LMFDB - Embedded newform 2061.1.bn.a.946.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2061,1,Mod(109,2061)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2061, base_ring=CyclotomicField(76))

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

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

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

chi := DirichletCharacter("2061.109");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2061 = 3^{2} \cdot 229 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2061.bn (of order $76$, degree $36$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.02857299104$
Analytic rank:	$0$
Dimension:	$36$
Coefficient field:	$\Q(\zeta_{76})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} + \cdots + 1 $ x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{76}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{76} - \cdots)$

Embedding invariants

Embedding label			946.1
Root			$0.837166 + 0.546948i$ of defining polynomial
Character	$\chi$	$=$	2061.946
Dual form			2061.1.bn.a.1000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.735724 - 0.677282i) q^{4} +(0.126633 + 1.01591i) q^{7} +O(q^{10})$	$q+(-0.735724 - 0.677282i) q^{4} +(0.126633 + 1.01591i) q^{7} +(-1.05198 + 0.751099i) q^{13} +(0.0825793 + 0.996584i) q^{16} +(-1.55676 + 0.259777i) q^{19} +(0.945817 + 0.324699i) q^{25} +(0.594889 - 0.833194i) q^{28} +(-0.546948 + 1.83717i) q^{31} +(-0.0903332 + 1.09016i) q^{37} +(0.151248 - 1.82529i) q^{43} +(-0.0466332 + 0.0118091i) q^{49} +(1.28267 + 0.159885i) q^{52} +(-0.778807 + 1.77550i) q^{61} +(0.614213 - 0.789141i) q^{64} +(-1.08265 + 0.422452i) q^{67} +(0.368727 - 0.180260i) q^{73} +(1.32128 + 0.863238i) q^{76} +(1.78298 + 0.222249i) q^{79} +(-0.896263 - 0.973601i) q^{91} +(0.647181 + 1.88517i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 2 q^{7}+O(q^{10}) $ 36 * q + 2 * q^7	$ 36 q + 2 q^{7} - 2 q^{13} + 2 q^{16} - 4 q^{19} - 2 q^{25} - 2 q^{28} + 2 q^{31} - 4 q^{37} - 2 q^{52} + 2 q^{67} + 2 q^{73} - 2 q^{79} + 4 q^{91}+O(q^{100}) $ 36 * q + 2 * q^7 - 2 * q^13 + 2 * q^16 - 4 * q^19 - 2 * q^25 - 2 * q^28 + 2 * q^31 - 4 * q^37 - 2 * q^52 + 2 * q^67 + 2 * q^73 - 2 * q^79 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2061\mathbb{Z}\right)^\times$.

$n$	$235$	$1604$
$\chi(n)$	$e\left(\frac{33}{76}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		0.363508	−	0.931591i	$-0.381579\pi$
$2$		0			0		−0.363508	+	0.931591i	$0.618421\pi$
$3$		0			0
$3$		0			0
$4$	−0.735724	−	0.677282i	−0.735724	−	0.677282i
$5$		0			0		−0.986361	−	0.164595i	$-0.947368\pi$
$5$		0			0		0.986361	+	0.164595i	$0.0526316\pi$
$6$		0			0
$7$	0.126633	+	1.01591i	0.126633	+	1.01591i	0.915773	+	0.401695i	$0.131579\pi$
$7$	0.126633	+	1.01591i	0.126633	+	1.01591i	−0.789141	+	0.614213i	$0.789474\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.879474	−	0.475947i	$-0.157895\pi$
$11$		0			0		−0.879474	+	0.475947i	$0.842105\pi$
$12$		0			0
$13$	−1.05198	+	0.751099i	−1.05198	+	0.751099i	−0.969400	−	0.245485i	$-0.921053\pi$
$13$	−1.05198	+	0.751099i	−1.05198	+	0.751099i	−0.0825793	+	0.996584i	$0.526316\pi$
$14$		0			0
$15$		0			0
$16$	0.0825793	+	0.996584i	0.0825793	+	0.996584i
$17$		0			0		0.164595	−	0.986361i	$-0.447368\pi$
$17$		0			0		−0.164595	+	0.986361i	$0.552632\pi$
$18$		0			0
$19$	−1.55676	+	0.259777i	−1.55676	+	0.259777i	−0.879474	−	0.475947i	$-0.842105\pi$
$19$	−1.55676	+	0.259777i	−1.55676	+	0.259777i	−0.677282	+	0.735724i	$0.736842\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.859054	−	0.511885i	$-0.171053\pi$
$23$		0			0		−0.859054	+	0.511885i	$0.828947\pi$
$24$		0			0
$25$	0.945817	+	0.324699i	0.945817	+	0.324699i
$26$		0			0
$27$		0			0
$28$	0.594889	−	0.833194i	0.594889	−	0.833194i
$29$		0			0		−0.123693	−	0.992321i	$-0.539474\pi$
$29$		0			0		0.123693	+	0.992321i	$0.460526\pi$
$30$		0			0
$31$	−0.546948	+	1.83717i	−0.546948	+	1.83717i			1.00000i	$0.5\pi$
$31$	−0.546948	+	1.83717i	−0.546948	+	1.83717i	−0.546948	+	0.837166i	$0.684211\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.0903332	+	1.09016i	−0.0903332	+	1.09016i	0.789141	+	0.614213i	$0.210526\pi$
$37$	−0.0903332	+	1.09016i	−0.0903332	+	1.09016i	−0.879474	+	0.475947i	$0.842105\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.931591	−	0.363508i	$-0.881579\pi$
$41$		0			0		0.931591	+	0.363508i	$0.118421\pi$
$42$		0			0
$43$	0.151248	−	1.82529i	0.151248	−	1.82529i	−0.324699	−	0.945817i	$-0.605263\pi$
$43$	0.151248	−	1.82529i	0.151248	−	1.82529i	0.475947	−	0.879474i	$-0.342105\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.363508	−	0.931591i	$-0.618421\pi$
$47$		0			0		0.363508	+	0.931591i	$0.381579\pi$
$48$		0			0
$49$	−0.0466332	+	0.0118091i	−0.0466332	+	0.0118091i
$50$		0			0
$51$		0			0
$52$	1.28267	+	0.159885i	1.28267	+	0.159885i
$53$		0			0		0.837166	−	0.546948i	$-0.184211\pi$
$53$		0			0		−0.837166	+	0.546948i	$0.815789\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.646299	−	0.763084i	$-0.276316\pi$
$59$		0			0		−0.646299	+	0.763084i	$0.723684\pi$
$60$		0			0
$61$	−0.778807	+	1.77550i	−0.778807	+	1.77550i	−0.164595	+	0.986361i	$0.552632\pi$
$61$	−0.778807	+	1.77550i	−0.778807	+	1.77550i	−0.614213	+	0.789141i	$0.710526\pi$
$62$		0			0
$63$		0			0
$64$	0.614213	−	0.789141i	0.614213	−	0.789141i
$65$		0			0
$66$		0			0
$67$	−1.08265	+	0.422452i	−1.08265	+	0.422452i	−0.837166	−	0.546948i	$-0.815789\pi$
$67$	−1.08265	+	0.422452i	−1.08265	+	0.422452i	−0.245485	+	0.969400i	$0.578947\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.879474	−	0.475947i	$-0.842105\pi$
$71$		0			0		0.879474	+	0.475947i	$0.157895\pi$
$72$		0			0
$73$	0.368727	−	0.180260i	0.368727	−	0.180260i	−0.245485	−	0.969400i	$-0.578947\pi$
$73$	0.368727	−	0.180260i	0.368727	−	0.180260i	0.614213	+	0.789141i	$0.289474\pi$
$74$		0			0
$75$		0			0
$76$	1.32128	+	0.863238i	1.32128	+	0.863238i
$77$		0			0
$78$		0			0
$79$	1.78298	+	0.222249i	1.78298	+	0.222249i	0.945817	−	0.324699i	$-0.105263\pi$
$79$	1.78298	+	0.222249i	1.78298	+	0.222249i	0.837166	+	0.546948i	$0.184211\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.996584	−	0.0825793i	$-0.0263158\pi$
$83$		0			0		−0.996584	+	0.0825793i	$0.973684\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$89$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$90$		0			0
$91$	−0.896263	−	0.973601i	−0.896263	−	0.973601i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.647181	+	1.88517i	0.647181	+	1.88517i	0.401695	+	0.915773i	$0.368421\pi$
$97$	0.647181	+	1.88517i	0.647181	+	1.88517i	0.245485	+	0.969400i	$0.421053\pi$
$98$		0			0
$99$		0			0
$100$	−0.475947	−	0.879474i	−0.475947	−	0.879474i
$101$		0			0		0.859054	−	0.511885i	$-0.171053\pi$
$101$		0			0		−0.859054	+	0.511885i	$0.828947\pi$
$102$		0			0
$103$	−0.831990	+	1.06894i	−0.831990	+	1.06894i	0.164595	+	0.986361i	$0.447368\pi$
$103$	−0.831990	+	1.06894i	−0.831990	+	1.06894i	−0.996584	+	0.0825793i	$0.973684\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$	−0.469870	+	0.554774i	−0.469870	+	0.554774i	−0.945817	−	0.324699i	$-0.894737\pi$
$109$	−0.469870	+	0.554774i	−0.469870	+	0.554774i	0.475947	+	0.879474i	$0.342105\pi$
$110$		0			0
$111$		0			0
$112$	−1.00198	+	0.210093i	−1.00198	+	0.210093i
$113$		0			0		−0.813849	−	0.581077i	$-0.802632\pi$
$113$		0			0		0.813849	+	0.581077i	$0.197368\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.546948	−	0.837166i	0.546948	−	0.837166i
$122$		0			0
$123$		0			0
$124$	1.64668	−	0.981209i	1.64668	−	0.981209i
$125$		0			0
$126$		0			0
$127$	−1.98295	+	0.0820152i	−1.98295	+	0.0820152i	−0.996584	−	0.0825793i	$-0.973684\pi$
$127$	−1.98295	+	0.0820152i	−1.98295	+	0.0820152i	−0.986361	+	0.164595i	$0.947368\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.992321	−	0.123693i	$-0.960526\pi$
$131$		0			0		0.992321	+	0.123693i	$0.0394737\pi$
$132$		0			0
$133$	−0.461045	−	1.54862i	−0.461045	−	1.54862i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.205215	−	0.978717i	$-0.434211\pi$
$137$		0			0		−0.205215	+	0.978717i	$0.565789\pi$
$138$		0			0
$139$	1.15096	−	0.821767i	1.15096	−	0.821767i	0.164595	−	0.986361i	$-0.447368\pi$
$139$	1.15096	−	0.821767i	1.15096	−	0.821767i	0.986361	+	0.164595i	$0.0526316\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.804806	−	0.740876i	0.804806	−	0.740876i
$149$		0			0		−0.996584	−	0.0825793i	$-0.973684\pi$
$149$		0			0		0.996584	+	0.0825793i	$0.0263158\pi$
$150$		0			0
$151$	1.28117	−	1.39172i	1.28117	−	1.39172i	0.401695	−	0.915773i	$-0.368421\pi$
$151$	1.28117	−	1.39172i	1.28117	−	1.39172i	0.879474	−	0.475947i	$-0.157895\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.0825793	−	1.99658i	−0.0825793	−	1.99658i	−0.0825793	−	0.996584i	$-0.526316\pi$
$157$	−0.0825793	−	1.99658i	−0.0825793	−	1.99658i		−	1.00000i	$-0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.0534166	+	0.0630689i	0.0534166	+	0.0630689i	0.789141	−	0.614213i	$-0.210526\pi$
$163$	0.0534166	+	0.0630689i	0.0534166	+	0.0630689i	−0.735724	+	0.677282i	$0.763158\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.614213	−	0.789141i	$-0.289474\pi$
$167$		0			0		−0.614213	+	0.789141i	$0.710526\pi$
$168$		0			0
$169$	0.217812	−	0.634464i	0.217812	−	0.634464i
$170$		0			0
$171$		0			0
$172$	−1.34751	+	1.24047i	−1.34751	+	1.24047i
$173$		0			0		0.837166	−	0.546948i	$-0.184211\pi$
$173$		0			0		−0.837166	+	0.546948i	$0.815789\pi$
$174$		0			0
$175$	−0.210093	+	1.00198i	−0.210093	+	1.00198i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.763084	−	0.646299i	$-0.776316\pi$
$179$		0			0		0.763084	+	0.646299i	$0.223684\pi$
$180$		0			0
$181$	0.101443	+	0.130333i	0.101443	+	0.130333i	0.837166	−	0.546948i	$-0.184211\pi$
$181$	0.101443	+	0.130333i	0.101443	+	0.130333i	−0.735724	+	0.677282i	$0.763158\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.978717	−	0.205215i	$-0.934211\pi$
$191$		0			0		0.978717	+	0.205215i	$0.0657895\pi$
$192$		0			0
$193$	−0.591074	−	1.34751i	−0.591074	−	1.34751i	−0.915773	−	0.401695i	$-0.868421\pi$
$193$	−0.591074	−	1.34751i	−0.591074	−	1.34751i	0.324699	−	0.945817i	$-0.394737\pi$
$194$		0			0
$195$		0			0
$196$	0.0423073	+	0.0228956i	0.0423073	+	0.0228956i
$197$		0			0		0.285336	−	0.958427i	$-0.407895\pi$
$197$		0			0		−0.285336	+	0.958427i	$0.592105\pi$
$198$		0			0
$199$	−0.0362996	−	0.877643i	−0.0362996	−	0.877643i	−0.915773	−	0.401695i	$-0.868421\pi$
$199$	−0.0362996	−	0.877643i	−0.0362996	−	0.877643i	0.879474	−	0.475947i	$-0.157895\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.835405	−	0.986361i	−0.835405	−	0.986361i
$209$		0			0
$210$		0			0
$211$	−1.15096	−	1.15096i	−1.15096	−	1.15096i	−0.986361	−	0.164595i	$-0.947368\pi$
$211$	−1.15096	−	1.15096i	−1.15096	−	1.15096i	−0.164595	−	0.986361i	$-0.552632\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.93565	−	0.323004i	−1.93565	−	0.323004i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.91577	+	0.401695i	−1.91577	+	0.401695i	−0.915773	+	0.401695i	$0.868421\pi$
$223$	−1.91577	+	0.401695i	−1.91577	+	0.401695i	−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.931591	−	0.363508i	$-0.881579\pi$
$227$		0			0		0.931591	+	0.363508i	$0.118421\pi$
$228$		0			0
$229$	0.969400	−	0.245485i	0.969400	−	0.245485i
$229$	0.969400	−	0.245485i	0.969400	−	0.245485i
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.677282	−	0.735724i	$-0.263158\pi$
$233$		0			0		−0.677282	+	0.735724i	$0.736842\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.859054	−	0.511885i	$-0.828947\pi$
$239$		0			0		0.859054	+	0.511885i	$0.171053\pi$
$240$		0			0
$241$	−0.0271842	+	0.162906i	−0.0271842	+	0.162906i	−0.996584	−	0.0825793i	$-0.973684\pi$
$241$	−0.0271842	+	0.162906i	−0.0271842	+	0.162906i	0.969400	+	0.245485i	$0.0789474\pi$
$242$		0			0
$243$		0			0
$244$	1.77550	−	0.778807i	1.77550	−	0.778807i
$245$		0			0
$246$		0			0
$247$	1.44256	−	1.44256i	1.44256	−	1.44256i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.992321	−	0.123693i	$-0.0394737\pi$
$251$		0			0		−0.992321	+	0.123693i	$0.960526\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.986361	+	0.164595i	−0.986361	+	0.164595i
$257$		0			0		−0.813849	−	0.581077i	$-0.802632\pi$
$257$		0			0		0.813849	+	0.581077i	$0.197368\pi$
$258$		0			0
$259$	−1.11894	+	0.0462798i	−1.11894	+	0.0462798i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.511885	−	0.859054i	$-0.328947\pi$
$263$		0			0		−0.511885	+	0.859054i	$0.671053\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.08265	+	0.422452i	1.08265	+	0.422452i
$269$		0			0		−0.285336	−	0.958427i	$-0.592105\pi$
$269$		0			0		0.285336	+	0.958427i	$0.407895\pi$
$270$		0			0
$271$	1.66364	−	0.571129i	1.66364	−	0.571129i	0.677282	−	0.735724i	$-0.263158\pi$
$271$	1.66364	−	0.571129i	1.66364	−	0.571129i	0.986361	+	0.164595i	$0.0526316\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.259777	−	0.202192i	0.259777	−	0.202192i	−0.475947	−	0.879474i	$-0.657895\pi$
$277$	0.259777	−	0.202192i	0.259777	−	0.202192i	0.735724	+	0.677282i	$0.236842\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.123693	−	0.992321i	$-0.460526\pi$
$281$		0			0		−0.123693	+	0.992321i	$0.539474\pi$
$282$		0			0
$283$	−1.38411	−	0.290218i	−1.38411	−	0.290218i	−0.546948	−	0.837166i	$-0.684211\pi$
$283$	−1.38411	−	0.290218i	−1.38411	−	0.290218i	−0.837166	+	0.546948i	$0.815789\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.945817	−	0.324699i	−0.945817	−	0.324699i
$290$		0			0
$291$		0			0
$292$	−0.393368	−	0.117111i	−0.393368	−	0.117111i
$293$		0			0		−0.789141	−	0.614213i	$-0.789474\pi$
$293$		0			0		0.789141	+	0.614213i	$0.210526\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.87348	−	0.0774877i	1.87348	−	0.0774877i
$302$		0			0
$303$		0			0
$304$	−0.387445	−	1.52999i	−0.387445	−	1.52999i
$305$		0			0
$306$		0			0
$307$	0.591074	+	0.544122i	0.591074	+	0.544122i	0.915773	−	0.401695i	$-0.131579\pi$
$307$	0.591074	+	0.544122i	0.591074	+	0.544122i	−0.324699	+	0.945817i	$0.605263\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.735724	−	0.677282i	$-0.763158\pi$
$311$		0			0		0.735724	+	0.677282i	$0.236842\pi$
$312$		0			0
$313$	1.98295	+	0.0820152i	1.98295	+	0.0820152i	0.996584	−	0.0825793i	$-0.0263158\pi$
$313$	1.98295	+	0.0820152i	1.98295	+	0.0820152i	0.986361	+	0.164595i	$0.0526316\pi$
$314$		0			0
$315$		0			0
$316$	−1.16126	−	1.37110i	−1.16126	−	1.37110i
$317$		0			0		−0.813849	−	0.581077i	$-0.802632\pi$
$317$		0			0		0.813849	+	0.581077i	$0.197368\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.23886	+	0.368825i	−1.23886	+	0.368825i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.0769960	+	1.86159i	0.0769960	+	1.86159i	0.401695	+	0.915773i	$0.368421\pi$
$331$	0.0769960	+	1.86159i	0.0769960	+	1.86159i	−0.324699	+	0.945817i	$0.605263\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.58361	+	1.03463i	1.58361	+	1.03463i	0.969400	+	0.245485i	$0.0789474\pi$
$337$	1.58361	+	1.03463i	1.58361	+	1.03463i	0.614213	+	0.789141i	$0.289474\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.354246	+	0.907855i	0.354246	+	0.907855i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.879474	−	0.475947i	$-0.157895\pi$
$347$		0			0		−0.879474	+	0.475947i	$0.842105\pi$
$348$		0			0
$349$	0.0630689	+	0.0534166i	0.0630689	+	0.0534166i	0.677282	−	0.735724i	$-0.263158\pi$
$349$	0.0630689	+	0.0534166i	0.0630689	+	0.0534166i	−0.614213	+	0.789141i	$0.710526\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.511885	−	0.859054i	$-0.671053\pi$
$353$		0			0		0.511885	+	0.859054i	$0.328947\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.996584	−	0.0825793i	$-0.0263158\pi$
$359$		0			0		−0.996584	+	0.0825793i	$0.973684\pi$
$360$		0			0
$361$	1.41019	−	0.484117i	1.41019	−	0.484117i
$362$		0			0
$363$		0			0
$364$			1.32332i			1.32332i
$365$		0			0
$366$		0			0
$367$	−1.23185	+	1.13399i	−1.23185	+	1.13399i	−0.245485	+	0.969400i	$0.578947\pi$
$367$	−1.23185	+	1.13399i	−1.23185	+	1.13399i	−0.986361	+	0.164595i	$0.947368\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.66364	+	0.571129i	1.66364	+	0.571129i	0.986361	−	0.164595i	$-0.0526316\pi$
$373$	1.66364	+	0.571129i	1.66364	+	0.571129i	0.677282	+	0.735724i	$0.263158\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0899265	+	0.721433i	−0.0899265	+	0.721433i	0.879474	+	0.475947i	$0.157895\pi$
$379$	−0.0899265	+	0.721433i	−0.0899265	+	0.721433i	−0.969400	+	0.245485i	$0.921053\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.245485	−	0.969400i	$-0.578947\pi$
$383$		0			0		0.245485	+	0.969400i	$0.421053\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.800647	−	1.82529i	0.800647	−	1.82529i
$389$		0			0		0.439197	−	0.898391i	$-0.355263\pi$
$389$		0			0		−0.439197	+	0.898391i	$0.644737\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.24047	+	0.544122i	1.24047	+	0.544122i	0.915773	−	0.401695i	$-0.131579\pi$
$397$	1.24047	+	0.544122i	1.24047	+	0.544122i	0.324699	+	0.945817i	$0.394737\pi$
$398$		0			0
$399$		0			0
$400$	−0.245485	+	0.969400i	−0.245485	+	0.969400i
$401$		0			0		−0.677282	−	0.735724i	$-0.736842\pi$
$401$		0			0		0.677282	+	0.735724i	$0.263158\pi$
$402$		0			0
$403$	−0.804516	−	2.34347i	−0.804516	−	2.34347i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.489294	−	1.93218i	−0.489294	−	1.93218i	−0.324699	−	0.945817i	$-0.605263\pi$
$409$	−0.489294	−	1.93218i	−0.489294	−	1.93218i	−0.164595	−	0.986361i	$-0.552632\pi$
$410$		0			0
$411$		0			0
$412$	1.33609	−	0.222954i	1.33609	−	0.222954i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.931591	−	0.363508i	$-0.881579\pi$
$419$		0			0		0.931591	+	0.363508i	$0.118421\pi$
$420$		0			0
$421$	1.34994	+	0.111859i	1.34994	+	0.111859i	0.735724	−	0.677282i	$-0.236842\pi$
$421$	1.34994	+	0.111859i	1.34994	+	0.111859i	0.614213	+	0.789141i	$0.289474\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.90237	−	0.566360i	−1.90237	−	0.566360i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.986361	−	0.164595i	$-0.0526316\pi$
$431$		0			0		−0.986361	+	0.164595i	$0.947368\pi$
$432$		0			0
$433$	−0.640542	+	1.86584i	−0.640542	+	1.86584i	−0.164595	+	0.986361i	$0.552632\pi$
$433$	−0.640542	+	1.86584i	−0.640542	+	1.86584i	−0.475947	+	0.879474i	$0.657895\pi$
$434$		0			0
$435$		0			0
$436$	0.721433	−	0.0899265i	0.721433	−	0.0899265i
$437$		0			0
$438$		0			0
$439$	0.0808112	+	0.484275i	0.0808112	+	0.484275i	0.996584	+	0.0825793i	$0.0263158\pi$
$439$	0.0808112	+	0.484275i	0.0808112	+	0.484275i	−0.915773	+	0.401695i	$0.868421\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.915773	−	0.401695i	$-0.131579\pi$
$443$		0			0		−0.915773	+	0.401695i	$0.868421\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.879474	+	0.524053i	0.879474	+	0.524053i
$449$		0			0		0.401695	−	0.915773i	$-0.368421\pi$
$449$		0			0		−0.401695	+	0.915773i	$0.631579\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$457$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.837166	−	0.546948i	$-0.184211\pi$
$461$		0			0		−0.837166	+	0.546948i	$0.815789\pi$
$462$		0			0
$463$	0.289513	−	1.73496i	0.289513	−	1.73496i	−0.324699	−	0.945817i	$-0.605263\pi$
$463$	0.289513	−	1.73496i	0.289513	−	1.73496i	0.614213	−	0.789141i	$-0.289474\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.915773	−	0.401695i	$-0.868421\pi$
$467$		0			0		0.915773	+	0.401695i	$0.131579\pi$
$468$		0			0
$469$	−0.566272	−	1.04638i	−0.566272	−	1.04638i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.55676	−	0.259777i	−1.55676	−	0.259777i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.646299	−	0.763084i	$-0.723684\pi$
$479$		0			0		0.646299	+	0.763084i	$0.276316\pi$
$480$		0			0
$481$	−0.723789	−	1.21468i	−0.723789	−	1.21468i
$482$		0			0
$483$		0			0
$484$	−0.969400	+	0.245485i	−0.969400	+	0.245485i
$485$		0			0
$486$		0			0
$487$	−0.245485	+	0.0305997i	−0.245485	+	0.0305997i	−0.245485	−	0.969400i	$-0.578947\pi$
$487$	−0.245485	+	0.0305997i	−0.245485	+	0.0305997i			1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.879474	−	0.475947i	$-0.842105\pi$
$491$		0			0		0.879474	+	0.475947i	$0.157895\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.87606	−	0.393368i	−1.87606	−	0.393368i
$497$		0			0
$498$		0			0
$499$	−0.653145	+	1.67387i	−0.653145	+	1.67387i	0.0825793	+	0.996584i	$0.473684\pi$
$499$	−0.653145	+	1.67387i	−0.653145	+	1.67387i	−0.735724	+	0.677282i	$0.763158\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.546948	−	0.837166i	$-0.315789\pi$
$503$		0			0		−0.546948	+	0.837166i	$0.684211\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1.51445	+	1.28267i	1.51445	+	1.28267i
$509$		0			0		0.915773	−	0.401695i	$-0.131579\pi$
$509$		0			0		−0.915773	+	0.401695i	$0.868421\pi$
$510$		0			0
$511$	0.229820	+	0.351766i	0.229820	+	0.351766i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.285336	−	0.958427i	$-0.407895\pi$
$521$		0			0		−0.285336	+	0.958427i	$0.592105\pi$
$522$		0			0
$523$	−0.714879	−	1.46231i	−0.714879	−	1.46231i	−0.879474	−	0.475947i	$-0.842105\pi$
$523$	−0.714879	−	1.46231i	−0.714879	−	1.46231i	0.164595	−	0.986361i	$-0.447368\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.475947	−	0.879474i	0.475947	−	0.879474i
$530$		0			0
$531$		0			0
$532$	−0.709652	+	1.45162i	−0.709652	+	1.45162i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.162906	+	1.96598i	0.162906	+	1.96598i	0.245485	+	0.969400i	$0.421053\pi$
$541$	0.162906	+	1.96598i	0.162906	+	1.96598i	−0.0825793	+	0.996584i	$0.526316\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.290218	+	0.290218i	−0.290218	+	0.290218i	−0.837166	−	0.546948i	$-0.815789\pi$
$547$	−0.290218	+	0.290218i	−0.290218	+	0.290218i	0.546948	+	0.837166i	$0.315789\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			1.83949i			1.83949i
$554$		0			0
$555$		0			0
$556$	−1.40335	−	0.174928i	−1.40335	−	0.174928i
$557$		0			0		−0.0825793	−	0.996584i	$-0.526316\pi$
$557$		0			0		0.0825793	+	0.996584i	$0.473684\pi$
$558$		0			0
$559$	1.21186	+	2.03377i	1.21186	+	2.03377i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.859054	−	0.511885i	$-0.171053\pi$
$563$		0			0		−0.859054	+	0.511885i	$0.828947\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.475947	−	0.879474i	$-0.657895\pi$
$569$		0			0		0.475947	+	0.879474i	$0.342105\pi$
$570$		0			0
$571$	0.238492	−	0.334028i	0.238492	−	0.334028i	−0.677282	−	0.735724i	$-0.736842\pi$
$571$	0.238492	−	0.334028i	0.238492	−	0.334028i	0.915773	+	0.401695i	$0.131579\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.0300439	+	0.726395i	−0.0300439	+	0.726395i	0.915773	+	0.401695i	$0.131579\pi$
$577$	−0.0300439	+	0.726395i	−0.0300439	+	0.726395i	−0.945817	+	0.324699i	$0.894737\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.475947	−	0.879474i	$-0.657895\pi$
$587$		0			0		0.475947	+	0.879474i	$0.342105\pi$
$588$		0			0
$589$	0.374212	−	3.00210i	0.374212	−	3.00210i
$590$		0			0
$591$		0			0
$592$	−1.09390			−1.09390
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.581077	−	0.813849i	$-0.302632\pi$
$599$		0			0		−0.581077	+	0.813849i	$0.697368\pi$
$600$		0			0
$601$	1.67387	+	0.818303i	1.67387	+	0.818303i	0.996584	+	0.0825793i	$0.0263158\pi$
$601$	1.67387	+	0.818303i	1.67387	+	0.818303i	0.677282	+	0.735724i	$0.263158\pi$
$602$		0			0
$603$		0			0
$604$	−1.88517	+	0.156210i	−1.88517	+	0.156210i
$605$		0			0
$606$		0			0
$607$	0.0271842	−	0.328065i	0.0271842	−	0.328065i	−0.969400	−	0.245485i	$-0.921053\pi$
$607$	0.0271842	−	0.328065i	0.0271842	−	0.328065i	0.996584	−	0.0825793i	$-0.0263158\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.789141	−	0.385787i	−0.789141	−	0.385787i		−	1.00000i	$-0.5\pi$
$613$	−0.789141	−	0.385787i	−0.789141	−	0.385787i	−0.789141	+	0.614213i	$0.789474\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.915773	−	0.401695i	$-0.868421\pi$
$617$		0			0		0.915773	+	0.401695i	$0.131579\pi$
$618$		0			0
$619$	−0.259777	+	0.202192i	−0.259777	+	0.202192i	−0.735724	−	0.677282i	$-0.763158\pi$
$619$	−0.259777	+	0.202192i	−0.259777	+	0.202192i	0.475947	+	0.879474i	$0.342105\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.789141	+	0.614213i	0.789141	+	0.614213i
$626$		0			0
$627$		0			0
$628$	−1.29149	+	1.52486i	−1.29149	+	1.52486i
$629$		0			0
$630$		0			0
$631$	−0.355188	−	0.543655i	−0.355188	−	0.543655i	0.614213	−	0.789141i	$-0.289474\pi$
$631$	−0.355188	−	0.543655i	−0.355188	−	0.543655i	−0.969400	+	0.245485i	$0.921053\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.0401874	−	0.0474491i	0.0401874	−	0.0474491i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.164595	−	0.986361i	$-0.552632\pi$
$641$		0			0		0.164595	+	0.986361i	$0.447368\pi$
$642$		0			0
$643$	0.629528	+	0.159418i	0.629528	+	0.159418i	0.546948	−	0.837166i	$-0.315789\pi$
$643$	0.629528	+	0.159418i	0.629528	+	0.159418i	0.0825793	+	0.996584i	$0.473684\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.958427	−	0.285336i	$-0.0921053\pi$
$647$		0			0		−0.958427	+	0.285336i	$0.907895\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.00341551	−	0.0825793i	0.00341551	−	0.0825793i
$653$		0			0		−0.859054	−	0.511885i	$-0.828947\pi$
$653$		0			0		0.859054	+	0.511885i	$0.171053\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.581077	−	0.813849i	$-0.302632\pi$
$659$		0			0		−0.581077	+	0.813849i	$0.697368\pi$
$660$		0			0
$661$	−0.411024	−	1.62310i	−0.411024	−	1.62310i	−0.735724	−	0.677282i	$-0.763158\pi$
$661$	−0.411024	−	1.62310i	−0.411024	−	1.62310i	0.324699	−	0.945817i	$-0.394737\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.87947	+	0.475947i	1.87947	+	0.475947i	1.00000			$0$
$673$	1.87947	+	0.475947i	1.87947	+	0.475947i	0.879474	+	0.475947i	$0.157895\pi$
$674$		0			0
$675$		0			0
$676$	−0.589961	+	0.319271i	−0.589961	+	0.319271i
$677$		0			0		0.511885	−	0.859054i	$-0.328947\pi$
$677$		0			0		−0.511885	+	0.859054i	$0.671053\pi$
$678$		0			0
$679$	−1.83321	+	0.896201i	−1.83321	+	0.896201i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.735724	−	0.677282i	$-0.763158\pi$
$683$		0			0		0.735724	+	0.677282i	$0.236842\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	1.83155			1.83155
$689$		0			0
$690$		0			0
$691$	0.903782	+	0.831990i	0.903782	+	0.831990i	0.986361	−	0.164595i	$-0.0526316\pi$
$691$	0.903782	+	0.831990i	0.903782	+	0.831990i	−0.0825793	+	0.996584i	$0.526316\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.833194	−	0.594889i	0.833194	−	0.594889i
$701$		0			0		−0.969400	−	0.245485i	$-0.921053\pi$
$701$		0			0		0.969400	+	0.245485i	$0.0789474\pi$
$702$		0			0
$703$	−0.142571	−	1.72058i	−0.142571	−	1.72058i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.242120	+	1.94240i	0.242120	+	1.94240i	0.324699	+	0.945817i	$0.394737\pi$
$709$	0.242120	+	1.94240i	0.242120	+	1.94240i	−0.0825793	+	0.996584i	$0.526316\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.958427	−	0.285336i	$-0.907895\pi$
$719$		0			0		0.958427	+	0.285336i	$0.0921053\pi$
$720$		0			0
$721$	−1.19130	−	0.709862i	−1.19130	−	0.709862i
$722$		0			0
$723$		0			0
$724$	0.0136387	−	0.164595i	0.0136387	−	0.164595i
$725$		0			0
$726$		0			0
$727$	−0.237101	+	0.0705880i	−0.237101	+	0.0705880i	−0.401695	−	0.915773i	$-0.631579\pi$
$727$	−0.237101	+	0.0705880i	−0.237101	+	0.0705880i	0.164595	+	0.986361i	$0.447368\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.202192	−	1.21167i	−0.202192	−	1.21167i	−0.879474	−	0.475947i	$-0.842105\pi$
$733$	−0.202192	−	1.21167i	−0.202192	−	1.21167i	0.677282	−	0.735724i	$-0.263158\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.96940	+	0.245485i	1.96940	+	0.245485i	1.00000			$0$
$739$	1.96940	+	0.245485i	1.96940	+	0.245485i	0.969400	+	0.245485i	$0.0789474\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.546948	−	0.837166i	$-0.684211\pi$
$743$		0			0		0.546948	+	0.837166i	$0.315789\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.19084	+	1.52999i	−1.19084	+	1.52999i	−0.401695	+	0.915773i	$0.631579\pi$
$751$	−1.19084	+	1.52999i	−1.19084	+	1.52999i	−0.789141	+	0.614213i	$0.789474\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.735724	+	0.322718i	0.735724	+	0.322718i	0.735724	−	0.677282i	$-0.236842\pi$
$757$	0.735724	+	0.322718i	0.735724	+	0.322718i			1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.898391	−	0.439197i	$-0.855263\pi$
$761$		0			0		0.898391	+	0.439197i	$0.144737\pi$
$762$		0			0
$763$	−0.623101	−	0.407092i	−0.623101	−	0.407092i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.19130	−	1.29410i	1.19130	−	1.29410i	0.245485	−	0.969400i	$-0.421053\pi$
$769$	1.19130	−	1.29410i	1.19130	−	1.29410i	0.945817	−	0.324699i	$-0.105263\pi$
$770$		0			0
$771$		0			0
$772$	−0.477778	+	1.39172i	−0.477778	+	1.39172i
$773$		0			0		−0.898391	−	0.439197i	$-0.855263\pi$
$773$		0			0		0.898391	+	0.439197i	$0.144737\pi$
$774$		0			0
$775$	−1.11384	+	1.56003i	−1.11384	+	1.56003i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.0156197	−	0.0454987i	−0.0156197	−	0.0454987i
$785$		0			0
$786$		0			0
$787$	0.700332	+	1.29410i	0.700332	+	1.29410i	0.945817	+	0.324699i	$0.105263\pi$
$787$	0.700332	+	1.29410i	0.700332	+	1.29410i	−0.245485	+	0.969400i	$0.578947\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.514288	−	2.45275i	−0.514288	−	2.45275i
$794$		0			0
$795$		0			0
$796$	−0.567705	+	0.670288i	−0.567705	+	0.670288i
$797$		0			0		0.0413250	−	0.999146i	$-0.486842\pi$
$797$		0			0		−0.0413250	+	0.999146i	$0.513158\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$809$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$810$		0			0
$811$	0.998353	−	0.594889i	0.998353	−	0.594889i	0.0825793	−	0.996584i	$-0.473684\pi$
$811$	0.998353	−	0.594889i	0.998353	−	0.594889i	0.915773	+	0.401695i	$0.131579\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.238712	+	2.88082i	0.238712	+	2.88082i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	0.464441	+	1.56003i	0.464441	+	1.56003i	0.789141	+	0.614213i	$0.210526\pi$
$823$	0.464441	+	1.56003i	0.464441	+	1.56003i	−0.324699	+	0.945817i	$0.605263\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	1.07916	−	0.914005i	1.07916	−	0.914005i	0.0825793	−	0.996584i	$-0.473684\pi$
$829$	1.07916	−	0.914005i	1.07916	−	0.914005i	0.996584	+	0.0825793i	$0.0263158\pi$
$830$		0			0
$831$		0			0
$832$	−0.0534166	+	1.29149i	−0.0534166	+	1.29149i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.813849	−	0.581077i	$-0.197368\pi$
$839$		0			0		−0.813849	+	0.581077i	$0.802632\pi$
$840$		0			0
$841$	−0.969400	+	0.245485i	−0.969400	+	0.245485i
$842$		0			0
$843$		0			0
$844$	0.0672646	+	1.62631i	0.0672646	+	1.62631i
$845$		0			0
$846$		0			0
$847$	0.919746	+	0.449636i	0.919746	+	0.449636i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.512687	+	1.72209i	−0.512687	+	1.72209i	0.164595	+	0.986361i	$0.447368\pi$
$853$	−0.512687	+	1.72209i	−0.512687	+	1.72209i	−0.677282	+	0.735724i	$0.736842\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.763084	−	0.646299i	$-0.776316\pi$
$857$		0			0		0.763084	+	0.646299i	$0.223684\pi$
$858$		0			0
$859$	−1.08258	+	0.996584i	−1.08258	+	0.996584i	−0.0825793	+	0.996584i	$0.526316\pi$
$859$	−1.08258	+	0.996584i	−1.08258	+	0.996584i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.546948	−	0.837166i	$-0.684211\pi$
$863$		0			0		0.546948	+	0.837166i	$0.315789\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	1.20534	+	1.54862i	1.20534	+	1.54862i
$869$		0			0
$870$		0			0
$871$	0.821624	−	1.25759i	0.821624	−	1.25759i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.207444	−	0.531633i	0.207444	−	0.531633i	−0.789141	−	0.614213i	$-0.789474\pi$
$877$	0.207444	−	0.531633i	0.207444	−	0.531633i	0.996584	+	0.0825793i	$0.0263158\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.999146	−	0.0413250i	$-0.986842\pi$
$881$		0			0		0.999146	+	0.0413250i	$0.0131579\pi$
$882$		0			0
$883$	−0.145253	−	0.0786068i	−0.145253	−	0.0786068i	0.401695	−	0.915773i	$-0.368421\pi$
$883$	−0.145253	−	0.0786068i	−0.145253	−	0.0786068i	−0.546948	+	0.837166i	$0.684211\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.992321	−	0.123693i	$-0.0394737\pi$
$887$		0			0		−0.992321	+	0.123693i	$0.960526\pi$
$888$		0			0
$889$	−0.334426	−	2.00410i	−0.334426	−	2.00410i
$890$		0			0
$891$		0			0
$892$	1.68154	+	1.00198i	1.68154	+	1.00198i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.871720	+	0.382372i	0.871720	+	0.382372i	0.789141	−	0.614213i	$-0.210526\pi$
$907$	0.871720	+	0.382372i	0.871720	+	0.382372i	0.0825793	+	0.996584i	$0.473684\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.164595	−	0.986361i	$-0.447368\pi$
$911$		0			0		−0.164595	+	0.986361i	$0.552632\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.879474	−	0.475947i	−0.879474	−	0.475947i
$917$		0			0
$918$		0			0
$919$	0.520637	+	0.796894i	0.520637	+	0.796894i	0.996584	−	0.0825793i	$-0.0263158\pi$
$919$	0.520637	+	0.796894i	0.520637	+	0.796894i	−0.475947	+	0.879474i	$0.657895\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.439413	+	1.00176i	−0.439413	+	1.00176i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.581077	−	0.813849i	$-0.697368\pi$
$929$		0			0		0.581077	+	0.813849i	$0.302632\pi$
$930$		0			0
$931$	0.0695288	−	0.0304981i	0.0695288	−	0.0304981i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.313193	−	0.265261i	0.313193	−	0.265261i	−0.475947	−	0.879474i	$-0.657895\pi$
$937$	0.313193	−	0.265261i	0.313193	−	0.265261i	0.789141	+	0.614213i	$0.210526\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.324699	−	0.945817i	$-0.394737\pi$
$941$		0			0		−0.324699	+	0.945817i	$0.605263\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.958427	−	0.285336i	$-0.907895\pi$
$947$		0			0		0.958427	+	0.285336i	$0.0921053\pi$
$948$		0			0
$949$	−0.252501	+	0.466580i	−0.252501	+	0.466580i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.996584	−	0.0825793i	$-0.973684\pi$
$953$		0			0		0.996584	+	0.0825793i	$0.0263158\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2.23886	−	1.46272i	−2.23886	−	1.46272i
$962$		0			0
$963$		0			0
$964$	0.130333	−	0.101443i	0.130333	−	0.101443i
$965$		0			0
$966$		0			0
$967$	0.544122	+	1.24047i	0.544122	+	1.24047i	0.945817	+	0.324699i	$0.105263\pi$
$967$	0.544122	+	1.24047i	0.544122	+	1.24047i	−0.401695	+	0.915773i	$0.631579\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.324699	−	0.945817i	$-0.605263\pi$
$971$		0			0		0.324699	+	0.945817i	$0.394737\pi$
$972$		0			0
$973$	0.980588	+	1.06520i	0.980588	+	1.06520i
$974$		0			0
$975$		0			0
$976$	−1.83375	−	0.629528i	−1.83375	−	0.629528i
$977$		0			0		−0.915773	−	0.401695i	$-0.868421\pi$
$977$		0			0		0.915773	+	0.401695i	$0.131579\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.363508	−	0.931591i	$-0.618421\pi$
$983$		0			0		0.363508	+	0.931591i	$0.381579\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−2.03834	+	0.0843064i	−2.03834	+	0.0843064i
$989$		0			0
$990$		0			0
$991$	0.159418	+	0.629528i	0.159418	+	0.629528i	0.996584	+	0.0825793i	$0.0263158\pi$
$991$	0.159418	+	0.629528i	0.159418	+	0.629528i	−0.837166	+	0.546948i	$0.815789\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.24047	−	1.34751i	−1.24047	−	1.34751i	−0.915773	−	0.401695i	$-0.868421\pi$
$997$	−1.24047	−	1.34751i	−1.24047	−	1.34751i	−0.324699	−	0.945817i	$-0.605263\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2061.1.bn.a.946.1	✓	36
3.2	odd	2	CM	2061.1.bn.a.946.1	✓	36
229.84	odd	76	inner	2061.1.bn.a.1000.1	yes	36
687.542	even	76	inner	2061.1.bn.a.1000.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2061.1.bn.a.946.1	✓	36	1.1	even	1	trivial
2061.1.bn.a.946.1	✓	36	3.2	odd	2	CM
2061.1.bn.a.1000.1	yes	36	229.84	odd	76	inner
2061.1.bn.a.1000.1	yes	36	687.542	even	76	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:	\( N \)	\(=\)	\( 2061 = 3^{2} \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2061.bn (of order \(76\), degree \(36\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.735724 - 0.677282i) q^{4} +(0.126633 + 1.01591i) q^{7} +O(q^{10})\)	\(q+(-0.735724 - 0.677282i) q^{4} +(0.126633 + 1.01591i) q^{7} +(-1.05198 + 0.751099i) q^{13} +(0.0825793 + 0.996584i) q^{16} +(-1.55676 + 0.259777i) q^{19} +(0.945817 + 0.324699i) q^{25} +(0.594889 - 0.833194i) q^{28} +(-0.546948 + 1.83717i) q^{31} +(-0.0903332 + 1.09016i) q^{37} +(0.151248 - 1.82529i) q^{43} +(-0.0466332 + 0.0118091i) q^{49} +(1.28267 + 0.159885i) q^{52} +(-0.778807 + 1.77550i) q^{61} +(0.614213 - 0.789141i) q^{64} +(-1.08265 + 0.422452i) q^{67} +(0.368727 - 0.180260i) q^{73} +(1.32128 + 0.863238i) q^{76} +(1.78298 + 0.222249i) q^{79} +(-0.896263 - 0.973601i) q^{91} +(0.647181 + 1.88517i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 2 q^{7}+O(q^{10}) \) 36 * q + 2 * q^7	\( 36 q + 2 q^{7} - 2 q^{13} + 2 q^{16} - 4 q^{19} - 2 q^{25} - 2 q^{28} + 2 q^{31} - 4 q^{37} - 2 q^{52} + 2 q^{67} + 2 q^{73} - 2 q^{79} + 4 q^{91}+O(q^{100}) \) 36 * q + 2 * q^7 - 2 * q^13 + 2 * q^16 - 4 * q^19 - 2 * q^25 - 2 * q^28 + 2 * q^31 - 4 * q^37 - 2 * q^52 + 2 * q^67 + 2 * q^73 - 2 * q^79 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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