LMFDB - Embedded newform 3789.1.du.a.1045.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3789,1,Mod(10,3789)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3789, base_ring=CyclotomicField(140))

chi = DirichletCharacter(H, H._module([0, 93]))

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

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

chi := DirichletCharacter("3789.10");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3789 = 3^{2} \cdot 421 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3789.du (of order $140$, degree $48$, 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:	$1.89095733287$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{140})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{46} - x^{38} - x^{36} - x^{34} - x^{32} + x^{28} + x^{26} + x^{24} + x^{22} + x^{20} + \cdots + 1 $ x^48 + x^46 - x^38 - x^36 - x^34 - x^32 + x^28 + x^26 + x^24 + x^22 + x^20 - x^16 - x^14 - x^12 - x^10 + x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{140}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{140} - \cdots)$

Embedding invariants

Embedding label			1045.1
Root			$-0.919528 - 0.393025i$ of defining polynomial
Character	$\chi$	$=$	3789.1045
Dual form			3789.1.du.a.2161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.722795 - 0.691063i) q^{4} +(-0.355676 - 1.95994i) q^{7} +O(q^{10})$	$q+(-0.722795 - 0.691063i) q^{4} +(-0.355676 - 1.95994i) q^{7} +(0.746362 - 1.46482i) q^{13} +(0.0448648 + 0.998993i) q^{16} +(1.52564 - 1.27369i) q^{19} +(-0.0448648 + 0.998993i) q^{25} +(-1.09736 + 1.66243i) q^{28} +(-0.258792 - 0.0714220i) q^{31} +(0.457642 + 1.52402i) q^{37} +(-1.39340 - 1.27362i) q^{43} +(-2.77861 + 1.04283i) q^{49} +(-1.55175 + 0.542980i) q^{52} +(0.388070 - 0.424565i) q^{61} +(0.657939 - 0.753071i) q^{64} +(-0.975592 - 0.316989i) q^{67} +(0.309017 - 1.95106i) q^{73} +(-1.98292 - 0.133692i) q^{76} +(-0.0825089 + 1.83720i) q^{79} +(-3.13642 - 0.941822i) q^{91} +(1.66243 - 0.149621i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 2 q^{13} - 2 q^{16} - 2 q^{19} + 2 q^{25} + 6 q^{28} + 4 q^{31} + 12 q^{37} - 2 q^{43} - 6 q^{49} + 2 q^{52} + 8 q^{61} - 10 q^{67} - 12 q^{73} - 2 q^{76} + 4 q^{91}+O(q^{100}) $ 48 * q - 2 * q^13 - 2 * q^16 - 2 * q^19 + 2 * q^25 + 6 * q^28 + 4 * q^31 + 12 * q^37 - 2 * q^43 - 6 * q^49 + 2 * q^52 + 8 * q^61 - 10 * q^67 - 12 * q^73 - 2 * q^76 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3789\mathbb{Z}\right)^\times$.

$n$	$1685$	$2107$
$\chi(n)$	$1$	$e\left(\frac{17}{140}\right)$

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			0		0.372294	−	0.928115i	$-0.378571\pi$
$2$		0			0		−0.372294	+	0.928115i	$0.621429\pi$
$3$		0			0
$3$		0			0
$4$	−0.722795	−	0.691063i	−0.722795	−	0.691063i
$5$		0			0		−0.691063	−	0.722795i	$-0.742857\pi$
$5$		0			0		0.691063	+	0.722795i	$0.257143\pi$
$6$		0			0
$7$	−0.355676	−	1.95994i	−0.355676	−	1.95994i	−0.266037	−	0.963963i	$-0.585714\pi$
$7$	−0.355676	−	1.95994i	−0.355676	−	1.95994i	−0.0896393	−	0.995974i	$-0.528571\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.880596	−	0.473869i	$-0.842857\pi$
$11$		0			0		0.880596	+	0.473869i	$0.157143\pi$
$12$		0			0
$13$	0.746362	−	1.46482i	0.746362	−	1.46482i	−0.134233	−	0.990950i	$-0.542857\pi$
$13$	0.746362	−	1.46482i	0.746362	−	1.46482i	0.880596	−	0.473869i	$-0.157143\pi$
$14$		0			0
$15$		0			0
$16$	0.0448648	+	0.998993i	0.0448648	+	0.998993i
$17$		0			0		−0.753071	−	0.657939i	$-0.771429\pi$
$17$		0			0		0.753071	+	0.657939i	$0.228571\pi$
$18$		0			0
$19$	1.52564	−	1.27369i	1.52564	−	1.27369i	0.691063	−	0.722795i	$-0.257143\pi$
$19$	1.52564	−	1.27369i	1.52564	−	1.27369i	0.834573	−	0.550897i	$-0.185714\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.795625	−	0.605790i	$-0.792857\pi$
$23$		0			0		0.795625	+	0.605790i	$0.207143\pi$
$24$		0			0
$25$	−0.0448648	+	0.998993i	−0.0448648	+	0.998993i
$26$		0			0
$27$		0			0
$28$	−1.09736	+	1.66243i	−1.09736	+	1.66243i
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$	−0.258792	−	0.0714220i	−0.258792	−	0.0714220i	0.134233	−	0.990950i	$-0.457143\pi$
$31$	−0.258792	−	0.0714220i	−0.258792	−	0.0714220i	−0.393025	+	0.919528i	$0.628571\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.457642	+	1.52402i	0.457642	+	1.52402i	0.809017	+	0.587785i	$0.200000\pi$
$37$	0.457642	+	1.52402i	0.457642	+	1.52402i	−0.351375	+	0.936235i	$0.614286\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.767645	−	0.640876i	$-0.221429\pi$
$41$		0			0		−0.767645	+	0.640876i	$0.778571\pi$
$42$		0			0
$43$	−1.39340	−	1.27362i	−1.39340	−	1.27362i	−0.919528	−	0.393025i	$-0.871429\pi$
$43$	−1.39340	−	1.27362i	−1.39340	−	1.27362i	−0.473869	−	0.880596i	$-0.657143\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.0224381	−	0.999748i	$-0.492857\pi$
$47$		0			0		−0.0224381	+	0.999748i	$0.507143\pi$
$48$		0			0
$49$	−2.77861	+	1.04283i	−2.77861	+	1.04283i
$50$		0			0
$51$		0			0
$52$	−1.55175	+	0.542980i	−1.55175	+	0.542980i
$53$		0			0		−0.822002	−	0.569484i	$-0.807143\pi$
$53$		0			0		0.822002	+	0.569484i	$0.192857\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.969690	−	0.244340i	$-0.0785714\pi$
$59$		0			0		−0.969690	+	0.244340i	$0.921429\pi$
$60$		0			0
$61$	0.388070	−	0.424565i	0.388070	−	0.424565i	−0.512899	−	0.858449i	$-0.671429\pi$
$61$	0.388070	−	0.424565i	0.388070	−	0.424565i	0.900969	+	0.433884i	$0.142857\pi$
$62$		0			0
$63$		0			0
$64$	0.657939	−	0.753071i	0.657939	−	0.753071i
$65$		0			0
$66$		0			0
$67$	−0.975592	−	0.316989i	−0.975592	−	0.316989i	−0.222521	−	0.974928i	$-0.571429\pi$
$67$	−0.975592	−	0.316989i	−0.975592	−	0.316989i	−0.753071	+	0.657939i	$0.771429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.969690	−	0.244340i	$-0.921429\pi$
$71$		0			0		0.969690	+	0.244340i	$0.0785714\pi$
$72$		0			0
$73$	0.309017	−	1.95106i	0.309017	−	1.95106i		−	1.00000i	$-0.5\pi$
$73$	0.309017	−	1.95106i	0.309017	−	1.95106i	0.309017	−	0.951057i	$-0.400000\pi$
$74$		0			0
$75$		0			0
$76$	−1.98292	−	0.133692i	−1.98292	−	0.133692i
$77$		0			0
$78$		0			0
$79$	−0.0825089	+	1.83720i	−0.0825089	+	1.83720i	0.351375	+	0.936235i	$0.385714\pi$
$79$	−0.0825089	+	1.83720i	−0.0825089	+	1.83720i	−0.433884	+	0.900969i	$0.642857\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.244340	−	0.969690i	$-0.578571\pi$
$83$		0			0		0.244340	+	0.969690i	$0.421429\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.869741	−	0.493508i	$-0.835714\pi$
$89$		0			0		0.869741	+	0.493508i	$0.164286\pi$
$90$		0			0
$91$	−3.13642	−	0.941822i	−3.13642	−	0.941822i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.66243	−	0.149621i	1.66243	−	0.149621i	0.781831	−	0.623490i	$-0.214286\pi$
$97$	1.66243	−	0.149621i	1.66243	−	0.149621i	0.880596	+	0.473869i	$0.157143\pi$
$98$		0			0
$99$		0			0
$100$	0.722795	−	0.691063i	0.722795	−	0.691063i
$101$		0			0		−0.0448648	−	0.998993i	$-0.514286\pi$
$101$		0			0		0.0448648	+	0.998993i	$0.485714\pi$
$102$		0			0
$103$	−0.635928	−	1.48783i	−0.635928	−	1.48783i	−0.858449	−	0.512899i	$-0.828571\pi$
$103$	−0.635928	−	1.48783i	−0.635928	−	1.48783i	0.222521	−	0.974928i	$-0.428571\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$107$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$108$		0			0
$109$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	0.623490	+	0.781831i	$0.285714\pi$
$109$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	−0.900969	+	0.433884i	$0.857143\pi$
$110$		0			0
$111$		0			0
$112$	1.94201	−	0.443250i	1.94201	−	0.443250i
$113$		0			0		−0.473869	−	0.880596i	$-0.657143\pi$
$113$		0			0		0.473869	+	0.880596i	$0.342857\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.550897	+	0.834573i	0.550897	+	0.834573i
$122$		0			0
$123$		0			0
$124$	0.137696	+	0.230465i	0.137696	+	0.230465i
$125$		0			0
$126$		0			0
$127$	−0.372340	+	1.81850i	−0.372340	+	1.81850i	0.178557	+	0.983930i	$0.442857\pi$
$127$	−0.372340	+	1.81850i	−0.372340	+	1.81850i	−0.550897	+	0.834573i	$0.685714\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.393025	−	0.919528i	$-0.371429\pi$
$131$		0			0		−0.393025	+	0.919528i	$0.628571\pi$
$132$		0			0
$133$	−3.03899	−	2.53713i	−3.03899	−	2.53713i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$137$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$138$		0			0
$139$	−0.293118	+	0.444054i	−0.293118	+	0.444054i	−0.951057	−	0.309017i	$-0.900000\pi$
$139$	−0.293118	+	0.444054i	−0.293118	+	0.444054i	0.657939	+	0.753071i	$0.271429\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.722412	−	1.41781i	0.722412	−	1.41781i
$149$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$149$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$150$		0			0
$151$	0.129390	−	0.816934i	0.129390	−	0.816934i	−0.834573	−	0.550897i	$-0.814286\pi$
$151$	0.129390	−	0.816934i	0.129390	−	0.816934i	0.963963	−	0.266037i	$-0.0857143\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.435418	+	1.72800i	−0.435418	+	1.72800i	0.222521	+	0.974928i	$0.428571\pi$
$157$	−0.435418	+	1.72800i	−0.435418	+	1.72800i	−0.657939	+	0.753071i	$0.728571\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.73700	−	0.479382i	1.73700	−	0.479382i	0.753071	−	0.657939i	$-0.228571\pi$
$163$	1.73700	−	0.479382i	1.73700	−	0.479382i	0.983930	+	0.178557i	$0.0571429\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.550897	−	0.834573i	$-0.314286\pi$
$167$		0			0		−0.550897	+	0.834573i	$0.685714\pi$
$168$		0			0
$169$	−1.00085	−	1.37755i	−1.00085	−	1.37755i
$170$		0			0
$171$		0			0
$172$	0.126988	+	1.88349i	0.126988	+	1.88349i
$173$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$173$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$174$		0			0
$175$	1.97392	−	0.267386i	1.97392	−	0.267386i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.200589	−	0.979675i	$-0.564286\pi$
$179$		0			0		0.200589	+	0.979675i	$0.435714\pi$
$180$		0			0
$181$	−1.06457	+	0.773453i	−1.06457	+	0.773453i	−0.974928	−	0.222521i	$-0.928571\pi$
$181$	−1.06457	+	0.773453i	−1.06457	+	0.773453i	−0.0896393	+	0.995974i	$0.528571\pi$
$182$		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		0.266037	−	0.963963i	$-0.414286\pi$
$191$		0			0		−0.266037	+	0.963963i	$0.585714\pi$
$192$		0			0
$193$	−0.259994	+	1.03181i	−0.259994	+	1.03181i	0.691063	+	0.722795i	$0.257143\pi$
$193$	−0.259994	+	1.03181i	−0.259994	+	1.03181i	−0.951057	+	0.309017i	$0.900000\pi$
$194$		0			0
$195$		0			0
$196$	2.72903	+	1.16644i	2.72903	+	1.16644i
$197$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$197$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$198$		0			0
$199$	1.33232	+	0.367696i	1.33232	+	0.367696i	0.858449	−	0.512899i	$-0.171429\pi$
$199$	1.33232	+	0.367696i	1.33232	+	0.367696i	0.473869	+	0.880596i	$0.342857\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.49683	+	0.679892i	1.49683	+	0.679892i
$209$		0			0
$210$		0			0
$211$	−0.124867	+	0.0500876i	−0.124867	+	0.0500876i	−0.433884	−	0.900969i	$-0.642857\pi$
$211$	−0.124867	+	0.0500876i	−0.124867	+	0.0500876i	0.309017	+	0.951057i	$0.400000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.0479366	+	0.532619i	−0.0479366	+	0.532619i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.467085	−	1.33485i	−0.467085	−	1.33485i	−0.900969	−	0.433884i	$-0.857143\pi$
$223$	−0.467085	−	1.33485i	−0.467085	−	1.33485i	0.433884	−	0.900969i	$-0.357143\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.957751	−	0.287599i	$-0.907143\pi$
$227$		0			0		0.957751	+	0.287599i	$0.0928571\pi$
$228$		0			0
$229$	−0.901310	−	1.13021i	−0.901310	−	1.13021i	−0.990950	−	0.134233i	$-0.957143\pi$
$229$	−0.901310	−	1.13021i	−0.901310	−	1.13021i	0.0896393	−	0.995974i	$-0.471429\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$239$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$240$		0			0
$241$	0.668355	−	1.78082i	0.668355	−	1.78082i	0.0448648	−	0.998993i	$-0.485714\pi$
$241$	0.668355	−	1.78082i	0.668355	−	1.78082i	0.623490	−	0.781831i	$-0.285714\pi$
$242$		0			0
$243$		0			0
$244$	−0.573896	+	0.0386930i	−0.573896	+	0.0386930i
$245$		0			0
$246$		0			0
$247$	−0.727050	−	3.18542i	−0.727050	−	3.18542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$251$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.995974	+	0.0896393i	−0.995974	+	0.0896393i
$257$		0			0		0.0672690	−	0.997735i	$-0.478571\pi$
$257$		0			0		−0.0672690	+	0.997735i	$0.521429\pi$
$258$		0			0
$259$	2.82421	−	1.43901i	2.82421	−	1.43901i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.413559	−	0.910478i	$-0.364286\pi$
$263$		0			0		−0.413559	+	0.910478i	$0.635714\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.486094	+	0.903314i	0.486094	+	0.903314i
$269$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$269$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$270$		0			0
$271$	1.64028	+	1.13639i	1.64028	+	1.13639i	0.858449	+	0.512899i	$0.171429\pi$
$271$	1.64028	+	1.13639i	1.64028	+	1.13639i	0.781831	+	0.623490i	$0.214286\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$277$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.178557	−	0.983930i	$-0.557143\pi$
$281$		0			0		0.178557	+	0.983930i	$0.442857\pi$
$282$		0			0
$283$	−0.702042	−	1.87058i	−0.702042	−	1.87058i	−0.393025	−	0.919528i	$-0.628571\pi$
$283$	−0.702042	−	1.87058i	−0.702042	−	1.87058i	−0.309017	−	0.951057i	$-0.600000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.134233	+	0.990950i	0.134233	+	0.990950i
$290$		0			0
$291$		0			0
$292$	−1.57166	+	1.19666i	−1.57166	+	1.19666i
$293$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$293$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−2.00062	+	3.18397i	−2.00062	+	3.18397i
$302$		0			0
$303$		0			0
$304$	1.34086	+	1.46696i	1.34086	+	1.46696i
$305$		0			0
$306$		0			0
$307$	−0.127218	−	0.939160i	−0.127218	−	0.939160i	−0.936235	−	0.351375i	$-0.885714\pi$
$307$	−0.127218	−	0.939160i	−0.127218	−	0.939160i	0.809017	−	0.587785i	$-0.200000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$311$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$312$		0			0
$313$	−1.80499	+	0.498146i	−1.80499	+	0.498146i	−0.995974	−	0.0896393i	$-0.971429\pi$
$313$	−1.80499	+	0.498146i	−1.80499	+	0.498146i	−0.809017	+	0.587785i	$0.800000\pi$
$314$		0			0
$315$		0			0
$316$	1.32926	−	1.27090i	1.32926	−	1.27090i
$317$		0			0		−0.998993	−	0.0448648i	$-0.985714\pi$
$317$		0			0		0.998993	+	0.0448648i	$0.0142857\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.42986	+	0.811330i	1.42986	+	0.811330i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.0302766	+	1.34900i	0.0302766	+	1.34900i	0.753071	+	0.657939i	$0.228571\pi$
$331$	0.0302766	+	1.34900i	0.0302766	+	1.34900i	−0.722795	+	0.691063i	$0.757143\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.627218	+	0.599682i	0.627218	+	0.599682i	0.936235	−	0.351375i	$-0.114286\pi$
$337$	0.627218	+	0.599682i	0.627218	+	0.599682i	−0.309017	+	0.951057i	$0.600000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	2.01050	+	3.36501i	2.01050	+	3.36501i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.0896393	−	0.995974i	$-0.471429\pi$
$347$		0			0		−0.0896393	+	0.995974i	$0.528571\pi$
$348$		0			0
$349$	0.0447745	−	0.00301877i	0.0447745	−	0.00301877i	−0.0448648	−	0.998993i	$-0.514286\pi$
$349$	0.0447745	−	0.00301877i	0.0447745	−	0.00301877i	0.0896393	+	0.995974i	$0.471429\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.995974	−	0.0896393i	$-0.0285714\pi$
$353$		0			0		−0.995974	+	0.0896393i	$0.971429\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.795625	−	0.605790i	$-0.792857\pi$
$359$		0			0		0.795625	+	0.605790i	$0.207143\pi$
$360$		0			0
$361$	0.526717	−	2.90245i	0.526717	−	2.90245i
$362$		0			0
$363$		0			0
$364$	1.61613	+	2.84820i	1.61613	+	2.84820i
$365$		0			0
$366$		0			0
$367$	0.388806	−	0.296038i	0.388806	−	0.296038i	−0.393025	−	0.919528i	$-0.628571\pi$
$367$	0.388806	−	0.296038i	0.388806	−	0.296038i	0.781831	+	0.623490i	$0.214286\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.93221	+	0.261736i	1.93221	+	0.261736i	0.995974	−	0.0896393i	$-0.0285714\pi$
$373$	1.93221	+	0.261736i	1.93221	+	0.261736i	0.936235	+	0.351375i	$0.114286\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.453552	+	0.181933i	0.453552	+	0.181933i	0.587785	−	0.809017i	$-0.300000\pi$
$379$	0.453552	+	0.181933i	0.453552	+	0.181933i	−0.134233	+	0.990950i	$0.542857\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$383$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.30499	−	1.04070i	−1.30499	−	1.04070i
$389$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$389$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.880460	+	0.221856i	−0.880460	+	0.221856i	−0.657939	−	0.753071i	$-0.728571\pi$
$397$	−0.880460	+	0.221856i	−0.880460	+	0.221856i	−0.222521	+	0.974928i	$0.571429\pi$
$398$		0			0
$399$		0			0
$400$	−1.00000			−1.00000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−0.297773	+	0.325776i	−0.297773	+	0.325776i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.26503	−	0.919098i	1.26503	−	0.919098i	0.266037	−	0.963963i	$-0.414286\pi$
$409$	1.26503	−	0.919098i	1.26503	−	0.919098i	0.998993	+	0.0448648i	$0.0142857\pi$
$410$		0			0
$411$		0			0
$412$	−0.568536	+	1.51486i	−0.568536	+	1.51486i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.928115	−	0.372294i	$-0.878571\pi$
$419$		0			0		0.928115	+	0.372294i	$0.121429\pi$
$420$		0			0
$421$	−0.178557	+	0.983930i	−0.178557	+	0.983930i
$421$	−0.178557	+	0.983930i	−0.178557	+	0.983930i
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.970148	−	0.609584i	−0.970148	−	0.609584i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.372294	−	0.928115i	$-0.621429\pi$
$431$		0			0		0.372294	+	0.928115i	$0.378571\pi$
$432$		0			0
$433$	−0.557066	−	0.766736i	−0.557066	−	0.766736i	0.433884	−	0.900969i	$-0.357143\pi$
$433$	−0.557066	−	0.766736i	−0.557066	−	0.766736i	−0.990950	+	0.134233i	$0.957143\pi$
$434$		0			0
$435$		0			0
$436$	1.04070	−	0.686957i	1.04070	−	0.686957i
$437$		0			0
$438$		0			0
$439$	0.296118	+	0.270664i	0.296118	+	0.270664i	0.809017	−	0.587785i	$-0.200000\pi$
$439$	0.296118	+	0.270664i	0.296118	+	0.270664i	−0.512899	+	0.858449i	$0.671429\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.640876	−	0.767645i	$-0.721429\pi$
$443$		0			0		0.640876	+	0.767645i	$0.278571\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.70999	−	1.02167i	−1.70999	−	1.02167i
$449$		0			0		−0.834573	−	0.550897i	$-0.814286\pi$
$449$		0			0		0.834573	+	0.550897i	$0.185714\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.478050	−	0.992682i	−0.478050	−	0.992682i	−0.990950	−	0.134233i	$-0.957143\pi$
$457$	−0.478050	−	0.992682i	−0.478050	−	0.992682i	0.512899	−	0.858449i	$-0.328571\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.372294	−	0.928115i	$-0.378571\pi$
$461$		0			0		−0.372294	+	0.928115i	$0.621429\pi$
$462$		0			0
$463$	−0.277205	+	0.691063i	−0.277205	+	0.691063i	0.722795	+	0.691063i	$0.242857\pi$
$463$	−0.277205	+	0.691063i	−0.277205	+	0.691063i	−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.512899	−	0.858449i	$-0.671429\pi$
$467$		0			0		0.512899	+	0.858449i	$0.328571\pi$
$468$		0			0
$469$	−0.274284	+	2.02485i	−0.274284	+	2.02485i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.20396	+	1.58124i	1.20396	+	1.58124i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.393025	−	0.919528i	$-0.628571\pi$
$479$		0			0		0.393025	+	0.919528i	$0.371429\pi$
$480$		0			0
$481$	2.57398	+	0.467108i	2.57398	+	0.467108i
$482$		0			0
$483$		0			0
$484$	0.178557	−	0.983930i	0.178557	−	0.983930i
$485$		0			0
$486$		0			0
$487$	−1.11479	+	0.632555i	−1.11479	+	0.632555i	−0.936235	−	0.351375i	$-0.885714\pi$
$487$	−1.11479	+	0.632555i	−1.11479	+	0.632555i	−0.178557	+	0.983930i	$0.557143\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$491$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.0597394	−	0.261736i	0.0597394	−	0.261736i
$497$		0			0
$498$		0			0
$499$	1.60742	−	0.960388i	1.60742	−	0.960388i	0.623490	−	0.781831i	$-0.285714\pi$
$499$	1.60742	−	0.960388i	1.60742	−	0.960388i	0.983930	−	0.178557i	$-0.0571429\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$503$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1.52582	−	1.05709i	1.52582	−	1.05709i
$509$		0			0		0.0672690	−	0.997735i	$-0.478571\pi$
$509$		0			0		−0.0672690	+	0.997735i	$0.521429\pi$
$510$		0			0
$511$	−3.93386	+	0.0882904i	−3.93386	+	0.0882904i
$512$		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		−0.691063	−	0.722795i	$-0.742857\pi$
$521$		0			0		0.691063	+	0.722795i	$0.257143\pi$
$522$		0			0
$523$	1.85744	−	0.557764i	1.85744	−	0.557764i	0.858449	−	0.512899i	$-0.171429\pi$
$523$	1.85744	−	0.557764i	1.85744	−	0.557764i	0.998993	−	0.0448648i	$-0.0142857\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.266037	+	0.963963i	0.266037	+	0.963963i
$530$		0			0
$531$		0			0
$532$	0.443250	+	3.93395i	0.443250	+	3.93395i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.835731	−	0.525124i	−0.835731	−	0.525124i	0.0448648	−	0.998993i	$-0.485714\pi$
$541$	−0.835731	−	0.525124i	−0.835731	−	0.525124i	−0.880596	+	0.473869i	$0.842857\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.785406	−	1.24997i	0.785406	−	1.24997i	−0.178557	−	0.983930i	$-0.557143\pi$
$547$	0.785406	−	1.24997i	0.785406	−	1.24997i	0.963963	−	0.266037i	$-0.0857143\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	3.63015	−	0.491737i	3.63015	−	0.491737i
$554$		0			0
$555$		0			0
$556$	0.518733	−	0.118398i	0.518733	−	0.118398i
$557$		0			0		0.287599	−	0.957751i	$-0.407143\pi$
$557$		0			0		−0.287599	+	0.957751i	$0.592857\pi$
$558$		0			0
$559$	−2.90560	+	1.09049i	−2.90560	+	1.09049i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$563$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$569$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$570$		0			0
$571$	0.562091	−	0.811330i	0.562091	−	0.811330i	−0.433884	−	0.900969i	$-0.642857\pi$
$571$	0.562091	−	0.811330i	0.562091	−	0.811330i	0.995974	+	0.0896393i	$0.0285714\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.67260	−	0.264914i	1.67260	−	0.264914i	0.753071	−	0.657939i	$-0.228571\pi$
$577$	1.67260	−	0.264914i	1.67260	−	0.264914i	0.919528	+	0.393025i	$0.128571\pi$
$578$		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		0.134233	−	0.990950i	$-0.457143\pi$
$587$		0			0		−0.134233	+	0.990950i	$0.542857\pi$
$588$		0			0
$589$	−0.485792	+	0.220657i	−0.485792	+	0.220657i
$590$		0			0
$591$		0			0
$592$	−1.50195	+	0.525556i	−1.50195	+	0.525556i
$593$		0			0		0.997735	−	0.0672690i	$-0.0214286\pi$
$593$		0			0		−0.997735	+	0.0672690i	$0.978571\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$599$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$600$		0			0
$601$	−0.735927	−	0.276198i	−0.735927	−	0.276198i	−0.0448648	−	0.998993i	$-0.514286\pi$
$601$	−0.735927	−	0.276198i	−0.735927	−	0.276198i	−0.691063	+	0.722795i	$0.742857\pi$
$602$		0			0
$603$		0			0
$604$	−0.658075	+	0.501059i	−0.658075	+	0.501059i
$605$		0			0
$606$		0			0
$607$	−0.211083	−	0.230935i	−0.211083	−	0.230935i	0.623490	−	0.781831i	$-0.285714\pi$
$607$	−0.211083	−	0.230935i	−0.211083	−	0.230935i	−0.834573	+	0.550897i	$0.814286\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.0701535	−	0.0559455i	0.0701535	−	0.0559455i	−0.587785	−	0.809017i	$-0.700000\pi$
$613$	0.0701535	−	0.0559455i	0.0701535	−	0.0559455i	0.657939	+	0.753071i	$0.271429\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.393025	−	0.919528i	$-0.371429\pi$
$617$		0			0		−0.393025	+	0.919528i	$0.628571\pi$
$618$		0			0
$619$	1.39340	−	0.487571i	1.39340	−	0.487571i	0.473869	−	0.880596i	$-0.342857\pi$
$619$	1.39340	−	0.487571i	1.39340	−	0.487571i	0.919528	+	0.393025i	$0.128571\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.995974	−	0.0896393i	−0.995974	−	0.0896393i
$626$		0			0
$627$		0			0
$628$	1.50887	−	0.948088i	1.50887	−	0.948088i
$629$		0			0
$630$		0			0
$631$	0.0500876	+	0.124867i	0.0500876	+	0.124867i	0.951057	−	0.309017i	$-0.100000\pi$
$631$	0.0500876	+	0.124867i	0.0500876	+	0.124867i	−0.900969	+	0.433884i	$0.857143\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.546294	+	4.84849i	−0.546294	+	4.84849i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.880596	−	0.473869i	$-0.842857\pi$
$641$		0			0		0.880596	+	0.473869i	$0.157143\pi$
$642$		0			0
$643$	0.209118	−	0.757723i	0.209118	−	0.757723i	−0.781831	−	0.623490i	$-0.785714\pi$
$643$	0.209118	−	0.757723i	0.209118	−	0.757723i	0.990950	−	0.134233i	$-0.0428571\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$647$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.58678	−	0.853882i	−1.58678	−	0.853882i
$653$		0			0		0.351375	−	0.936235i	$-0.385714\pi$
$653$		0			0		−0.351375	+	0.936235i	$0.614286\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.795625	−	0.605790i	$-0.207143\pi$
$659$		0			0		−0.795625	+	0.605790i	$0.792857\pi$
$660$		0			0
$661$	0.261589	+	0.360046i	0.261589	+	0.360046i	0.919528	−	0.393025i	$-0.128571\pi$
$661$	0.261589	+	0.360046i	0.261589	+	0.360046i	−0.657939	+	0.753071i	$0.728571\pi$
$662$		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
$673$	−0.360046	+	0.261589i	−0.360046	+	0.261589i	−0.753071	−	0.657939i	$-0.771429\pi$
$673$	−0.360046	+	0.261589i	−0.360046	+	0.261589i	0.393025	+	0.919528i	$0.371429\pi$
$674$		0			0
$675$		0			0
$676$	−0.228566	+	1.68734i	−0.228566	+	1.68734i
$677$		0			0		−0.0896393	−	0.995974i	$-0.528571\pi$
$677$		0			0		0.0896393	+	0.995974i	$0.471429\pi$
$678$		0			0
$679$	−0.884534	−	3.20504i	−0.884534	−	3.20504i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	1.20982	−	1.44913i	1.20982	−	1.44913i
$689$		0			0
$690$		0			0
$691$	1.33273	+	0.211083i	1.33273	+	0.211083i	0.781831	−	0.623490i	$-0.214286\pi$
$691$	1.33273	+	0.211083i	1.33273	+	0.211083i	0.550897	+	0.834573i	$0.314286\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.61152	−	1.17084i	−1.61152	−	1.17084i
$701$		0			0		−0.979675	−	0.200589i	$-0.935714\pi$
$701$		0			0		0.979675	+	0.200589i	$0.0642857\pi$
$702$		0			0
$703$	2.63933	+	1.74220i	2.63933	+	1.74220i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.681933	−	0.816823i	0.681933	−	0.816823i	−0.309017	−	0.951057i	$-0.600000\pi$
$709$	0.681933	−	0.816823i	0.681933	−	0.816823i	0.990950	+	0.134233i	$0.0428571\pi$
$710$		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		−0.493508	−	0.869741i	$-0.664286\pi$
$719$		0			0		0.493508	+	0.869741i	$0.335714\pi$
$720$		0			0
$721$	−2.68986	+	1.77556i	−2.68986	+	1.77556i
$722$		0			0
$723$		0			0
$724$	1.30397	+	0.176635i	1.30397	+	0.176635i
$725$		0			0
$726$		0			0
$727$	−1.90346	+	0.214468i	−1.90346	+	0.214468i	−0.983930	−	0.178557i	$-0.942857\pi$
$727$	−1.90346	+	0.214468i	−1.90346	+	0.214468i	−0.919528	+	0.393025i	$0.871429\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.07417	+	0.245172i	1.07417	+	0.245172i	0.722795	−	0.691063i	$-0.242857\pi$
$733$	1.07417	+	0.245172i	1.07417	+	0.245172i	0.351375	+	0.936235i	$0.385714\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.871471	+	0.372484i	−0.871471	+	0.372484i	−0.781831	−	0.623490i	$-0.785714\pi$
$739$	−0.871471	+	0.372484i	−0.871471	+	0.372484i	−0.0896393	+	0.995974i	$0.528571\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.753071	−	0.657939i	$-0.228571\pi$
$743$		0			0		−0.753071	+	0.657939i	$0.771429\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.368631	−	1.22760i	0.368631	−	1.22760i	−0.550897	−	0.834573i	$-0.685714\pi$
$751$	0.368631	−	1.22760i	0.368631	−	1.22760i	0.919528	−	0.393025i	$-0.128571\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.90020	−	0.0853380i	1.90020	−	0.0853380i	0.936235	−	0.351375i	$-0.114286\pi$
$757$	1.90020	−	0.0853380i	1.90020	−	0.0853380i	0.963963	+	0.266037i	$0.0857143\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.753071	−	0.657939i	$-0.771429\pi$
$761$		0			0		0.753071	+	0.657939i	$0.228571\pi$
$762$		0			0
$763$	2.48142	+	0.111441i	2.48142	+	0.111441i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.26597	−	0.200510i	−1.26597	−	0.200510i	−0.512899	−	0.858449i	$-0.671429\pi$
$769$	−1.26597	−	0.200510i	−1.26597	−	0.200510i	−0.753071	+	0.657939i	$0.771429\pi$
$770$		0			0
$771$		0			0
$772$	0.900969	−	0.566116i	0.900969	−	0.566116i
$773$		0			0		0.738119	−	0.674671i	$-0.235714\pi$
$773$		0			0		−0.738119	+	0.674671i	$0.764286\pi$
$774$		0			0
$775$	0.0829607	−	0.255327i	0.0829607	−	0.255327i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.16644	−	2.72903i	−1.16644	−	2.72903i
$785$		0			0
$786$		0			0
$787$	0.677425	−	0.186957i	0.677425	−	0.186957i	0.0896393	−	0.995974i	$-0.471429\pi$
$787$	0.677425	−	0.186957i	0.677425	−	0.186957i	0.587785	+	0.809017i	$0.300000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.332270	−	0.885331i	−0.332270	−	0.885331i
$794$		0			0
$795$		0			0
$796$	−0.708891	−	1.18648i	−0.708891	−	1.18648i
$797$		0			0		0.919528	−	0.393025i	$-0.128571\pi$
$797$		0			0		−0.919528	+	0.393025i	$0.871429\pi$
$798$		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		0.623490	−	0.781831i	$-0.285714\pi$
$809$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$810$		0			0
$811$	0.0238714	−	0.0864961i	0.0238714	−	0.0864961i	−0.951057	−	0.309017i	$-0.900000\pi$
$811$	0.0238714	−	0.0864961i	0.0238714	−	0.0864961i	0.974928	+	0.222521i	$0.0714286\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−3.74802	−	0.168324i	−3.74802	−	0.168324i
$818$		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
$823$	−0.200510	−	0.240172i	−0.200510	−	0.240172i	0.657939	−	0.753071i	$-0.271429\pi$
$823$	−0.200510	−	0.240172i	−0.200510	−	0.240172i	−0.858449	+	0.512899i	$0.828571\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.834573	−	0.550897i	$-0.185714\pi$
$827$		0			0		−0.834573	+	0.550897i	$0.814286\pi$
$828$		0			0
$829$	−0.278768	−	0.142040i	−0.278768	−	0.142040i	0.309017	−	0.951057i	$-0.400000\pi$
$829$	−0.278768	−	0.142040i	−0.278768	−	0.142040i	−0.587785	+	0.809017i	$0.700000\pi$
$830$		0			0
$831$		0			0
$832$	−0.612052	−	1.52582i	−0.612052	−	1.52582i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.983930	−	0.178557i	$-0.942857\pi$
$839$		0			0		0.983930	+	0.178557i	$0.0571429\pi$
$840$		0			0
$841$			1.00000i			1.00000i
$842$		0			0
$843$		0			0
$844$	0.124867	+	0.0500876i	0.124867	+	0.0500876i
$845$		0			0
$846$		0			0
$847$	1.43977	−	1.37656i	1.43977	−	1.37656i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.623553	−	1.15876i	0.623553	−	1.15876i	−0.351375	−	0.936235i	$-0.614286\pi$
$853$	0.623553	−	1.15876i	0.623553	−	1.15876i	0.974928	−	0.222521i	$-0.0714286\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.550897	−	0.834573i	$-0.685714\pi$
$857$		0			0		0.550897	+	0.834573i	$0.314286\pi$
$858$		0			0
$859$	0.990950	−	1.13423i	0.990950	−	1.13423i		−	1.00000i	$-0.5\pi$
$859$	0.990950	−	1.13423i	0.990950	−	1.13423i	0.990950	−	0.134233i	$-0.0428571\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	0.402721	−	0.351847i	0.402721	−	0.351847i
$869$		0			0
$870$		0			0
$871$	−1.19248	+	1.19248i	−1.19248	+	1.19248i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.61640	+	1.17439i	1.61640	+	1.17439i	0.834573	+	0.550897i	$0.185714\pi$
$877$	1.61640	+	1.17439i	1.61640	+	1.17439i	0.781831	+	0.623490i	$0.214286\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.795625	−	0.605790i	$-0.792857\pi$
$881$		0			0		0.795625	+	0.605790i	$0.207143\pi$
$882$		0			0
$883$	−1.14205	−	1.36795i	−1.14205	−	1.36795i	−0.919528	−	0.393025i	$-0.871429\pi$
$883$	−1.14205	−	1.36795i	−1.14205	−	1.36795i	−0.222521	−	0.974928i	$-0.571429\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.393025	−	0.919528i	$-0.628571\pi$
$887$		0			0		0.393025	+	0.919528i	$0.371429\pi$
$888$		0			0
$889$	3.69658	+	0.0829651i	3.69658	+	0.0829651i
$890$		0			0
$891$		0			0
$892$	−0.584860	+	1.28761i	−0.584860	+	1.28761i
$893$		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
$907$	−0.0129063	−	0.0429801i	−0.0129063	−	0.0429801i	0.951057	−	0.309017i	$-0.100000\pi$
$907$	−0.0129063	−	0.0429801i	−0.0129063	−	0.0429801i	−0.963963	+	0.266037i	$0.914286\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.674671	−	0.738119i	$-0.735714\pi$
$911$		0			0		0.674671	+	0.738119i	$0.264286\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.129582	+	1.43977i	−0.129582	+	1.43977i
$917$		0			0
$918$		0			0
$919$	−0.339635	−	0.110354i	−0.339635	−	0.110354i	0.134233	−	0.990950i	$-0.457143\pi$
$919$	−0.339635	−	0.110354i	−0.339635	−	0.110354i	−0.473869	+	0.880596i	$0.657143\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.54302	+	0.388806i	−1.54302	+	0.388806i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.569484	−	0.822002i	$-0.692857\pi$
$929$		0			0		0.569484	+	0.822002i	$0.307143\pi$
$930$		0			0
$931$	−2.91091	+	5.13008i	−2.91091	+	5.13008i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.0926077	+	0.821916i	−0.0926077	+	0.821916i	0.858449	+	0.512899i	$0.171429\pi$
$937$	−0.0926077	+	0.821916i	−0.0926077	+	0.821916i	−0.951057	+	0.309017i	$0.900000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.657939	−	0.753071i	$-0.728571\pi$
$941$		0			0		0.657939	+	0.753071i	$0.271429\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.722795	−	0.691063i	$-0.242857\pi$
$947$		0			0		−0.722795	+	0.691063i	$0.757143\pi$
$948$		0			0
$949$	−2.62730	−	1.90885i	−2.62730	−	1.90885i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$953$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.796577	−	0.475932i	−0.796577	−	0.475932i
$962$		0			0
$963$		0			0
$964$	−1.71374	+	0.825296i	−1.71374	+	0.825296i
$965$		0			0
$966$		0			0
$967$	−0.350072	−	0.400690i	−0.350072	−	0.400690i	0.550897	−	0.834573i	$-0.314286\pi$
$967$	−0.350072	−	0.400690i	−0.350072	−	0.400690i	−0.900969	+	0.433884i	$0.857143\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.795625	−	0.605790i	$-0.207143\pi$
$971$		0			0		−0.795625	+	0.605790i	$0.792857\pi$
$972$		0			0
$973$	0.974574	+	0.416553i	0.974574	+	0.416553i
$974$		0			0
$975$		0			0
$976$	0.441548	+	0.368631i	0.441548	+	0.368631i
$977$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$977$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.200589	−	0.979675i	$-0.435714\pi$
$983$		0			0		−0.200589	+	0.979675i	$0.564286\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.67581	+	2.80484i	−1.67581	+	2.80484i
$989$		0			0
$990$		0			0
$991$	−0.820436	+	1.02879i	−0.820436	+	1.02879i	0.178557	+	0.983930i	$0.442857\pi$
$991$	−0.820436	+	1.02879i	−0.820436	+	1.02879i	−0.998993	+	0.0448648i	$0.985714\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.03520	−	0.557066i	1.03520	−	0.557066i	0.134233	−	0.990950i	$-0.457143\pi$
$997$	1.03520	−	0.557066i	1.03520	−	0.557066i	0.900969	+	0.433884i	$0.142857\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3789.1.du.a.1045.1	✓	48
3.2	odd	2	CM	3789.1.du.a.1045.1	✓	48
421.56	odd	140	inner	3789.1.du.a.2161.1	yes	48
1263.56	even	140	inner	3789.1.du.a.2161.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3789.1.du.a.1045.1	✓	48	1.1	even	1	trivial
3789.1.du.a.1045.1	✓	48	3.2	odd	2	CM
3789.1.du.a.2161.1	yes	48	421.56	odd	140	inner
3789.1.du.a.2161.1	yes	48	1263.56	even	140	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:	\( N \)	\(=\)	\( 3789 = 3^{2} \cdot 421 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3789.du (of order \(140\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.722795 - 0.691063i) q^{4} +(-0.355676 - 1.95994i) q^{7} +O(q^{10})\)	\(q+(-0.722795 - 0.691063i) q^{4} +(-0.355676 - 1.95994i) q^{7} +(0.746362 - 1.46482i) q^{13} +(0.0448648 + 0.998993i) q^{16} +(1.52564 - 1.27369i) q^{19} +(-0.0448648 + 0.998993i) q^{25} +(-1.09736 + 1.66243i) q^{28} +(-0.258792 - 0.0714220i) q^{31} +(0.457642 + 1.52402i) q^{37} +(-1.39340 - 1.27362i) q^{43} +(-2.77861 + 1.04283i) q^{49} +(-1.55175 + 0.542980i) q^{52} +(0.388070 - 0.424565i) q^{61} +(0.657939 - 0.753071i) q^{64} +(-0.975592 - 0.316989i) q^{67} +(0.309017 - 1.95106i) q^{73} +(-1.98292 - 0.133692i) q^{76} +(-0.0825089 + 1.83720i) q^{79} +(-3.13642 - 0.941822i) q^{91} +(1.66243 - 0.149621i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 2 q^{13} - 2 q^{16} - 2 q^{19} + 2 q^{25} + 6 q^{28} + 4 q^{31} + 12 q^{37} - 2 q^{43} - 6 q^{49} + 2 q^{52} + 8 q^{61} - 10 q^{67} - 12 q^{73} - 2 q^{76} + 4 q^{91}+O(q^{100}) \) 48 * q - 2 * q^13 - 2 * q^16 - 2 * q^19 + 2 * q^25 + 6 * q^28 + 4 * q^31 + 12 * q^37 - 2 * q^43 - 6 * q^49 + 2 * q^52 + 8 * q^61 - 10 * q^67 - 12 * q^73 - 2 * q^76 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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