i	$0.938272\pi$
$450$	−0.875558	+	0.483113i	−0.875558	+	0.483113i
$451$	−0.233973	−	0.781525i	−0.233973	−	0.781525i
$452$	0.952978	+	1.51200i	0.952978	+	1.51200i
$453$		0			0
$454$	−0.896427	+	0.879210i	−0.896427	+	0.879210i
$455$		0			0
$456$	−0.567291	+	0.650043i	−0.567291	+	0.650043i
$457$	0.165773	+	0.100044i	0.165773	+	0.100044i	0.597159	−	0.802123i	$-0.296296\pi$
$457$	0.165773	+	0.100044i	0.165773	+	0.100044i	−0.431386	+	0.902167i	$0.641975\pi$
$458$		0			0
$459$	−0.608421	−	0.221447i	−0.608421	−	0.221447i
$460$		0			0
$461$		0			0		0.657521	−	0.753436i	$-0.271605\pi$
$461$		0			0		−0.657521	+	0.753436i	$0.728395\pi$
$462$		0			0
$463$		0			0		0.790393	−	0.612601i	$-0.209877\pi$
$463$		0			0		−0.790393	+	0.612601i	$0.790123\pi$
$464$		0			0
$465$		0			0
$466$	−0.143385	+	0.130115i	−0.143385	+	0.130115i
$467$	−0.675514	−	0.907373i	−0.675514	−	0.907373i	0.323734	−	0.946148i	$-0.395062\pi$
$467$	−0.675514	−	0.907373i	−0.675514	−	0.907373i	−0.999248	+	0.0387754i	$0.987654\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.0315517	+	1.62680i	0.0315517	+	1.62680i
$473$	0.0987402	−	0.288579i	0.0987402	−	0.288579i
$474$		0			0
$475$	−0.860178	−	0.0668583i	−0.860178	−	0.0668583i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.135331	−	0.990800i	$-0.456790\pi$
$479$		0			0		−0.135331	+	0.990800i	$0.543210\pi$
$480$		0			0
$481$		0			0
$482$	1.76304	−	0.565295i	1.76304	−	0.565295i
$483$		0			0
$484$	−0.810869	+	0.126818i	−0.810869	+	0.126818i
$485$		0			0
$486$	0.893633	−	0.448799i	0.893633	−	0.448799i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−1.19296	−	1.17005i	−1.19296	−	1.17005i
$490$		0			0
$491$	−1.55885	+	0.708582i	−1.55885	+	0.708582i	−0.993238	−	0.116093i	$-0.962963\pi$
$491$	−1.55885	+	0.708582i	−1.55885	+	0.708582i	−0.565607	+	0.824675i	$0.691358\pi$
$492$	1.75403	−	0.797306i	1.75403	−	0.797306i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.512593	−	0.257434i	−0.512593	−	0.257434i
$499$	0.425724	−	1.24423i	0.425724	−	1.24423i	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	0.425724	−	1.24423i	0.425724	−	1.24423i	0.925724	−	0.378200i	$-0.123457\pi$
$500$		0			0
$501$		0			0
$502$	1.60756	−	1.14902i	1.60756	−	1.14902i
$503$		0			0		−0.396080	−	0.918216i	$-0.629630\pi$
$503$		0			0		0.396080	+	0.918216i	$0.370370\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.790393	+	0.612601i	−0.790393	+	0.612601i
$508$		0			0
$509$		0			0		0.790393	−	0.612601i	$-0.209877\pi$
$509$		0			0		−0.790393	+	0.612601i	$0.790123\pi$
$510$		0			0
$511$		0			0
$512$	0.173648	−	0.984808i	0.173648	−	0.984808i
$513$	0.856938	+	0.100162i	0.856938	+	0.100162i
$514$	0.321501	+	1.82332i	0.321501	+	1.82332i
$515$		0			0
$516$	0.706853	+	0.138821i	0.706853	+	0.138821i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.360117	+	1.20287i	0.360117	+	1.20287i	0.925724	+	0.378200i	$0.123457\pi$
$521$	0.360117	+	1.20287i	0.360117	+	1.20287i	−0.565607	+	0.824675i	$0.691358\pi$
$522$		0			0
$523$	−1.22650	−	1.30001i	−1.22650	−	1.30001i	−0.939693	−	0.342020i	$-0.888889\pi$
$523$	−1.22650	−	1.30001i	−1.22650	−	1.30001i	−0.286803	−	0.957990i	$-0.592593\pi$
$524$	−0.0830226	−	0.0814280i	−0.0830226	−	0.0814280i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.121435	+	0.405620i	0.121435	+	0.405620i
$529$	0.249441	−	0.968390i	0.249441	−	0.968390i
$530$		0			0
$531$	1.32373	−	0.946148i	1.32373	−	0.946148i
$532$		0			0
$533$		0			0
$534$	1.17997	+	0.184544i	1.17997	+	0.184544i
$535$		0			0
$536$	−0.144318	+	0.228977i	−0.144318	+	0.228977i
$537$	−0.497327	−	0.616589i	−0.497327	−	0.616589i
$538$		0			0
$539$	0.397873	+	0.144814i	0.397873	+	0.144814i
$540$		0			0
$541$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$541$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$542$		0			0
$543$		0			0
$544$	0.599377	+	0.244872i	0.599377	+	0.244872i
$545$		0			0
$546$		0			0
$547$	−1.86880	−	0.599206i	−1.86880	−	0.599206i	−0.993238	−	0.116093i	$-0.962963\pi$
$547$	−1.86880	−	0.599206i	−1.86880	−	0.599206i	−0.875558	−	0.483113i	$-0.839506\pi$
$548$	−0.925785	+	0.108209i	−0.925785	+	0.108209i
$549$		0			0
$550$	−0.252842	+	0.339625i	−0.252842	+	0.339625i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.460034	−	0.962079i	−0.460034	−	0.962079i
$557$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$557$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.273319	+	0.0212440i	−0.273319	+	0.0212440i
$562$	−1.11659	−	0.798088i	−1.11659	−	0.798088i
$563$	−0.225763	+	1.04224i	−0.225763	+	1.04224i	0.713930	+	0.700217i	$0.246914\pi$
$563$	−0.225763	+	1.04224i	−0.225763	+	1.04224i	−0.939693	+	0.342020i	$0.888889\pi$
$564$		0			0
$565$		0			0
$566$	−0.973045	+	1.68536i	−0.973045	+	1.68536i
$567$		0			0
$568$		0			0
$569$	0.655785	−	1.69853i	0.655785	−	1.69853i	−0.0581448	−	0.998308i	$-0.518519\pi$
$569$	0.655785	−	1.69853i	0.655785	−	1.69853i	0.713930	−	0.700217i	$-0.246914\pi$
$570$		0			0
$571$	1.29678	+	1.17676i	1.29678	+	1.17676i	0.973045	+	0.230616i	$0.0740741\pi$
$571$	1.29678	+	1.17676i	1.29678	+	1.17676i	0.323734	+	0.946148i	$0.395062\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.910363	+	0.413811i	−0.910363	+	0.413811i
$577$	0.832943	+	0.418320i	0.832943	+	0.418320i	0.813552	−	0.581492i	$-0.197531\pi$
$577$	0.832943	+	0.418320i	0.832943	+	0.418320i	0.0193913	+	0.999812i	$0.493827\pi$
$578$	0.328496	−	0.478959i	0.328496	−	0.478959i
$579$	−0.342354	−	0.362874i	−0.342354	−	0.362874i
$580$		0			0
$581$		0			0
$582$	0.999248	−	1.73075i	0.999248	−	1.73075i
$583$		0			0
$584$	−0.0385204	−	0.00450239i	−0.0385204	−	0.00450239i
$585$		0			0
$586$		0			0
$587$	−0.391959	−	1.80948i	−0.391959	−	1.80948i	−0.565607	−	0.824675i	$-0.691358\pi$
$587$	−0.391959	−	1.80948i	−0.391959	−	1.80948i	0.173648	−	0.984808i	$-0.444444\pi$
$588$	−0.211704	+	0.977334i	−0.211704	+	0.977334i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.52173	−	1.27688i	−1.52173	−	1.27688i	−0.835488	−	0.549509i	$-0.814815\pi$
$593$	−1.52173	−	1.27688i	−1.52173	−	1.27688i	−0.686242	−	0.727374i	$-0.740741\pi$
$594$	0.265821	−	0.329566i	0.265821	−	0.329566i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.249441	−	0.968390i	$-0.580247\pi$
$599$		0			0		0.249441	+	0.968390i	$0.419753\pi$
$600$	−0.875558	−	0.483113i	−0.875558	−	0.483113i
$601$	0.887578	−	1.68503i	0.887578	−	1.68503i	0.173648	−	0.984808i	$-0.444444\pi$
$601$	0.887578	−	1.68503i	0.887578	−	1.68503i	0.713930	−	0.700217i	$-0.246914\pi$
$602$		0			0
$603$	0.270459	−	0.0104950i	0.270459	−	0.0104950i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.987990	−	0.154519i	$-0.950617\pi$
$607$		0			0		0.987990	+	0.154519i	$0.0493827\pi$
$608$	−0.852410	−	0.133315i	−0.852410	−	0.133315i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.137071	−	0.632792i	−0.137071	−	0.632792i
$613$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$613$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$614$	0.0902362	−	0.171309i	0.0902362	−	0.171309i
$615$		0			0
$616$		0			0
$617$	−0.931386	−	1.76819i	−0.931386	−	1.76819i	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	−0.931386	−	1.76819i	−0.931386	−	1.76819i	−0.431386	−	0.902167i	$-0.641975\pi$
$618$		0			0
$619$	−0.436800	+	0.241016i	−0.436800	+	0.241016i	−0.686242	−	0.727374i	$-0.740741\pi$
$619$	−0.436800	+	0.241016i	−0.436800	+	0.241016i	0.249441	+	0.968390i	$0.419753\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.135331	−	0.990800i	−0.135331	−	0.990800i
$626$	−0.807607	−	1.17752i	−0.807607	−	1.17752i
$627$	0.347860	−	0.111537i	0.347860	−	0.111537i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$631$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$632$		0			0
$633$	0.686242	+	1.18861i	0.686242	+	1.18861i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.84588	−	0.754125i	1.84588	−	0.754125i	0.893633	−	0.448799i	$-0.148148\pi$
$641$	1.84588	−	0.754125i	1.84588	−	0.754125i	0.952248	−	0.305326i	$-0.0987654\pi$
$642$	−1.25017	+	1.22616i	−1.25017	+	1.22616i
$643$	−0.143385	+	1.47413i	−0.143385	+	1.47413i	0.597159	+	0.802123i	$0.296296\pi$
$643$	−0.143385	+	1.47413i	−0.143385	+	1.47413i	−0.740544	+	0.672008i	$0.765432\pi$
$644$		0			0
$645$		0			0
$646$	0.201201	−	0.521125i	0.201201	−	0.521125i
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	0.856167	+	0.516699i	0.856167	+	0.516699i
$649$	0.344464	−	0.596630i	0.344464	−	0.596630i
$650$		0			0
$651$		0			0
$652$	0.353752	−	1.63310i	0.353752	−	1.63310i
$653$		0			0		−0.813552	−	0.581492i	$-0.802469\pi$
$653$		0			0		0.813552	+	0.581492i	$0.197531\pi$
$654$		0			0
$655$		0			0
$656$	1.60977	+	1.05876i	1.60977	+	1.05876i
$657$	0.0180744	+	0.0343134i	0.0180744	+	0.0343134i
$658$		0			0
$659$	0.592070	+	1.23821i	0.592070	+	1.23821i	0.952248	+	0.305326i	$0.0987654\pi$
$659$	0.592070	+	1.23821i	0.592070	+	1.23821i	−0.360178	+	0.932884i	$0.617284\pi$
$660$		0			0
$661$		0			0		−0.657521	−	0.753436i	$-0.728395\pi$
$661$		0			0		0.657521	+	0.753436i	$0.271605\pi$
$662$	0.678221	−	0.409308i	0.678221	−	0.409308i
$663$		0			0
$664$	−0.0555313	−	0.570912i	−0.0555313	−	0.570912i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.630015	+	1.84129i	0.630015	+	1.84129i	0.533204	+	0.845986i	$0.320988\pi$
$673$	0.630015	+	1.84129i	0.630015	+	1.84129i	0.0968109	+	0.995303i	$0.469136\pi$
$674$	−1.85681	+	0.675825i	−1.85681	+	0.675825i
$675$	0.0968109	+	0.995303i	0.0968109	+	0.995303i
$676$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$677$		0			0		0.0193913	−	0.999812i	$-0.493827\pi$
$677$		0			0		−0.0193913	+	0.999812i	$0.506173\pi$
$678$	1.76580	−	0.276166i	1.76580	−	0.276166i
$679$		0			0
$680$		0			0
$681$	0.452248	+	1.17135i	0.452248	+	1.17135i
$682$		0			0
$683$	1.08480	−	1.14982i	1.08480	−	1.14982i	0.0968109	−	0.995303i	$-0.469136\pi$
$683$	1.08480	−	1.14982i	1.08480	−	1.14982i	0.987990	−	0.154519i	$-0.0493827\pi$
$684$	0.372188	+	0.778365i	0.372188	+	0.778365i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.259456	+	0.672008i	0.259456	+	0.672008i
$689$		0			0
$690$		0			0
$691$	0.852659	+	0.836282i	0.852659	+	0.836282i	0.987990	−	0.154519i	$-0.0493827\pi$
$691$	0.852659	+	0.836282i	0.852659	+	0.836282i	−0.135331	+	0.990800i	$0.543210\pi$
$692$		0			0
$693$		0			0
$694$	−0.571882	−	1.91022i	−0.571882	−	1.91022i
$695$		0			0
$696$		0			0
$697$	−0.890630	+	0.873524i	−0.890630	+	0.873524i
$698$		0			0
$699$	0.0626819	+	0.183195i	0.0626819	+	0.183195i
$700$		0			0
$701$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$701$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$702$		0			0
$703$		0			0
$704$	−0.278400	+	0.319011i	−0.278400	+	0.319011i
$705$		0			0
$706$	−1.35342	+	1.04898i	−1.35342	+	1.04898i
$707$		0			0
$708$	1.50625	+	0.615371i	1.50625	+	0.615371i
$709$		0			0		0.740544	−	0.672008i	$-0.234568\pi$
$709$		0			0		−0.740544	+	0.672008i	$0.765432\pi$
$710$		0			0
$711$		0			0
$712$	0.473045	+	1.09664i	0.473045	+	1.09664i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.256449	−	0.749500i	0.256449	−	0.749500i
$717$		0			0
$718$		0			0
$719$		0			0		0.835488	−	0.549509i	$-0.185185\pi$
$719$		0			0		−0.835488	+	0.549509i	$0.814815\pi$
$720$		0			0
$721$		0			0
$722$	0.0345940	−	0.253273i	0.0345940	−	0.253273i
$723$	0.179240	−	1.84275i	0.179240	−	1.84275i
$724$		0			0
$725$		0			0
$726$	−0.204723	+	0.794783i	−0.204723	+	0.794783i
$727$		0			0		0.987990	−	0.154519i	$-0.0493827\pi$
$727$		0			0		−0.987990	+	0.154519i	$0.950617\pi$
$728$		0			0
$729$	−0.0581448	−	0.998308i	−0.0581448	−	0.998308i
$730$		0			0
$731$	−0.460805	+	0.0720686i	−0.460805	+	0.0720686i
$732$		0			0
$733$		0			0		0.952248	−	0.305326i	$-0.0987654\pi$
$733$		0			0		−0.952248	+	0.305326i	$0.901235\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.102411	−	0.0514327i	0.102411	−	0.0514327i
$738$	−0.0373621	−	1.92638i	−0.0373621	−	1.92638i
$739$	−1.59118	+	1.04654i	−1.59118	+	1.04654i	−0.627812	+	0.778365i	$0.716049\pi$
$739$	−1.59118	+	1.04654i	−1.59118	+	1.04654i	−0.963371	+	0.268173i	$0.913580\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.0193913	−	0.999812i	$-0.506173\pi$
$743$		0			0		0.0193913	+	0.999812i	$0.493827\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.439408	+	0.368707i	−0.439408	+	0.368707i
$748$	−0.163707	−	0.219896i	−0.163707	−	0.219896i
$749$		0			0
$750$		0			0
$751$		0			0		0.996993	−	0.0774924i	$-0.0246914\pi$
$751$		0			0		−0.996993	+	0.0774924i	$0.975309\pi$
$752$		0			0
$753$	−0.418323	−	1.93119i	−0.418323	−	1.93119i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$757$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$758$	1.22249	+	0.737774i	1.22249	+	0.737774i
$759$		0			0
$760$		0			0
$761$	0.247945	−	0.243183i	0.247945	−	0.243183i	−0.565607	−	0.824675i	$-0.691358\pi$
$761$	0.247945	−	0.243183i	0.247945	−	0.243183i	0.813552	+	0.581492i	$0.197531\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.835488	−	0.549509i	−0.835488	−	0.549509i
$769$	−1.12207	+	1.39114i	−1.12207	+	1.39114i	−0.211704	+	0.977334i	$0.567901\pi$
$769$	−1.12207	+	1.39114i	−1.12207	+	1.39114i	−0.910363	+	0.413811i	$0.864198\pi$
$770$		0			0
$771$	1.80154	+	0.426973i	1.80154	+	0.426973i
$772$	0.124442	−	0.483113i	0.124442	−	0.483113i
$773$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$773$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$774$	0.407438	−	0.594059i	0.407438	−	0.594059i
$775$		0			0
$776$	1.99699	−	0.0774924i	1.99699	−	0.0774924i
$777$		0			0
$778$		0			0
$779$	0.886367	−	1.40632i	0.886367	−	1.40632i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$785$		0			0
$786$	−0.107652	+	0.0439807i	−0.107652	+	0.0439807i
$787$	−0.531001	−	0.216938i	−0.531001	−	0.216938i	0.0968109	−	0.995303i	$-0.469136\pi$
$787$	−0.531001	−	0.216938i	−0.531001	−	0.216938i	−0.627812	+	0.778365i	$0.716049\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.420545	+	0.0491546i	0.420545	+	0.0491546i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.657521	−	0.753436i	$-0.728395\pi$
$797$		0			0		0.657521	+	0.753436i	$0.271605\pi$
$798$		0			0
$799$		0			0
$800$	−0.0581448	−	0.998308i	−0.0581448	−	0.998308i
$801$	0.636815	−	1.01038i	0.636815	−	1.01038i
$802$	0.945115	+	0.621612i	0.945115	+	0.621612i
$803$	0.0129789	+	0.0100594i	0.0129789	+	0.0100594i
$804$	0.153089	+	0.223209i	0.153089	+	0.223209i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.0193913	+	0.0335868i	−0.0193913	+	0.0335868i	−0.875558	−	0.483113i	$-0.839506\pi$
$809$	−0.0193913	+	0.0335868i	−0.0193913	+	0.0335868i	0.856167	+	0.516699i	$0.172840\pi$
$810$		0			0
$811$	0.790393	+	1.36900i	0.790393	+	1.36900i	0.925724	+	0.378200i	$0.123457\pi$
$811$	0.790393	+	1.36900i	0.790393	+	1.36900i	−0.135331	+	0.990800i	$0.543210\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	0.462247	−	0.453368i	0.462247	−	0.453368i
$817$	0.575340	−	0.235052i	0.575340	−	0.235052i
$818$	0.114110	−	1.95919i	0.114110	−	1.95919i
$819$		0			0
$820$		0			0
$821$		0			0		0.565607	−	0.824675i	$-0.308642\pi$
$821$		0			0		−0.565607	+	0.824675i	$0.691358\pi$
$822$	−0.267326	+	0.892930i	−0.267326	+	0.892930i
$823$		0			0		0.981255	−	0.192712i	$-0.0617284\pi$
$823$		0			0		−0.981255	+	0.192712i	$0.938272\pi$
$824$		0			0
$825$	0.211704	+	0.366682i	0.211704	+	0.366682i
$826$		0			0
$827$	1.86668	+	0.218183i	1.86668	+	0.218183i	0.973045	−	0.230616i	$-0.0740741\pi$
$827$	1.86668	+	0.218183i	1.86668	+	0.218183i	0.893633	+	0.448799i	$0.148148\pi$
$828$		0			0
$829$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$829$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.0876227	−	0.641511i	−0.0876227	−	0.641511i
$834$	−1.06561	+	0.0413504i	−1.06561	+	0.0413504i
$835$		0			0
$836$	0.279839	+	0.234813i	0.279839	+	0.234813i
$837$		0			0
$838$	−1.28004	+	1.07408i	−1.28004	+	1.07408i
$839$		0			0		0.875558	−	0.483113i	$-0.160494\pi$
$839$		0			0		−0.875558	+	0.483113i	$0.839506\pi$
$840$		0			0
$841$	0.466044	+	0.884762i	0.466044	+	0.884762i
$842$		0			0
$843$	−1.17507	+	0.709161i	−1.17507	+	0.709161i
$844$	−0.639637	+	1.21432i	−0.639637	+	1.21432i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.16212	+	1.56100i	1.16212	+	1.56100i
$850$	0.639692	+	0.100046i	0.639692	+	0.100046i
$851$		0			0
$852$		0			0
$853$		0			0		0.963371	−	0.268173i	$-0.0864198\pi$
$853$		0			0		−0.963371	+	0.268173i	$0.913580\pi$
$854$		0			0
$855$		0			0
$856$	−1.70391	−	0.403835i	−1.70391	−	0.403835i
$857$	0.714020	−	1.35553i	0.714020	−	1.35553i	−0.211704	−	0.977334i	$-0.567901\pi$
$857$	0.714020	−	1.35553i	0.714020	−	1.35553i	0.925724	−	0.378200i	$-0.123457\pi$
$858$		0			0
$859$	0.475060	+	1.84429i	0.475060	+	1.84429i	0.533204	+	0.845986i	$0.320988\pi$
$859$	0.475060	+	1.84429i	0.475060	+	1.84429i	−0.0581448	+	0.998308i	$0.518519\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$863$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$864$	0.0193913	+	0.999812i	0.0193913	+	0.999812i
$865$		0			0
$866$	−1.29039	−	0.253425i	−1.29039	−	0.253425i
$867$	−0.309677	−	0.491337i	−0.309677	−	0.491337i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.19342	−	1.60304i	−1.19342	−	1.60304i
$874$		0			0
$875$		0			0
$876$	−0.0193913	+	0.0335868i	−0.0193913	+	0.0335868i
$877$		0			0		−0.875558	−	0.483113i	$-0.839506\pi$
$877$		0			0		0.875558	+	0.483113i	$0.160494\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.707900	+	0.355521i	0.707900	+	0.355521i	0.766044	−	0.642788i	$-0.222222\pi$
$881$	0.707900	+	0.355521i	0.707900	+	0.355521i	−0.0581448	+	0.998308i	$0.518519\pi$
$882$	0.813552	+	0.581492i	0.813552	+	0.581492i
$883$	0.114110	−	1.95919i	0.114110	−	1.95919i	−0.135331	−	0.990800i	$-0.543210\pi$
$883$	0.114110	−	1.95919i	0.114110	−	1.95919i	0.249441	−	0.968390i	$-0.419753\pi$
$884$		0			0
$885$		0			0
$886$	0.0482972	−	0.496539i	0.0482972	−	0.496539i
$887$		0			0		−0.740544	−	0.672008i	$-0.765432\pi$
$887$		0			0		0.740544	+	0.672008i	$0.234568\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.197326	−	0.374615i	−0.197326	−	0.374615i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.842882	−	0.653283i	−0.842882	−	0.653283i
$899$		0			0
$900$	−0.790393	+	0.612601i	−0.790393	+	0.612601i
$901$		0			0
$902$	−0.351924	−	0.735986i	−0.351924	−	0.735986i
$903$		0			0
$904$	1.17517	+	1.34659i	1.17517	+	1.34659i
$905$		0			0
$906$		0			0
$907$	0.179240	+	1.84275i	0.179240	+	1.84275i	0.466044	+	0.884762i	$0.345679\pi$
$907$	0.179240	+	1.84275i	0.179240	+	1.84275i	−0.286803	+	0.957990i	$0.592593\pi$
$908$	−0.749807	+	1.00717i	−0.749807	+	1.00717i
$909$		0			0
$910$		0			0
$911$		0			0		−0.952248	−	0.305326i	$-0.901235\pi$
$911$		0			0		0.952248	+	0.305326i	$0.0987654\pi$
$912$	−0.460034	+	0.729894i	−0.460034	+	0.729894i
$913$	−0.104770	+	0.219109i	−0.104770	+	0.219109i
$914$	0.179240	+	0.0732277i	0.179240	+	0.0732277i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−0.635331	−	0.124775i	−0.635331	−	0.124775i
$919$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$919$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$920$		0			0
$921$	−0.121558	−	0.150708i	−0.121558	−	0.150708i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.382166	−	1.48366i	0.382166	−	1.48366i	−0.431386	−	0.902167i	$-0.641975\pi$
$929$	0.382166	−	1.48366i	0.382166	−	1.48366i	0.813552	−	0.581492i	$-0.197531\pi$
$930$		0			0
$931$	0.310751	+	0.804866i	0.310751	+	0.804866i
$932$	−0.121558	+	0.150708i	−0.121558	+	0.150708i
$933$		0			0
$934$	−0.807607	−	0.792095i	−0.807607	−	0.792095i
$935$		0			0
$936$		0			0
$937$	0.569728	+	1.90302i	0.569728	+	1.90302i	0.396080	+	0.918216i	$0.370370\pi$
$937$	0.569728	+	1.90302i	0.569728	+	1.90302i	0.173648	+	0.984808i	$0.444444\pi$
$938$		0			0
$939$	−1.40109	+	0.275166i	−1.40109	+	0.275166i
$940$		0			0
$941$		0			0		−0.999248	−	0.0387754i	$-0.987654\pi$
$941$		0			0		0.999248	+	0.0387754i	$0.0123457\pi$
$942$		0			0
$943$		0			0
$944$	0.282544	+	1.60238i	0.282544	+	1.60238i
$945$		0			0
$946$	0.0529634	−	0.300370i	0.0529634	−	0.300370i
$947$	0.0255004	−	0.0292203i	0.0255004	−	0.0292203i	−0.740544	−	0.672008i	$-0.765432\pi$
$947$	0.0255004	−	0.0292203i	0.0255004	−	0.0292203i	0.766044	+	0.642788i	$0.222222\pi$
$948$		0			0
$949$		0			0
$950$	−0.860178	+	0.0668583i	−0.860178	+	0.0668583i
$951$		0			0
$952$		0			0
$953$	−0.161628	−	0.217105i	−0.161628	−	0.217105i	0.713930	−	0.700217i	$-0.246914\pi$
$953$	−0.161628	−	0.217105i	−0.161628	−	0.217105i	−0.875558	+	0.483113i	$0.839506\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.996993	+	0.0774924i	0.996993	+	0.0774924i
$962$		0			0
$963$	0.630713	+	1.63359i	0.630713	+	1.63359i
$964$	1.65451	−	0.830928i	1.65451	−	0.830928i
$965$		0			0
$966$		0			0
$967$		0			0		0.910363	−	0.413811i	$-0.135802\pi$
$967$		0			0		−0.910363	+	0.413811i	$0.864198\pi$
$968$	−0.781534	+	0.250589i	−0.781534	+	0.250589i
$969$	−0.398813	−	0.391153i	−0.398813	−	0.391153i
$970$		0			0
$971$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$971$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$972$	0.813552	−	0.581492i	0.813552	−	0.581492i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.65752	−	0.753436i	1.65752	−	0.753436i	0.657521	−	0.753436i	$-0.271605\pi$
$977$	1.65752	−	0.753436i	1.65752	−	0.753436i	1.00000			$0$
$978$	−1.35943	−	0.971659i	−1.35943	−	0.971659i
$979$	0.0684348	−	0.501031i	0.0684348	−	0.501031i
$980$		0			0
$981$		0			0
$982$	−1.43063	+	0.940943i	−1.43063	+	0.940943i
$983$		0			0		−0.996993	−	0.0774924i	$-0.975309\pi$
$983$		0			0		0.996993	+	0.0774924i	$0.0246914\pi$
$984$	1.60977	−	1.05876i	1.60977	−	1.05876i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.597159	−	0.802123i	$-0.703704\pi$
$991$		0			0		0.597159	+	0.802123i	$0.296296\pi$
$992$		0			0
$993$	−0.107204	−	0.784872i	−0.107204	−	0.784872i
$994$		0			0
$995$		0			0
$996$	−0.546216	−	0.175137i	−0.546216	−	0.175137i
$997$		0			0		0.657521	−	0.753436i	$-0.271605\pi$
$997$		0			0		−0.657521	+	0.753436i	$0.728395\pi$
$998$	0.228355	−	1.29506i	0.228355	−	1.29506i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1944.1.bh.a.187.1	✓	54
8.3	odd	2	CM	1944.1.bh.a.187.1	✓	54
243.13	even	81	inner	1944.1.bh.a.499.1	yes	54
1944.499	odd	162	inner	1944.1.bh.a.499.1	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1944.1.bh.a.187.1	✓	54	1.1	even	1	trivial
1944.1.bh.a.187.1	✓	54	8.3	odd	2	CM
1944.1.bh.a.499.1	yes	54	243.13	even	81	inner
1944.1.bh.a.499.1	yes	54	1944.499	odd	162	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:	\( N \)	\(=\)	\( 1944 = 2^{3} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1944.bh (of order \(162\), degree \(54\), minimal)

\(f(q)\)	\(=\)	\(q+(0.987990 - 0.154519i) q^{2} +(0.249441 - 0.968390i) q^{3} +(0.952248 - 0.305326i) q^{4} +(0.0968109 - 0.995303i) q^{6} +(0.893633 - 0.448799i) q^{8} +(-0.875558 - 0.483113i) q^{9} +O(q^{10})\)	\(q+(0.987990 - 0.154519i) q^{2} +(0.249441 - 0.968390i) q^{3} +(0.952248 - 0.305326i) q^{4} +(0.0968109 - 0.995303i) q^{6} +(0.893633 - 0.448799i) q^{8} +(-0.875558 - 0.483113i) q^{9} +(-0.422135 - 0.0328109i) q^{11} +(-0.0581448 - 0.998308i) q^{12} +(0.813552 - 0.581492i) q^{16} +(0.256449 + 0.594515i) q^{17} +(-0.939693 - 0.342020i) q^{18} +(-0.515212 - 0.692050i) q^{19} +(-0.422135 + 0.0328109i) q^{22} +(-0.211704 - 0.977334i) q^{24} +(0.657521 - 0.753436i) q^{25} +(-0.686242 + 0.727374i) q^{27} +(0.713930 - 0.700217i) q^{32} +(-0.137071 + 0.400606i) q^{33} +(0.345233 + 0.547749i) q^{34} +(-0.981255 - 0.192712i) q^{36} +(-0.615959 - 0.604128i) q^{38} +(0.693969 + 1.79743i) q^{41} +(-0.179686 + 0.697585i) q^{43} +(-0.411995 + 0.0976445i) q^{44} +(-0.360178 - 0.932884i) q^{48} +(-0.963371 - 0.268173i) q^{49} +(0.533204 - 0.845986i) q^{50} +(0.639692 - 0.100046i) q^{51} +(-0.565607 + 0.824675i) q^{54} +(-0.798689 + 0.326300i) q^{57} +(-0.701910 + 1.46792i) q^{59} +(0.597159 - 0.802123i) q^{64} +(-0.0735240 + 0.416975i) q^{66} +(-0.231732 + 0.139851i) q^{67} +(0.425724 + 0.487826i) q^{68} +(-0.999248 - 0.0387754i) q^{72} +(-0.0324024 - 0.0213114i) q^{73} +(-0.565607 - 0.824675i) q^{75} +(-0.701910 - 0.501695i) q^{76} +(0.533204 + 0.845986i) q^{81} +(0.963371 + 1.66861i) q^{82} +(0.206600 - 0.535108i) q^{83} +(-0.0697382 + 0.716972i) q^{86} +(-0.391959 + 0.160133i) q^{88} +(-0.0694434 + 1.19230i) q^{89} +(-0.500000 - 0.866025i) q^{96} +(1.81936 + 0.826999i) q^{97} +(-0.993238 - 0.116093i) q^{98} +(0.353752 + 0.232666i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q+O(q^{10}) \) 54 * q	\( 54 q - 27 q^{68} - 27 q^{96}+O(q^{100}) \) 54 * q - 27 * q^68 - 27 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more