LMFDB - Embedded newform 2061.1.bn.a.208.1

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			208.1
Root			$0.915773 + 0.401695i$ of defining polynomial
Character	$\chi$	$=$	2061.208
Dual form			2061.1.bn.a.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.996584 + 0.0825793i) q^{4} +(-0.981209 - 1.64668i) q^{7} +O(q^{10})$	$q+(0.996584 + 0.0825793i) q^{4} +(-0.981209 - 1.64668i) q^{7} +(-1.46231 - 0.714879i) q^{13} +(0.986361 + 0.164595i) q^{16} +(0.464369 + 0.159418i) q^{19} +(0.789141 - 0.614213i) q^{25} +(-0.841876 - 1.72209i) q^{28} +(0.401695 - 1.91577i) q^{31} +(0.792434 - 0.132234i) q^{37} +(-1.45138 + 0.242192i) q^{43} +(-1.27284 + 2.35201i) q^{49} +(-1.39828 - 0.833194i) q^{52} +(-0.644701 + 0.700332i) q^{61} +(0.969400 + 0.245485i) q^{64} +(-0.0362996 - 0.877643i) q^{67} +(1.84887 + 0.230462i) q^{73} +(0.449618 + 0.197221i) q^{76} +(1.70491 + 1.01591i) q^{79} +(0.257653 + 3.10940i) q^{91} +(-0.202192 + 0.259777i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2061\mathbb{Z}\right)^\times$.

$n$	$235$	$1604$
$\chi(n)$	$e\left(\frac{67}{76}\right)$	$1$

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		−0.999146	−	0.0413250i	$-0.986842\pi$
$2$		0			0		0.999146	+	0.0413250i	$0.0131579\pi$
$3$		0			0
$3$		0			0
$4$	0.996584	+	0.0825793i	0.996584	+	0.0825793i
$5$		0			0		0.945817	−	0.324699i	$-0.105263\pi$
$5$		0			0		−0.945817	+	0.324699i	$0.894737\pi$
$6$		0			0
$7$	−0.981209	−	1.64668i	−0.981209	−	1.64668i	−0.735724	−	0.677282i	$-0.763158\pi$
$7$	−0.981209	−	1.64668i	−0.981209	−	1.64668i	−0.245485	−	0.969400i	$-0.578947\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.546948	−	0.837166i	$-0.684211\pi$
$11$		0			0		0.546948	+	0.837166i	$0.315789\pi$
$12$		0			0
$13$	−1.46231	−	0.714879i	−1.46231	−	0.714879i	−0.475947	−	0.879474i	$-0.657895\pi$
$13$	−1.46231	−	0.714879i	−1.46231	−	0.714879i	−0.986361	+	0.164595i	$0.947368\pi$
$14$		0			0
$15$		0			0
$16$	0.986361	+	0.164595i	0.986361	+	0.164595i
$17$		0			0		−0.324699	−	0.945817i	$-0.605263\pi$
$17$		0			0		0.324699	+	0.945817i	$0.394737\pi$
$18$		0			0
$19$	0.464369	+	0.159418i	0.464369	+	0.159418i	0.546948	−	0.837166i	$-0.315789\pi$
$19$	0.464369	+	0.159418i	0.464369	+	0.159418i	−0.0825793	+	0.996584i	$0.526316\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.285336	−	0.958427i	$-0.407895\pi$
$23$		0			0		−0.285336	+	0.958427i	$0.592105\pi$
$24$		0			0
$25$	0.789141	−	0.614213i	0.789141	−	0.614213i
$26$		0			0
$27$		0			0
$28$	−0.841876	−	1.72209i	−0.841876	−	1.72209i
$29$		0			0		−0.511885	−	0.859054i	$-0.671053\pi$
$29$		0			0		0.511885	+	0.859054i	$0.328947\pi$
$30$		0			0
$31$	0.401695	−	1.91577i	0.401695	−	1.91577i		−	1.00000i	$-0.5\pi$
$31$	0.401695	−	1.91577i	0.401695	−	1.91577i	0.401695	−	0.915773i	$-0.368421\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.792434	−	0.132234i	0.792434	−	0.132234i	0.245485	−	0.969400i	$-0.421053\pi$
$37$	0.792434	−	0.132234i	0.792434	−	0.132234i	0.546948	+	0.837166i	$0.315789\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.0413250	−	0.999146i	$-0.486842\pi$
$41$		0			0		−0.0413250	+	0.999146i	$0.513158\pi$
$42$		0			0
$43$	−1.45138	+	0.242192i	−1.45138	+	0.242192i	−0.837166	−	0.546948i	$-0.815789\pi$
$43$	−1.45138	+	0.242192i	−1.45138	+	0.242192i	−0.614213	+	0.789141i	$0.710526\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.999146	−	0.0413250i	$-0.0131579\pi$
$47$		0			0		−0.999146	+	0.0413250i	$0.986842\pi$
$48$		0			0
$49$	−1.27284	+	2.35201i	−1.27284	+	2.35201i
$50$		0			0
$51$		0			0
$52$	−1.39828	−	0.833194i	−1.39828	−	0.833194i
$53$		0			0		0.915773	−	0.401695i	$-0.131579\pi$
$53$		0			0		−0.915773	+	0.401695i	$0.868421\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.813849	−	0.581077i	$-0.197368\pi$
$59$		0			0		−0.813849	+	0.581077i	$0.802632\pi$
$60$		0			0
$61$	−0.644701	+	0.700332i	−0.644701	+	0.700332i	−0.969400	−	0.245485i	$-0.921053\pi$
$61$	−0.644701	+	0.700332i	−0.644701	+	0.700332i	0.324699	+	0.945817i	$0.394737\pi$
$62$		0			0
$63$		0			0
$64$	0.969400	+	0.245485i	0.969400	+	0.245485i
$65$		0			0
$66$		0			0
$67$	−0.0362996	−	0.877643i	−0.0362996	−	0.877643i	−0.915773	−	0.401695i	$-0.868421\pi$
$67$	−0.0362996	−	0.877643i	−0.0362996	−	0.877643i	0.879474	−	0.475947i	$-0.157895\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.546948	−	0.837166i	$-0.315789\pi$
$71$		0			0		−0.546948	+	0.837166i	$0.684211\pi$
$72$		0			0
$73$	1.84887	+	0.230462i	1.84887	+	0.230462i	0.969400	−	0.245485i	$-0.0789474\pi$
$73$	1.84887	+	0.230462i	1.84887	+	0.230462i	0.879474	+	0.475947i	$0.157895\pi$
$74$		0			0
$75$		0			0
$76$	0.449618	+	0.197221i	0.449618	+	0.197221i
$77$		0			0
$78$		0			0
$79$	1.70491	+	1.01591i	1.70491	+	1.01591i	0.915773	+	0.401695i	$0.131579\pi$
$79$	1.70491	+	1.01591i	1.70491	+	1.01591i	0.789141	+	0.614213i	$0.210526\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.164595	−	0.986361i	$-0.447368\pi$
$83$		0			0		−0.164595	+	0.986361i	$0.552632\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$	0.257653	+	3.10940i	0.257653	+	3.10940i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.202192	+	0.259777i	−0.202192	+	0.259777i	−0.879474	−	0.475947i	$-0.842105\pi$
$97$	−0.202192	+	0.259777i	−0.202192	+	0.259777i	0.677282	+	0.735724i	$0.263158\pi$
$98$		0			0
$99$		0			0
$100$	0.837166	−	0.546948i	0.837166	−	0.546948i
$101$		0			0		0.285336	−	0.958427i	$-0.407895\pi$
$101$		0			0		−0.285336	+	0.958427i	$0.592105\pi$
$102$		0			0
$103$	−0.160105	−	0.0405441i	−0.160105	−	0.0405441i	0.164595	−	0.986361i	$-0.447368\pi$
$103$	−0.160105	−	0.0405441i	−0.160105	−	0.0405441i	−0.324699	+	0.945817i	$0.605263\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	−1.62631	+	1.16116i	−1.62631	+	1.16116i	−0.789141	+	0.614213i	$0.789474\pi$
$109$	−1.62631	+	1.16116i	−1.62631	+	1.16116i	−0.837166	+	0.546948i	$0.815789\pi$
$110$		0			0
$111$		0			0
$112$	−0.696792	−	1.78573i	−0.696792	−	1.78573i
$113$		0			0		0.898391	−	0.439197i	$-0.144737\pi$
$113$		0			0		−0.898391	+	0.439197i	$0.855263\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.401695	+	0.915773i	−0.401695	+	0.915773i
$122$		0			0
$123$		0			0
$124$	0.558527	−	1.87606i	0.558527	−	1.87606i
$125$		0			0
$126$		0			0
$127$	1.11041	+	1.31106i	1.11041	+	1.31106i	0.945817	+	0.324699i	$0.105263\pi$
$127$	1.11041	+	1.31106i	1.11041	+	1.31106i	0.164595	+	0.986361i	$0.447368\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.859054	−	0.511885i	$-0.828947\pi$
$131$		0			0		0.859054	+	0.511885i	$0.171053\pi$
$132$		0			0
$133$	−0.193132	−	0.921090i	−0.193132	−	0.921090i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.931591	−	0.363508i	$-0.881579\pi$
$137$		0			0		0.931591	+	0.363508i	$0.118421\pi$
$138$		0			0
$139$	−1.27052	−	0.621118i	−1.27052	−	0.621118i	−0.324699	−	0.945817i	$-0.605263\pi$
$139$	−1.27052	−	0.621118i	−1.27052	−	0.621118i	−0.945817	+	0.324699i	$0.894737\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.800647	−	0.0663435i	0.800647	−	0.0663435i
$149$		0			0		−0.164595	−	0.986361i	$-0.552632\pi$
$149$		0			0		0.164595	+	0.986361i	$0.447368\pi$
$150$		0			0
$151$	0.130333	−	1.57289i	0.130333	−	1.57289i	−0.546948	−	0.837166i	$-0.684211\pi$
$151$	0.130333	−	1.57289i	0.130333	−	1.57289i	0.677282	−	0.735724i	$-0.263158\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.986361	+	0.835405i	−0.986361	+	0.835405i	−0.986361	−	0.164595i	$-0.947368\pi$
$157$	−0.986361	+	0.835405i	−0.986361	+	0.835405i			1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.24207	+	0.886821i	1.24207	+	0.886821i	0.996584	−	0.0825793i	$-0.0263158\pi$
$163$	1.24207	+	0.886821i	1.24207	+	0.886821i	0.245485	+	0.969400i	$0.421053\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.969400	−	0.245485i	$-0.921053\pi$
$167$		0			0		0.969400	+	0.245485i	$0.0789474\pi$
$168$		0			0
$169$	1.01308	+	1.30161i	1.01308	+	1.30161i
$170$		0			0
$171$		0			0
$172$	−1.46642	+	0.121511i	−1.46642	+	0.121511i
$173$		0			0		0.915773	−	0.401695i	$-0.131579\pi$
$173$		0			0		−0.915773	+	0.401695i	$0.868421\pi$
$174$		0			0
$175$	−1.78573	−	0.696792i	−1.78573	−	0.696792i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.581077	−	0.813849i	$-0.697368\pi$
$179$		0			0		0.581077	+	0.813849i	$0.302632\pi$
$180$		0			0
$181$	1.91236	−	0.484275i	1.91236	−	0.484275i	0.915773	−	0.401695i	$-0.131579\pi$
$181$	1.91236	−	0.484275i	1.91236	−	0.484275i	0.996584	−	0.0825793i	$-0.0263158\pi$
$182$		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		0.363508	−	0.931591i	$-0.381579\pi$
$191$		0			0		−0.363508	+	0.931591i	$0.618421\pi$
$192$		0			0
$193$	1.34994	+	1.46642i	1.34994	+	1.46642i	0.735724	+	0.677282i	$0.236842\pi$
$193$	1.34994	+	1.46642i	1.34994	+	1.46642i	0.614213	+	0.789141i	$0.289474\pi$
$194$		0			0
$195$		0			0
$196$	−1.46272	+	2.23886i	−1.46272	+	2.23886i
$197$		0			0		0.205215	−	0.978717i	$-0.434211\pi$
$197$		0			0		−0.205215	+	0.978717i	$0.565789\pi$
$198$		0			0
$199$	0.188776	−	0.159885i	0.188776	−	0.159885i	−0.546948	−	0.837166i	$-0.684211\pi$
$199$	0.188776	−	0.159885i	0.188776	−	0.159885i	0.735724	+	0.677282i	$0.236842\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.32470	−	0.945817i	−1.32470	−	0.945817i
$209$		0			0
$210$		0			0
$211$	1.27052	−	1.27052i	1.27052	−	1.27052i	0.324699	−	0.945817i	$-0.394737\pi$
$211$	1.27052	−	1.27052i	1.27052	−	1.27052i	0.945817	−	0.324699i	$-0.105263\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.54882	+	1.21831i	−3.54882	+	1.21831i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.264276	−	0.677282i	−0.264276	−	0.677282i	0.735724	−	0.677282i	$-0.236842\pi$
$223$	−0.264276	−	0.677282i	−0.264276	−	0.677282i	−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.0413250	−	0.999146i	$-0.486842\pi$
$227$		0			0		−0.0413250	+	0.999146i	$0.513158\pi$
$228$		0			0
$229$	0.475947	−	0.879474i	0.475947	−	0.879474i
$229$	0.475947	−	0.879474i	0.475947	−	0.879474i
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.0825793	−	0.996584i	$-0.473684\pi$
$233$		0			0		−0.0825793	+	0.996584i	$0.526316\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.285336	−	0.958427i	$-0.592105\pi$
$239$		0			0		0.285336	+	0.958427i	$0.407895\pi$
$240$		0			0
$241$	0.640542	+	1.86584i	0.640542	+	1.86584i	0.475947	+	0.879474i	$0.342105\pi$
$241$	0.640542	+	1.86584i	0.640542	+	1.86584i	0.164595	+	0.986361i	$0.447368\pi$
$242$		0			0
$243$		0			0
$244$	−0.700332	+	0.644701i	−0.700332	+	0.644701i
$245$		0			0
$246$		0			0
$247$	−0.565086	−	0.565086i	−0.565086	−	0.565086i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.859054	−	0.511885i	$-0.171053\pi$
$251$		0			0		−0.859054	+	0.511885i	$0.828947\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.945817	+	0.324699i	0.945817	+	0.324699i
$257$		0			0		0.898391	−	0.439197i	$-0.144737\pi$
$257$		0			0		−0.898391	+	0.439197i	$0.855263\pi$
$258$		0			0
$259$	−0.995290	−	1.17514i	−0.995290	−	1.17514i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.958427	−	0.285336i	$-0.0921053\pi$
$263$		0			0		−0.958427	+	0.285336i	$0.907895\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.0362996	−	0.877643i	0.0362996	−	0.877643i
$269$		0			0		−0.205215	−	0.978717i	$-0.565789\pi$
$269$		0			0		0.205215	+	0.978717i	$0.434211\pi$
$270$		0			0
$271$	−0.863238	−	0.671885i	−0.863238	−	0.671885i	0.0825793	−	0.996584i	$-0.473684\pi$
$271$	−0.863238	−	0.671885i	−0.863238	−	0.671885i	−0.945817	+	0.324699i	$0.894737\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.159418	−	0.629528i	−0.159418	−	0.629528i	−0.996584	−	0.0825793i	$-0.973684\pi$
$277$	−0.159418	−	0.629528i	−0.159418	−	0.629528i	0.837166	−	0.546948i	$-0.184211\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.511885	−	0.859054i	$-0.328947\pi$
$281$		0			0		−0.511885	+	0.859054i	$0.671053\pi$
$282$		0			0
$283$	−0.514078	+	1.31747i	−0.514078	+	1.31747i	0.401695	+	0.915773i	$0.368421\pi$
$283$	−0.514078	+	1.31747i	−0.514078	+	1.31747i	−0.915773	+	0.401695i	$0.868421\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.789141	+	0.614213i	−0.789141	+	0.614213i
$290$		0			0
$291$		0			0
$292$	1.82353	+	0.382354i	1.82353	+	0.382354i
$293$		0			0		0.245485	−	0.969400i	$-0.421053\pi$
$293$		0			0		−0.245485	+	0.969400i	$0.578947\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.82292	+	2.15232i	1.82292	+	2.15232i
$302$		0			0
$303$		0			0
$304$	0.431796	+	0.233676i	0.431796	+	0.233676i
$305$		0			0
$306$		0			0
$307$	−1.34994	−	0.111859i	−1.34994	−	0.111859i	−0.614213	−	0.789141i	$-0.710526\pi$
$307$	−1.34994	−	0.111859i	−1.34994	−	0.111859i	−0.735724	+	0.677282i	$0.763158\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.996584	−	0.0825793i	$-0.973684\pi$
$311$		0			0		0.996584	+	0.0825793i	$0.0263158\pi$
$312$		0			0
$313$	−1.11041	+	1.31106i	−1.11041	+	1.31106i	−0.164595	+	0.986361i	$0.552632\pi$
$313$	−1.11041	+	1.31106i	−1.11041	+	1.31106i	−0.945817	+	0.324699i	$0.894737\pi$
$314$		0			0
$315$		0			0
$316$	1.61520	+	1.15323i	1.61520	+	1.15323i
$317$		0			0		0.898391	−	0.439197i	$-0.144737\pi$
$317$		0			0		−0.898391	+	0.439197i	$0.855263\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.59305	+	0.334028i	−1.59305	+	0.334028i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.0630689	−	0.0534166i	0.0630689	−	0.0534166i	−0.614213	−	0.789141i	$-0.710526\pi$
$331$	0.0630689	−	0.0534166i	0.0630689	−	0.0534166i	0.677282	+	0.735724i	$0.263158\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.44535	+	0.633988i	1.44535	+	0.633988i	0.969400	−	0.245485i	$-0.0789474\pi$
$337$	1.44535	+	0.633988i	1.44535	+	0.633988i	0.475947	+	0.879474i	$0.342105\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	3.20671	−	0.132631i	3.20671	−	0.132631i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.546948	−	0.837166i	$-0.684211\pi$
$347$		0			0		0.546948	+	0.837166i	$0.315789\pi$
$348$		0			0
$349$	−0.886821	−	1.24207i	−0.886821	−	1.24207i	−0.969400	−	0.245485i	$-0.921053\pi$
$349$	−0.886821	−	1.24207i	−0.886821	−	1.24207i	0.0825793	−	0.996584i	$-0.473684\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.958427	−	0.285336i	$-0.907895\pi$
$353$		0			0		0.958427	+	0.285336i	$0.0921053\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.164595	−	0.986361i	$-0.447368\pi$
$359$		0			0		−0.164595	+	0.986361i	$0.552632\pi$
$360$		0			0
$361$	−0.598916	−	0.466155i	−0.598916	−	0.466155i
$362$		0			0
$363$		0			0
$364$			3.12006i			3.12006i
$365$		0			0
$366$		0			0
$367$	1.82529	−	0.151248i	1.82529	−	0.151248i	0.879474	−	0.475947i	$-0.157895\pi$
$367$	1.82529	−	0.151248i	1.82529	−	0.151248i	0.945817	+	0.324699i	$0.105263\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.863238	+	0.671885i	−0.863238	+	0.671885i	−0.945817	−	0.324699i	$-0.894737\pi$
$373$	−0.863238	+	0.671885i	−0.863238	+	0.671885i	0.0825793	+	0.996584i	$0.473684\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.02290	+	1.71664i	−1.02290	+	1.71664i	−0.475947	+	0.879474i	$0.657895\pi$
$379$	−1.02290	+	1.71664i	−1.02290	+	1.71664i	−0.546948	+	0.837166i	$0.684211\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.879474	−	0.475947i	$-0.842105\pi$
$383$		0			0		0.879474	+	0.475947i	$0.157895\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.222954	+	0.242192i	−0.222954	+	0.242192i
$389$		0			0		−0.123693	−	0.992321i	$-0.539474\pi$
$389$		0			0		0.123693	+	0.992321i	$0.460526\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.121511	−	0.111859i	−0.121511	−	0.111859i	0.614213	−	0.789141i	$-0.289474\pi$
$397$	−0.121511	−	0.111859i	−0.121511	−	0.111859i	−0.735724	+	0.677282i	$0.763158\pi$
$398$		0			0
$399$		0			0
$400$	0.879474	−	0.475947i	0.879474	−	0.475947i
$401$		0			0		−0.0825793	−	0.996584i	$-0.526316\pi$
$401$		0			0		0.0825793	+	0.996584i	$0.473684\pi$
$402$		0			0
$403$	−1.95695	+	2.51429i	−1.95695	+	2.51429i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.289513	−	0.156677i	−0.289513	−	0.156677i	0.324699	−	0.945817i	$-0.394737\pi$
$409$	−0.289513	−	0.156677i	−0.289513	−	0.156677i	−0.614213	+	0.789141i	$0.710526\pi$
$410$		0			0
$411$		0			0
$412$	−0.156210	−	0.0536269i	−0.156210	−	0.0536269i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.0413250	−	0.999146i	$-0.486842\pi$
$419$		0			0		−0.0413250	+	0.999146i	$0.513158\pi$
$420$		0			0
$421$	−0.0271842	−	0.162906i	−0.0271842	−	0.162906i	0.969400	−	0.245485i	$-0.0789474\pi$
$421$	−0.0271842	−	0.162906i	−0.0271842	−	0.162906i	−0.996584	+	0.0825793i	$0.973684\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.78581	+	0.374445i	1.78581	+	0.374445i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.945817	−	0.324699i	$-0.894737\pi$
$431$		0			0		0.945817	+	0.324699i	$0.105263\pi$
$432$		0			0
$433$	1.16187	+	1.49277i	1.16187	+	1.49277i	0.837166	+	0.546948i	$0.184211\pi$
$433$	1.16187	+	1.49277i	1.16187	+	1.49277i	0.324699	+	0.945817i	$0.394737\pi$
$434$		0			0
$435$		0			0
$436$	−1.71664	+	1.02290i	−1.71664	+	1.02290i
$437$		0			0
$438$		0			0
$439$	0.571129	−	1.66364i	0.571129	−	1.66364i	−0.164595	−	0.986361i	$-0.552632\pi$
$439$	0.571129	−	1.66364i	0.571129	−	1.66364i	0.735724	−	0.677282i	$-0.236842\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.735724	−	0.677282i	$-0.236842\pi$
$443$		0			0		−0.735724	+	0.677282i	$0.763158\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.546948	−	1.83717i	−0.546948	−	1.83717i
$449$		0			0		0.677282	−	0.735724i	$-0.263158\pi$
$449$		0			0		−0.677282	+	0.735724i	$0.736842\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$457$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.915773	−	0.401695i	$-0.131579\pi$
$461$		0			0		−0.915773	+	0.401695i	$0.868421\pi$
$462$		0			0
$463$	0.355188	+	1.03463i	0.355188	+	1.03463i	0.969400	+	0.245485i	$0.0789474\pi$
$463$	0.355188	+	1.03463i	0.355188	+	1.03463i	−0.614213	+	0.789141i	$0.710526\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.735724	−	0.677282i	$-0.763158\pi$
$467$		0			0		0.735724	+	0.677282i	$0.236842\pi$
$468$		0			0
$469$	−1.40958	+	0.920925i	−1.40958	+	0.920925i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.464369	−	0.159418i	0.464369	−	0.159418i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.813849	−	0.581077i	$-0.802632\pi$
$479$		0			0		0.813849	+	0.581077i	$0.197368\pi$
$480$		0			0
$481$	−1.25331	−	0.373128i	−1.25331	−	0.373128i
$482$		0			0
$483$		0			0
$484$	−0.475947	+	0.879474i	−0.475947	+	0.879474i
$485$		0			0
$486$		0			0
$487$	0.879474	−	0.524053i	0.879474	−	0.524053i		−	1.00000i	$-0.5\pi$
$487$	0.879474	−	0.524053i	0.879474	−	0.524053i	0.879474	+	0.475947i	$0.157895\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.546948	−	0.837166i	$-0.315789\pi$
$491$		0			0		−0.546948	+	0.837166i	$0.684211\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.711543	−	1.82353i	0.711543	−	1.82353i
$497$		0			0
$498$		0			0
$499$	1.98295	+	0.0820152i	1.98295	+	0.0820152i	0.996584	−	0.0825793i	$-0.0263158\pi$
$499$	1.98295	+	0.0820152i	1.98295	+	0.0820152i	0.986361	+	0.164595i	$0.0526316\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.401695	−	0.915773i	$-0.368421\pi$
$503$		0			0		−0.401695	+	0.915773i	$0.631579\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.998353	+	1.39828i	0.998353	+	1.39828i
$509$		0			0		0.735724	−	0.677282i	$-0.236842\pi$
$509$		0			0		−0.735724	+	0.677282i	$0.763158\pi$
$510$		0			0
$511$	−1.43464	−	3.27064i	−1.43464	−	3.27064i
$512$		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		0.205215	−	0.978717i	$-0.434211\pi$
$521$		0			0		−0.205215	+	0.978717i	$0.565789\pi$
$522$		0			0
$523$	0.222249	−	1.78298i	0.222249	−	1.78298i	−0.324699	−	0.945817i	$-0.605263\pi$
$523$	0.222249	−	1.78298i	0.222249	−	1.78298i	0.546948	−	0.837166i	$-0.315789\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.837166	−	0.546948i	−0.837166	−	0.546948i
$530$		0			0
$531$		0			0
$532$	−0.116410	−	0.933893i	−0.116410	−	0.933893i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.86584	−	0.311353i	−1.86584	−	0.311353i	−0.879474	−	0.475947i	$-0.842105\pi$
$541$	−1.86584	−	0.311353i	−1.86584	−	0.311353i	−0.986361	+	0.164595i	$0.947368\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.31747	−	1.31747i	−1.31747	−	1.31747i	−0.915773	−	0.401695i	$-0.868421\pi$
$547$	−1.31747	−	1.31747i	−1.31747	−	1.31747i	−0.401695	−	0.915773i	$-0.631579\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	3.80427i		−	3.80427i
$554$		0			0
$555$		0			0
$556$	−1.21489	−	0.723915i	−1.21489	−	0.723915i
$557$		0			0		−0.986361	−	0.164595i	$-0.947368\pi$
$557$		0			0		0.986361	+	0.164595i	$0.0526316\pi$
$558$		0			0
$559$	2.29550	+	0.683401i	2.29550	+	0.683401i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.285336	−	0.958427i	$-0.407895\pi$
$563$		0			0		−0.285336	+	0.958427i	$0.592105\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.837166	−	0.546948i	$-0.184211\pi$
$569$		0			0		−0.837166	+	0.546948i	$0.815789\pi$
$570$		0			0
$571$	−0.818303	−	1.67387i	−0.818303	−	1.67387i	−0.735724	−	0.677282i	$-0.763158\pi$
$571$	−0.818303	−	1.67387i	−0.818303	−	1.67387i	−0.0825793	−	0.996584i	$-0.526316\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.52486	−	1.29149i	−1.52486	−	1.29149i	−0.789141	−	0.614213i	$-0.789474\pi$
$577$	−1.52486	−	1.29149i	−1.52486	−	1.29149i	−0.735724	−	0.677282i	$-0.763158\pi$
$578$		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		0.837166	−	0.546948i	$-0.184211\pi$
$587$		0			0		−0.837166	+	0.546948i	$0.815789\pi$
$588$		0			0
$589$	0.491944	−	0.825588i	0.491944	−	0.825588i
$590$		0			0
$591$		0			0
$592$	0.803391			0.803391
$593$		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		−0.439197	−	0.898391i	$-0.644737\pi$
$599$		0			0		0.439197	+	0.898391i	$0.355263\pi$
$600$		0			0
$601$	−0.0820152	+	0.0102232i	−0.0820152	+	0.0102232i	−0.164595	−	0.986361i	$-0.552632\pi$
$601$	−0.0820152	+	0.0102232i	−0.0820152	+	0.0102232i	0.0825793	+	0.996584i	$0.473684\pi$
$602$		0			0
$603$		0			0
$604$	0.259777	−	1.55676i	0.259777	−	1.55676i
$605$		0			0
$606$		0			0
$607$	−0.640542	+	0.106888i	−0.640542	+	0.106888i	−0.475947	−	0.879474i	$-0.657895\pi$
$607$	−0.640542	+	0.106888i	−0.640542	+	0.106888i	−0.164595	+	0.986361i	$0.552632\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.245485	+	0.0305997i	−0.245485	+	0.0305997i	−0.245485	−	0.969400i	$-0.578947\pi$
$613$	−0.245485	+	0.0305997i	−0.245485	+	0.0305997i			1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.735724	−	0.677282i	$-0.763158\pi$
$617$		0			0		0.735724	+	0.677282i	$0.236842\pi$
$618$		0			0
$619$	0.159418	+	0.629528i	0.159418	+	0.629528i	0.996584	+	0.0825793i	$0.0263158\pi$
$619$	0.159418	+	0.629528i	0.159418	+	0.629528i	−0.837166	+	0.546948i	$0.815789\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.245485	−	0.969400i	0.245485	−	0.969400i
$626$		0			0
$627$		0			0
$628$	−1.05198	+	0.751099i	−1.05198	+	0.751099i
$629$		0			0
$630$		0			0
$631$	0.493453	+	1.12496i	0.493453	+	1.12496i	0.969400	+	0.245485i	$0.0789474\pi$
$631$	0.493453	+	1.12496i	0.493453	+	1.12496i	−0.475947	+	0.879474i	$0.657895\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.54269	−	2.52943i	3.54269	−	2.52943i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.324699	−	0.945817i	$-0.394737\pi$
$641$		0			0		−0.324699	+	0.945817i	$0.605263\pi$
$642$		0			0
$643$	0.584666	+	1.08037i	0.584666	+	1.08037i	0.986361	+	0.164595i	$0.0526316\pi$
$643$	0.584666	+	1.08037i	0.584666	+	1.08037i	−0.401695	+	0.915773i	$0.631579\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.978717	−	0.205215i	$-0.0657895\pi$
$647$		0			0		−0.978717	+	0.205215i	$0.934211\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.16459	+	0.986361i	1.16459	+	0.986361i
$653$		0			0		−0.285336	−	0.958427i	$-0.592105\pi$
$653$		0			0		0.285336	+	0.958427i	$0.407895\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.439197	−	0.898391i	$-0.644737\pi$
$659$		0			0		0.439197	+	0.898391i	$0.355263\pi$
$660$		0			0
$661$	1.61080	+	0.871720i	1.61080	+	0.871720i	0.996584	+	0.0825793i	$0.0263158\pi$
$661$	1.61080	+	0.871720i	1.61080	+	0.871720i	0.614213	+	0.789141i	$0.289474\pi$
$662$		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
$673$	0.453052	+	0.837166i	0.453052	+	0.837166i	1.00000			$0$
$673$	0.453052	+	0.837166i	0.453052	+	0.837166i	−0.546948	+	0.837166i	$0.684211\pi$
$674$		0			0
$675$		0			0
$676$	0.902136	+	1.38082i	0.902136	+	1.38082i
$677$		0			0		0.958427	−	0.285336i	$-0.0921053\pi$
$677$		0			0		−0.958427	+	0.285336i	$0.907895\pi$
$678$		0			0
$679$	0.626162	+	0.0780510i	0.626162	+	0.0780510i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.996584	−	0.0825793i	$-0.973684\pi$
$683$		0			0		0.996584	+	0.0825793i	$0.0263158\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.47145			−1.47145
$689$		0			0
$690$		0			0
$691$	−1.93218	−	0.160105i	−1.93218	−	0.160105i	−0.945817	−	0.324699i	$-0.894737\pi$
$691$	−1.93218	−	0.160105i	−1.93218	−	0.160105i	−0.986361	+	0.164595i	$0.947368\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.72209	−	0.841876i	−1.72209	−	0.841876i
$701$		0			0		−0.475947	−	0.879474i	$-0.657895\pi$
$701$		0			0		0.475947	+	0.879474i	$0.342105\pi$
$702$		0			0
$703$	0.389062	+	0.0649230i	0.389062	+	0.0649230i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.372149	−	0.624546i	−0.372149	−	0.624546i	0.614213	−	0.789141i	$-0.289474\pi$
$709$	−0.372149	−	0.624546i	−0.372149	−	0.624546i	−0.986361	+	0.164595i	$0.947368\pi$
$710$		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		−0.978717	−	0.205215i	$-0.934211\pi$
$719$		0			0		0.978717	+	0.205215i	$0.0657895\pi$
$720$		0			0
$721$	0.0903332	+	0.303424i	0.0903332	+	0.303424i
$722$		0			0
$723$		0			0
$724$	1.94582	−	0.324699i	1.94582	−	0.324699i
$725$		0			0
$726$		0			0
$727$	−1.00198	+	0.210093i	−1.00198	+	0.210093i	−0.677282	−	0.735724i	$-0.736842\pi$
$727$	−1.00198	+	0.210093i	−1.00198	+	0.210093i	−0.324699	+	0.945817i	$0.605263\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.629528	−	1.83375i	0.629528	−	1.83375i	0.0825793	−	0.996584i	$-0.473684\pi$
$733$	0.629528	−	1.83375i	0.629528	−	1.83375i	0.546948	−	0.837166i	$-0.315789\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.47595	+	0.879474i	1.47595	+	0.879474i	1.00000			$0$
$739$	1.47595	+	0.879474i	1.47595	+	0.879474i	0.475947	+	0.879474i	$0.342105\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.401695	−	0.915773i	$-0.631579\pi$
$743$		0			0		0.401695	+	0.915773i	$0.368421\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.922767	−	0.233676i	−0.922767	−	0.233676i	−0.245485	−	0.969400i	$-0.578947\pi$
$751$	−0.922767	−	0.233676i	−0.922767	−	0.233676i	−0.677282	+	0.735724i	$0.736842\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.996584	−	0.917421i	−0.996584	−	0.917421i		−	1.00000i	$-0.5\pi$
$757$	−0.996584	−	0.917421i	−0.996584	−	0.917421i	−0.996584	+	0.0825793i	$0.973684\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.992321	−	0.123693i	$-0.0394737\pi$
$761$		0			0		−0.992321	+	0.123693i	$0.960526\pi$
$762$		0			0
$763$	3.50781	+	1.53867i	3.50781	+	1.53867i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.0903332	+	1.09016i	−0.0903332	+	1.09016i	0.789141	+	0.614213i	$0.210526\pi$
$769$	−0.0903332	+	1.09016i	−0.0903332	+	1.09016i	−0.879474	+	0.475947i	$0.842105\pi$
$770$		0			0
$771$		0			0
$772$	1.22423	+	1.57289i	1.22423	+	1.57289i
$773$		0			0		0.992321	−	0.123693i	$-0.0394737\pi$
$773$		0			0		−0.992321	+	0.123693i	$0.960526\pi$
$774$		0			0
$775$	−0.859698	−	1.75854i	−0.859698	−	1.75854i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.64261	+	2.11042i	−1.64261	+	2.11042i
$785$		0			0
$786$		0			0
$787$	1.66861	−	1.09016i	1.66861	−	1.09016i	0.789141	−	0.614213i	$-0.210526\pi$
$787$	1.66861	−	1.09016i	1.66861	−	1.09016i	0.879474	−	0.475947i	$-0.157895\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.44340	−	0.563218i	1.44340	−	0.563218i
$794$		0			0
$795$		0			0
$796$	0.201334	−	0.143750i	0.201334	−	0.143750i
$797$		0			0		−0.763084	−	0.646299i	$-0.776316\pi$
$797$		0			0		0.763084	+	0.646299i	$0.223684\pi$
$798$		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	$-0.250000\pi$
$809$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$810$		0			0
$811$	0.250637	−	0.841876i	0.250637	−	0.841876i	−0.735724	−	0.677282i	$-0.763158\pi$
$811$	0.250637	−	0.841876i	0.250637	−	0.841876i	0.986361	−	0.164595i	$-0.0526316\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.712585	−	0.118909i	−0.712585	−	0.118909i
$818$		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
$823$	−0.368727	−	1.75854i	−0.368727	−	1.75854i	−0.614213	−	0.789141i	$-0.710526\pi$
$823$	−0.368727	−	1.75854i	−0.368727	−	1.75854i	0.245485	−	0.969400i	$-0.421053\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	0.821767	−	1.15096i	0.821767	−	1.15096i	−0.164595	−	0.986361i	$-0.552632\pi$
$829$	0.821767	−	1.15096i	0.821767	−	1.15096i	0.986361	−	0.164595i	$-0.0526316\pi$
$830$		0			0
$831$		0			0
$832$	−1.24207	−	1.05198i	−1.24207	−	1.05198i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.898391	−	0.439197i	$-0.855263\pi$
$839$		0			0		0.898391	+	0.439197i	$0.144737\pi$
$840$		0			0
$841$	−0.475947	+	0.879474i	−0.475947	+	0.879474i
$842$		0			0
$843$		0			0
$844$	1.37110	−	1.16126i	1.37110	−	1.16126i
$845$		0			0
$846$		0			0
$847$	1.90213	−	0.237101i	1.90213	−	0.237101i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.407279	+	1.94240i	−0.407279	+	1.94240i	−0.0825793	+	0.996584i	$0.526316\pi$
$853$	−0.407279	+	1.94240i	−0.407279	+	1.94240i	−0.324699	+	0.945817i	$0.605263\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.581077	−	0.813849i	$-0.697368\pi$
$857$		0			0		0.581077	+	0.813849i	$0.302632\pi$
$858$		0			0
$859$	−1.98636	+	0.164595i	−1.98636	+	0.164595i	−0.986361	+	0.164595i	$0.947368\pi$
$859$	−1.98636	+	0.164595i	−1.98636	+	0.164595i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.401695	−	0.915773i	$-0.631579\pi$
$863$		0			0		0.401695	+	0.915773i	$0.368421\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	−3.63730	+	0.921090i	−3.63730	+	0.921090i
$869$		0			0
$870$		0			0
$871$	−0.574327	+	1.30933i	−0.574327	+	1.30933i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.410080	−	0.0169610i	−0.410080	−	0.0169610i	−0.164595	−	0.986361i	$-0.552632\pi$
$877$	−0.410080	−	0.0169610i	−0.410080	−	0.0169610i	−0.245485	+	0.969400i	$0.578947\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.646299	−	0.763084i	$-0.276316\pi$
$881$		0			0		−0.646299	+	0.763084i	$0.723684\pi$
$882$		0			0
$883$	1.07898	−	1.65150i	1.07898	−	1.65150i	0.401695	−	0.915773i	$-0.368421\pi$
$883$	1.07898	−	1.65150i	1.07898	−	1.65150i	0.677282	−	0.735724i	$-0.263158\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.859054	−	0.511885i	$-0.171053\pi$
$887$		0			0		−0.859054	+	0.511885i	$0.828947\pi$
$888$		0			0
$889$	1.06935	−	3.11492i	1.06935	−	3.11492i
$890$		0			0
$891$		0			0
$892$	−0.207444	−	0.696792i	−0.207444	−	0.696792i
$893$		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
$907$	1.23185	+	1.13399i	1.23185	+	1.13399i	0.986361	+	0.164595i	$0.0526316\pi$
$907$	1.23185	+	1.13399i	1.23185	+	1.13399i	0.245485	+	0.969400i	$0.421053\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.324699	−	0.945817i	$-0.605263\pi$
$911$		0			0		0.324699	+	0.945817i	$0.394737\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.546948	−	0.837166i	0.546948	−	0.837166i
$917$		0			0
$918$		0			0
$919$	0.672572	+	1.53331i	0.672572	+	1.53331i	0.837166	+	0.546948i	$0.184211\pi$
$919$	0.672572	+	1.53331i	0.672572	+	1.53331i	−0.164595	+	0.986361i	$0.552632\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.544122	−	0.591074i	0.544122	−	0.591074i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.439197	−	0.898391i	$-0.355263\pi$
$929$		0			0		−0.439197	+	0.898391i	$0.644737\pi$
$930$		0			0
$931$	−0.966020	+	0.889284i	−0.966020	+	0.889284i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.08265	−	1.51635i	1.08265	−	1.51635i	0.245485	−	0.969400i	$-0.421053\pi$
$937$	1.08265	−	1.51635i	1.08265	−	1.51635i	0.837166	−	0.546948i	$-0.184211\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.614213	−	0.789141i	$-0.710526\pi$
$941$		0			0		0.614213	+	0.789141i	$0.289474\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.978717	−	0.205215i	$-0.934211\pi$
$947$		0			0		0.978717	+	0.205215i	$0.0657895\pi$
$948$		0			0
$949$	−2.53887	−	1.65873i	−2.53887	−	1.65873i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.164595	−	0.986361i	$-0.552632\pi$
$953$		0			0		0.164595	+	0.986361i	$0.447368\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2.59305	−	1.13742i	−2.59305	−	1.13742i
$962$		0			0
$963$		0			0
$964$	0.484275	+	1.91236i	0.484275	+	1.91236i
$965$		0			0
$966$		0			0
$967$	0.111859	+	0.121511i	0.111859	+	0.121511i	0.789141	−	0.614213i	$-0.210526\pi$
$967$	0.111859	+	0.121511i	0.111859	+	0.121511i	−0.677282	+	0.735724i	$0.736842\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.614213	−	0.789141i	$-0.289474\pi$
$971$		0			0		−0.614213	+	0.789141i	$0.710526\pi$
$972$		0			0
$973$	0.223860	+	2.70158i	0.223860	+	2.70158i
$974$		0			0
$975$		0			0
$976$	−0.751179	+	0.584666i	−0.751179	+	0.584666i
$977$		0			0		−0.735724	−	0.677282i	$-0.763158\pi$
$977$		0			0		0.735724	+	0.677282i	$0.236842\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.999146	−	0.0413250i	$-0.0131579\pi$
$983$		0			0		−0.999146	+	0.0413250i	$0.986842\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.516491	−	0.609820i	−0.516491	−	0.609820i
$989$		0			0
$990$		0			0
$991$	−1.08037	−	0.584666i	−1.08037	−	0.584666i	−0.164595	−	0.986361i	$-0.552632\pi$
$991$	−1.08037	−	0.584666i	−1.08037	−	0.584666i	−0.915773	+	0.401695i	$0.868421\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.121511	+	1.46642i	0.121511	+	1.46642i	0.735724	+	0.677282i	$0.236842\pi$
$997$	0.121511	+	1.46642i	0.121511	+	1.46642i	−0.614213	+	0.789141i	$0.710526\pi$
$998$		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.208.1	yes	36
3.2	odd	2	CM	2061.1.bn.a.208.1	yes	36
229.109	odd	76	inner	2061.1.bn.a.109.1	✓	36
687.338	even	76	inner	2061.1.bn.a.109.1	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2061.1.bn.a.109.1	✓	36	229.109	odd	76	inner
2061.1.bn.a.109.1	✓	36	687.338	even	76	inner
2061.1.bn.a.208.1	yes	36	1.1	even	1	trivial
2061.1.bn.a.208.1	yes	36	3.2	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(0.996584 + 0.0825793i) q^{4} +(-0.981209 - 1.64668i) q^{7} +O(q^{10})\)	\(q+(0.996584 + 0.0825793i) q^{4} +(-0.981209 - 1.64668i) q^{7} +(-1.46231 - 0.714879i) q^{13} +(0.986361 + 0.164595i) q^{16} +(0.464369 + 0.159418i) q^{19} +(0.789141 - 0.614213i) q^{25} +(-0.841876 - 1.72209i) q^{28} +(0.401695 - 1.91577i) q^{31} +(0.792434 - 0.132234i) q^{37} +(-1.45138 + 0.242192i) q^{43} +(-1.27284 + 2.35201i) q^{49} +(-1.39828 - 0.833194i) q^{52} +(-0.644701 + 0.700332i) q^{61} +(0.969400 + 0.245485i) q^{64} +(-0.0362996 - 0.877643i) q^{67} +(1.84887 + 0.230462i) q^{73} +(0.449618 + 0.197221i) q^{76} +(1.70491 + 1.01591i) q^{79} +(0.257653 + 3.10940i) q^{91} +(-0.202192 + 0.259777i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 2 q^{7}+O(q^{10}) \) 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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