LMFDB - Embedded newform 3789.1.du.a.685.1

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			685.1
Root			$0.351375 + 0.936235i$ of defining polynomial
Character	$\chi$	$=$	3789.685
Dual form			3789.1.du.a.2467.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.657939 + 0.753071i) q^{4} +(-0.988832 - 1.65503i) q^{7} +O(q^{10})$	$q+(-0.657939 + 0.753071i) q^{4} +(-0.988832 - 1.65503i) q^{7} +(0.303386 + 1.91550i) q^{13} +(-0.134233 - 0.990950i) q^{16} +(-0.574515 + 0.325991i) q^{19} +(0.134233 - 0.990950i) q^{25} +(1.89694 + 0.344244i) q^{28} +(0.543210 + 0.568153i) q^{31} +(-1.18961 + 1.42493i) q^{37} +(1.34735 + 1.02587i) q^{43} +(-1.28745 + 2.39249i) q^{49} +(-1.64212 - 1.03181i) q^{52} +(-0.776472 + 1.01979i) q^{61} +(0.834573 + 0.550897i) q^{64} +(1.17439 + 1.61640i) q^{67} +(-0.809017 - 0.412215i) q^{73} +(0.132501 - 0.647133i) q^{76} +(-0.0943324 + 0.696390i) q^{79} +(2.87021 - 2.39622i) q^{91} +(0.344244 - 0.0950054i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3789\mathbb{Z}\right)^\times$.

$n$	$1685$	$2107$
$\chi(n)$	$1$	$e\left(\frac{121}{140}\right)$

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		−0.413559	−	0.910478i	$-0.635714\pi$
$2$		0			0		0.413559	+	0.910478i	$0.364286\pi$
$3$		0			0
$3$		0			0
$4$	−0.657939	+	0.753071i	−0.657939	+	0.753071i
$5$		0			0		0.753071	−	0.657939i	$-0.228571\pi$
$5$		0			0		−0.753071	+	0.657939i	$0.771429\pi$
$6$		0			0
$7$	−0.988832	−	1.65503i	−0.988832	−	1.65503i	−0.722795	−	0.691063i	$-0.757143\pi$
$7$	−0.988832	−	1.65503i	−0.988832	−	1.65503i	−0.266037	−	0.963963i	$-0.585714\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.0896393	−	0.995974i	$-0.528571\pi$
$11$		0			0		0.0896393	+	0.995974i	$0.471429\pi$
$12$		0			0
$13$	0.303386	+	1.91550i	0.303386	+	1.91550i	0.393025	+	0.919528i	$0.371429\pi$
$13$	0.303386	+	1.91550i	0.303386	+	1.91550i	−0.0896393	+	0.995974i	$0.528571\pi$
$14$		0			0
$15$		0			0
$16$	−0.134233	−	0.990950i	−0.134233	−	0.990950i
$17$		0			0		0.550897	−	0.834573i	$-0.314286\pi$
$17$		0			0		−0.550897	+	0.834573i	$0.685714\pi$
$18$		0			0
$19$	−0.574515	+	0.325991i	−0.574515	+	0.325991i	−0.753071	−	0.657939i	$-0.771429\pi$
$19$	−0.574515	+	0.325991i	−0.574515	+	0.325991i	0.178557	+	0.983930i	$0.442857\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.928115	−	0.372294i	$-0.878571\pi$
$23$		0			0		0.928115	+	0.372294i	$0.121429\pi$
$24$		0			0
$25$	0.134233	−	0.990950i	0.134233	−	0.990950i
$26$		0			0
$27$		0			0
$28$	1.89694	+	0.344244i	1.89694	+	0.344244i
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$	0.543210	+	0.568153i	0.543210	+	0.568153i	0.936235	−	0.351375i	$-0.114286\pi$
$31$	0.543210	+	0.568153i	0.543210	+	0.568153i	−0.393025	+	0.919528i	$0.628571\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.18961	+	1.42493i	−1.18961	+	1.42493i	−0.309017	+	0.951057i	$0.600000\pi$
$37$	−1.18961	+	1.42493i	−1.18961	+	1.42493i	−0.880596	+	0.473869i	$0.842857\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.869741	−	0.493508i	$-0.164286\pi$
$41$		0			0		−0.869741	+	0.493508i	$0.835714\pi$
$42$		0			0
$43$	1.34735	+	1.02587i	1.34735	+	1.02587i	0.995974	+	0.0896393i	$0.0285714\pi$
$43$	1.34735	+	1.02587i	1.34735	+	1.02587i	0.351375	+	0.936235i	$0.385714\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.997735	−	0.0672690i	$-0.978571\pi$
$47$		0			0		0.997735	+	0.0672690i	$0.0214286\pi$
$48$		0			0
$49$	−1.28745	+	2.39249i	−1.28745	+	2.39249i
$50$		0			0
$51$		0			0
$52$	−1.64212	−	1.03181i	−1.64212	−	1.03181i
$53$		0			0		−0.969690	−	0.244340i	$-0.921429\pi$
$53$		0			0		0.969690	+	0.244340i	$0.0785714\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.674671	−	0.738119i	$-0.735714\pi$
$59$		0			0		0.674671	+	0.738119i	$0.264286\pi$
$60$		0			0
$61$	−0.776472	+	1.01979i	−0.776472	+	1.01979i	0.222521	+	0.974928i	$0.428571\pi$
$61$	−0.776472	+	1.01979i	−0.776472	+	1.01979i	−0.998993	+	0.0448648i	$0.985714\pi$
$62$		0			0
$63$		0			0
$64$	0.834573	+	0.550897i	0.834573	+	0.550897i
$65$		0			0
$66$		0			0
$67$	1.17439	+	1.61640i	1.17439	+	1.61640i	0.623490	+	0.781831i	$0.285714\pi$
$67$	1.17439	+	1.61640i	1.17439	+	1.61640i	0.550897	+	0.834573i	$0.314286\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.674671	−	0.738119i	$-0.264286\pi$
$71$		0			0		−0.674671	+	0.738119i	$0.735714\pi$
$72$		0			0
$73$	−0.809017	−	0.412215i	−0.809017	−	0.412215i		−	1.00000i	$-0.5\pi$
$73$	−0.809017	−	0.412215i	−0.809017	−	0.412215i	−0.809017	+	0.587785i	$0.800000\pi$
$74$		0			0
$75$		0			0
$76$	0.132501	−	0.647133i	0.132501	−	0.647133i
$77$		0			0
$78$		0			0
$79$	−0.0943324	+	0.696390i	−0.0943324	+	0.696390i	0.880596	+	0.473869i	$0.157143\pi$
$79$	−0.0943324	+	0.696390i	−0.0943324	+	0.696390i	−0.974928	+	0.222521i	$0.928571\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.738119	−	0.674671i	$-0.235714\pi$
$83$		0			0		−0.738119	+	0.674671i	$0.764286\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.999748	−	0.0224381i	$-0.00714286\pi$
$89$		0			0		−0.999748	+	0.0224381i	$0.992857\pi$
$90$		0			0
$91$	2.87021	−	2.39622i	2.87021	−	2.39622i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.344244	−	0.0950054i	0.344244	−	0.0950054i	−0.0896393	−	0.995974i	$-0.528571\pi$
$97$	0.344244	−	0.0950054i	0.344244	−	0.0950054i	0.433884	+	0.900969i	$0.357143\pi$
$98$		0			0
$99$		0			0
$100$	0.657939	+	0.753071i	0.657939	+	0.753071i
$101$		0			0		−0.134233	−	0.990950i	$-0.542857\pi$
$101$		0			0		0.134233	+	0.990950i	$0.457143\pi$
$102$		0			0
$103$	−0.578625	−	0.217162i	−0.578625	−	0.217162i	0.0448648	−	0.998993i	$-0.485714\pi$
$103$	−0.578625	−	0.217162i	−0.578625	−	0.217162i	−0.623490	+	0.781831i	$0.714286\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$107$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$108$		0			0
$109$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.222521	+	0.974928i	$0.571429\pi$
$109$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$110$		0			0
$111$		0			0
$112$	−1.50731	+	1.20204i	−1.50731	+	1.20204i
$113$		0			0		−0.995974	−	0.0896393i	$-0.971429\pi$
$113$		0			0		0.995974	+	0.0896393i	$0.0285714\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.983930	+	0.178557i	−0.983930	+	0.178557i
$122$		0			0
$123$		0			0
$124$	−0.785259	+	0.0352660i	−0.785259	+	0.0352660i
$125$		0			0
$126$		0			0
$127$	1.49683	+	1.03701i	1.49683	+	1.03701i	0.983930	+	0.178557i	$0.0571429\pi$
$127$	1.49683	+	1.03701i	1.49683	+	1.03701i	0.512899	+	0.858449i	$0.328571\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.936235	−	0.351375i	$-0.114286\pi$
$131$		0			0		−0.936235	+	0.351375i	$0.885714\pi$
$132$		0			0
$133$	1.10762	+	0.628486i	1.10762	+	0.628486i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$137$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$138$		0			0
$139$	1.42236	+	0.258120i	1.42236	+	0.258120i	0.834573	−	0.550897i	$-0.185714\pi$
$139$	1.42236	+	0.258120i	1.42236	+	0.258120i	0.587785	+	0.809017i	$0.300000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.290378	−	1.83338i	−0.290378	−	1.83338i
$149$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$149$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$150$		0			0
$151$	0.512506	+	0.261135i	0.512506	+	0.261135i	0.691063	−	0.722795i	$-0.257143\pi$
$151$	0.512506	+	0.261135i	0.512506	+	0.261135i	−0.178557	+	0.983930i	$0.557143\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.45806	−	1.33273i	−1.45806	−	1.33273i	−0.834573	−	0.550897i	$-0.814286\pi$
$157$	−1.45806	−	1.33273i	−1.45806	−	1.33273i	−0.623490	−	0.781831i	$-0.714286\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.307552	−	0.321674i	0.307552	−	0.321674i	−0.550897	−	0.834573i	$-0.685714\pi$
$163$	0.307552	−	0.321674i	0.307552	−	0.321674i	0.858449	+	0.512899i	$0.171429\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.983930	−	0.178557i	$-0.942857\pi$
$167$		0			0		0.983930	+	0.178557i	$0.0571429\pi$
$168$		0			0
$169$	−2.62605	+	0.853255i	−2.62605	+	0.853255i
$170$		0			0
$171$		0			0
$172$	−1.65903	+	0.339688i	−1.65903	+	0.339688i
$173$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$173$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$174$		0			0
$175$	−1.77278	+	0.757723i	−1.77278	+	0.757723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.822002	−	0.569484i	$-0.192857\pi$
$179$		0			0		−0.822002	+	0.569484i	$0.807143\pi$
$180$		0			0
$181$	0.515795	+	1.58745i	0.515795	+	1.58745i	0.781831	+	0.623490i	$0.214286\pi$
$181$	0.515795	+	1.58745i	0.515795	+	1.58745i	−0.266037	+	0.963963i	$0.585714\pi$
$182$		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		0.722795	−	0.691063i	$-0.242857\pi$
$191$		0			0		−0.722795	+	0.691063i	$0.757143\pi$
$192$		0			0
$193$	−0.165286	−	0.151078i	−0.165286	−	0.151078i	0.587785	−	0.809017i	$-0.300000\pi$
$193$	−0.165286	−	0.151078i	−0.165286	−	0.151078i	−0.753071	+	0.657939i	$0.771429\pi$
$194$		0			0
$195$		0			0
$196$	−0.954649	−	2.54365i	−0.954649	−	2.54365i
$197$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$197$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$198$		0			0
$199$	−1.04084	−	1.08863i	−1.04084	−	1.08863i	−0.995974	−	0.0896393i	$-0.971429\pi$
$199$	−1.04084	−	1.08863i	−1.04084	−	1.08863i	−0.0448648	−	0.998993i	$-0.514286\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.85744	−	0.557764i	1.85744	−	0.557764i
$209$		0			0
$210$		0			0
$211$	−1.78394	−	0.810306i	−1.78394	−	0.810306i	−0.974928	−	0.222521i	$-0.928571\pi$
$211$	−1.78394	−	0.810306i	−1.78394	−	0.810306i	−0.809017	−	0.587785i	$-0.800000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.403165	−	1.46083i	0.403165	−	1.46083i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.752407	−	1.19745i	0.752407	−	1.19745i	−0.222521	−	0.974928i	$-0.571429\pi$
$223$	0.752407	−	1.19745i	0.752407	−	1.19745i	0.974928	−	0.222521i	$-0.0714286\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.767645	−	0.640876i	$-0.221429\pi$
$227$		0			0		−0.767645	+	0.640876i	$0.778571\pi$
$228$		0			0
$229$	1.18556	−	0.570938i	1.18556	−	0.570938i	0.266037	−	0.963963i	$-0.414286\pi$
$229$	1.18556	−	0.570938i	1.18556	−	0.570938i	0.919528	+	0.393025i	$0.128571\pi$
$230$		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		0.433884	−	0.900969i	$-0.357143\pi$
$239$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$240$		0			0
$241$	−1.03520	+	0.557066i	−1.03520	+	0.557066i	−0.900969	−	0.433884i	$-0.857143\pi$
$241$	−1.03520	+	0.557066i	−1.03520	+	0.557066i	−0.134233	+	0.990950i	$0.542857\pi$
$242$		0			0
$243$		0			0
$244$	−0.257106	−	1.25570i	−0.257106	−	1.25570i
$245$		0			0
$246$		0			0
$247$	−0.798736	−	1.00158i	−0.798736	−	1.00158i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$251$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.963963	+	0.266037i	−0.963963	+	0.266037i
$257$		0			0		−0.979675	−	0.200589i	$-0.935714\pi$
$257$		0			0		0.979675	+	0.200589i	$0.0642857\pi$
$258$		0			0
$259$	3.53461	+	0.559828i	3.53461	+	0.559828i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.287599	−	0.957751i	$-0.592857\pi$
$263$		0			0		0.287599	+	0.957751i	$0.407143\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.98994	−	0.179098i	−1.98994	−	0.179098i
$269$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$269$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$270$		0			0
$271$	0.389019	+	0.0980242i	0.389019	+	0.0980242i	0.433884	−	0.900969i	$-0.357143\pi$
$271$	0.389019	+	0.0980242i	0.389019	+	0.0980242i	−0.0448648	+	0.998993i	$0.514286\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$277$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.512899	−	0.858449i	$-0.671429\pi$
$281$		0			0		0.512899	+	0.858449i	$0.328571\pi$
$282$		0			0
$283$	1.74525	+	0.939160i	1.74525	+	0.939160i	0.936235	+	0.351375i	$0.114286\pi$
$283$	1.74525	+	0.939160i	1.74525	+	0.939160i	0.809017	+	0.587785i	$0.200000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.393025	−	0.919528i	−0.393025	−	0.919528i
$290$		0			0
$291$		0			0
$292$	0.842711	−	0.338036i	0.842711	−	0.338036i
$293$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$293$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	0.365546	−	3.24431i	0.365546	−	3.24431i
$302$		0			0
$303$		0			0
$304$	0.400160	+	0.525556i	0.400160	+	0.525556i
$305$		0			0
$306$		0			0
$307$	−0.782886	−	1.83165i	−0.782886	−	1.83165i	−0.473869	−	0.880596i	$-0.657143\pi$
$307$	−0.782886	−	1.83165i	−0.782886	−	1.83165i	−0.309017	−	0.951057i	$-0.600000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$311$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$312$		0			0
$313$	−0.654946	+	0.685020i	−0.654946	+	0.685020i	−0.963963	−	0.266037i	$-0.914286\pi$
$313$	−0.654946	+	0.685020i	−0.654946	+	0.685020i	0.309017	+	0.951057i	$0.400000\pi$
$314$		0			0
$315$		0			0
$316$	−0.462366	−	0.529221i	−0.462366	−	0.529221i
$317$		0			0		−0.990950	−	0.134233i	$-0.957143\pi$
$317$		0			0		0.990950	+	0.134233i	$0.0428571\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.93889	−	0.0435159i	1.93889	−	0.0435159i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.20884	+	0.0815018i	−1.20884	+	0.0815018i	−0.657939	−	0.753071i	$-0.728571\pi$
$331$	−1.20884	+	0.0815018i	−1.20884	+	0.0815018i	−0.550897	+	0.834573i	$0.685714\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.28289	−	1.46838i	1.28289	−	1.46838i	0.473869	−	0.880596i	$-0.342857\pi$
$337$	1.28289	−	1.46838i	1.28289	−	1.46838i	0.809017	−	0.587785i	$-0.200000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	3.30672	−	0.148505i	3.30672	−	0.148505i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.266037	−	0.963963i	$-0.414286\pi$
$347$		0			0		−0.266037	+	0.963963i	$0.585714\pi$
$348$		0			0
$349$	0.400270	+	1.95491i	0.400270	+	1.95491i	0.266037	+	0.963963i	$0.414286\pi$
$349$	0.400270	+	1.95491i	0.400270	+	1.95491i	0.134233	+	0.990950i	$0.457143\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.963963	−	0.266037i	$-0.0857143\pi$
$353$		0			0		−0.963963	+	0.266037i	$0.914286\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.928115	−	0.372294i	$-0.878571\pi$
$359$		0			0		0.928115	+	0.372294i	$0.121429\pi$
$360$		0			0
$361$	−0.289102	+	0.483876i	−0.289102	+	0.483876i
$362$		0			0
$363$		0			0
$364$	−0.0838954	+	3.73804i	−0.0838954	+	3.73804i
$365$		0			0
$366$		0			0
$367$	1.37012	−	0.549594i	1.37012	−	0.549594i	0.433884	−	0.900969i	$-0.357143\pi$
$367$	1.37012	−	0.549594i	1.37012	−	0.549594i	0.936235	+	0.351375i	$0.114286\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.43783	+	0.614559i	1.43783	+	0.614559i	0.963963	−	0.266037i	$-0.0857143\pi$
$373$	1.43783	+	0.614559i	1.43783	+	0.614559i	0.473869	+	0.880596i	$0.342857\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.34408	−	0.610511i	1.34408	−	0.610511i	0.393025	−	0.919528i	$-0.371429\pi$
$379$	1.34408	−	0.610511i	1.34408	−	0.610511i	0.951057	+	0.309017i	$0.100000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$383$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.154946	+	0.321748i	−0.154946	+	0.321748i
$389$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$389$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.211083	−	0.230935i	−0.211083	−	0.230935i	0.623490	−	0.781831i	$-0.285714\pi$
$397$	−0.211083	−	0.230935i	−0.211083	−	0.230935i	−0.834573	+	0.550897i	$0.814286\pi$
$398$		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
$403$	−0.923496	+	1.21289i	−0.923496	+	1.21289i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.268155	−	0.825296i	−0.268155	−	0.825296i	−0.990950	−	0.134233i	$-0.957143\pi$
$409$	−0.268155	−	0.825296i	−0.268155	−	0.825296i	0.722795	−	0.691063i	$-0.242857\pi$
$410$		0			0
$411$		0			0
$412$	0.544238	−	0.292867i	0.544238	−	0.292867i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.910478	−	0.413559i	$-0.135714\pi$
$419$		0			0		−0.910478	+	0.413559i	$0.864286\pi$
$420$		0			0
$421$	−0.512899	+	0.858449i	−0.512899	+	0.858449i
$421$	−0.512899	+	0.858449i	−0.512899	+	0.858449i
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.45558	+	0.276678i	2.45558	+	0.276678i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.413559	−	0.910478i	$-0.364286\pi$
$431$		0			0		−0.413559	+	0.910478i	$0.635714\pi$
$432$		0			0
$433$	1.89446	−	0.615546i	1.89446	−	0.615546i	0.919528	−	0.393025i	$-0.128571\pi$
$433$	1.89446	−	0.615546i	1.89446	−	0.615546i	0.974928	−	0.222521i	$-0.0714286\pi$
$434$		0			0
$435$		0			0
$436$	−0.321748	−	1.77298i	−0.321748	−	1.77298i
$437$		0			0
$438$		0			0
$439$	−1.30801	−	0.995921i	−1.30801	−	0.995921i	−0.998993	−	0.0448648i	$-0.985714\pi$
$439$	−1.30801	−	0.995921i	−1.30801	−	0.995921i	−0.309017	−	0.951057i	$-0.600000\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.493508	−	0.869741i	$-0.664286\pi$
$443$		0			0		0.493508	+	0.869741i	$0.335714\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.0864961	−	1.92598i	0.0864961	−	1.92598i
$449$		0			0		0.178557	−	0.983930i	$-0.442857\pi$
$449$		0			0		−0.178557	+	0.983930i	$0.557143\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.91852	+	0.437890i	1.91852	+	0.437890i	0.998993	+	0.0448648i	$0.0142857\pi$
$457$	1.91852	+	0.437890i	1.91852	+	0.437890i	0.919528	+	0.393025i	$0.128571\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.413559	−	0.910478i	$-0.635714\pi$
$461$		0			0		0.413559	+	0.910478i	$0.364286\pi$
$462$		0			0
$463$	−0.342061	−	0.753071i	−0.342061	−	0.753071i	0.657939	−	0.753071i	$-0.271429\pi$
$463$	−0.342061	−	0.753071i	−0.342061	−	0.753071i	−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.998993	−	0.0448648i	$-0.0142857\pi$
$467$		0			0		−0.998993	+	0.0448648i	$0.985714\pi$
$468$		0			0
$469$	1.51392	−	3.54199i	1.51392	−	3.54199i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.245922	+	0.613074i	0.245922	+	0.613074i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.936235	−	0.351375i	$-0.885714\pi$
$479$		0			0		0.936235	+	0.351375i	$0.114286\pi$
$480$		0			0
$481$	−3.09036	−	1.84640i	−3.09036	−	1.84640i
$482$		0			0
$483$		0			0
$484$	0.512899	−	0.858449i	0.512899	−	0.858449i
$485$		0			0
$486$		0			0
$487$	−0.986768	−	0.0221467i	−0.986768	−	0.0221467i	−0.473869	−	0.880596i	$-0.657143\pi$
$487$	−0.986768	−	0.0221467i	−0.986768	−	0.0221467i	−0.512899	+	0.858449i	$0.671429\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$491$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.490094	−	0.614559i	0.490094	−	0.614559i
$497$		0			0
$498$		0			0
$499$	−0.0425201	−	0.946783i	−0.0425201	−	0.946783i	−0.900969	−	0.433884i	$-0.857143\pi$
$499$	−0.0425201	−	0.946783i	−0.0425201	−	0.946783i	0.858449	−	0.512899i	$-0.171429\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$503$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1.76576	+	0.444933i	−1.76576	+	0.444933i
$509$		0			0		−0.979675	−	0.200589i	$-0.935714\pi$
$509$		0			0		0.979675	+	0.200589i	$0.0642857\pi$
$510$		0			0
$511$	0.117756	+	1.74655i	0.117756	+	1.74655i
$512$		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		0.753071	−	0.657939i	$-0.228571\pi$
$521$		0			0		−0.753071	+	0.657939i	$0.771429\pi$
$522$		0			0
$523$	−1.03581	−	0.864760i	−1.03581	−	0.864760i	−0.0448648	−	0.998993i	$-0.514286\pi$
$523$	−1.03581	−	0.864760i	−1.03581	−	0.864760i	−0.990950	+	0.134233i	$0.957143\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.722795	+	0.691063i	0.722795	+	0.691063i
$530$		0			0
$531$		0			0
$532$	−1.20204	+	0.420613i	−1.20204	+	0.420613i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.0445940	−	0.00502453i	−0.0445940	−	0.00502453i	0.0896393	−	0.995974i	$-0.471429\pi$
$541$	−0.0445940	−	0.00502453i	−0.0445940	−	0.00502453i	−0.134233	+	0.990950i	$0.542857\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.178163	−	1.58124i	0.178163	−	1.58124i	−0.512899	−	0.858449i	$-0.671429\pi$
$547$	0.178163	−	1.58124i	0.178163	−	1.58124i	0.691063	−	0.722795i	$-0.257143\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.24582	−	0.532490i	1.24582	−	0.532490i
$554$		0			0
$555$		0			0
$556$	−1.13021	+	0.901310i	−1.13021	+	0.901310i
$557$		0			0		−0.640876	−	0.767645i	$-0.721429\pi$
$557$		0			0		0.640876	+	0.767645i	$0.278571\pi$
$558$		0			0
$559$	−1.55630	+	2.89209i	−1.55630	+	2.89209i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$563$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$569$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$570$		0			0
$571$	−0.0109651	+	0.0435159i	−0.0109651	+	0.0435159i	−0.974928	−	0.222521i	$-0.928571\pi$
$571$	−0.0109651	+	0.0435159i	−0.0109651	+	0.0435159i	0.963963	+	0.266037i	$0.0857143\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.902272	−	1.77081i	−0.902272	−	1.77081i	−0.550897	−	0.834573i	$-0.685714\pi$
$577$	−0.902272	−	1.77081i	−0.902272	−	1.77081i	−0.351375	−	0.936235i	$-0.614286\pi$
$578$		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		0.393025	−	0.919528i	$-0.371429\pi$
$587$		0			0		−0.393025	+	0.919528i	$0.628571\pi$
$588$		0			0
$589$	−0.497295	−	0.149331i	−0.497295	−	0.149331i
$590$		0			0
$591$		0			0
$592$	1.57171	+	0.987574i	1.57171	+	0.987574i
$593$		0			0		−0.200589	−	0.979675i	$-0.564286\pi$
$593$		0			0		0.200589	+	0.979675i	$0.435714\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$599$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$600$		0			0
$601$	0.887305	+	1.64889i	0.887305	+	1.64889i	0.753071	+	0.657939i	$0.228571\pi$
$601$	0.887305	+	1.64889i	0.887305	+	1.64889i	0.134233	+	0.990950i	$0.457143\pi$
$602$		0			0
$603$		0			0
$604$	−0.533850	+	0.214143i	−0.533850	+	0.214143i
$605$		0			0
$606$		0			0
$607$	−1.07953	−	1.41781i	−1.07953	−	1.41781i	−0.900969	−	0.433884i	$-0.857143\pi$
$607$	−1.07953	−	1.41781i	−1.07953	−	1.41781i	−0.178557	−	0.983930i	$-0.557143\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.116483	−	0.241880i	−0.116483	−	0.241880i	0.834573	−	0.550897i	$-0.185714\pi$
$613$	−0.116483	−	0.241880i	−0.116483	−	0.241880i	−0.951057	+	0.309017i	$0.900000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.936235	−	0.351375i	$-0.114286\pi$
$617$		0			0		−0.936235	+	0.351375i	$0.885714\pi$
$618$		0			0
$619$	−1.34735	−	0.846596i	−1.34735	−	0.846596i	−0.351375	−	0.936235i	$-0.614286\pi$
$619$	−1.34735	−	0.846596i	−1.34735	−	0.846596i	−0.995974	+	0.0896393i	$0.971429\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.963963	−	0.266037i	−0.963963	−	0.266037i
$626$		0			0
$627$		0			0
$628$	1.96296	−	0.221172i	1.96296	−	0.221172i
$629$		0			0
$630$		0			0
$631$	−0.810306	+	1.78394i	−0.810306	+	1.78394i	−0.222521	+	0.974928i	$0.571429\pi$
$631$	−0.810306	+	1.78394i	−0.810306	+	1.78394i	−0.587785	+	0.809017i	$0.700000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−4.97341	−	1.74027i	−4.97341	−	1.74027i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.0896393	−	0.995974i	$-0.528571\pi$
$641$		0			0		0.0896393	+	0.995974i	$0.471429\pi$
$642$		0			0
$643$	−1.35341	+	1.29399i	−1.35341	+	1.29399i	−0.433884	+	0.900969i	$0.642857\pi$
$643$	−1.35341	+	1.29399i	−1.35341	+	1.29399i	−0.919528	+	0.393025i	$0.871429\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$647$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.0398932	+	0.443250i	0.0398932	+	0.443250i
$653$		0			0		0.880596	−	0.473869i	$-0.157143\pi$
$653$		0			0		−0.880596	+	0.473869i	$0.842857\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.928115	−	0.372294i	$-0.121429\pi$
$659$		0			0		−0.928115	+	0.372294i	$0.878571\pi$
$660$		0			0
$661$	−1.18595	+	0.385338i	−1.18595	+	0.385338i	−0.834573	−	0.550897i	$-0.814286\pi$
$661$	−1.18595	+	0.385338i	−1.18595	+	0.385338i	−0.351375	+	0.936235i	$0.614286\pi$
$662$		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
$673$	−0.385338	−	1.18595i	−0.385338	−	1.18595i	−0.936235	−	0.351375i	$-0.885714\pi$
$673$	−0.385338	−	1.18595i	−0.385338	−	1.18595i	0.550897	−	0.834573i	$-0.314286\pi$
$674$		0			0
$675$		0			0
$676$	1.08522	−	2.53899i	1.08522	−	2.53899i
$677$		0			0		−0.266037	−	0.963963i	$-0.585714\pi$
$677$		0			0		0.266037	+	0.963963i	$0.414286\pi$
$678$		0			0
$679$	−0.497636	−	0.475789i	−0.497636	−	0.475789i
$680$		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
$688$	0.835731	−	1.47286i	0.835731	−	1.47286i
$689$		0			0
$690$		0			0
$691$	−0.550046	+	1.07953i	−0.550046	+	1.07953i	0.433884	+	0.900969i	$0.357143\pi$
$691$	−0.550046	+	1.07953i	−0.550046	+	1.07953i	−0.983930	+	0.178557i	$0.942857\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.595762	−	1.83357i	0.595762	−	1.83357i
$701$		0			0		0.569484	−	0.822002i	$-0.307143\pi$
$701$		0			0		−0.569484	+	0.822002i	$0.692857\pi$
$702$		0			0
$703$	0.218937	−	1.20644i	0.218937	−	1.20644i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.110511	+	0.194760i	−0.110511	+	0.194760i	−0.919528	−	0.393025i	$-0.871429\pi$
$709$	−0.110511	+	0.194760i	−0.110511	+	0.194760i	0.809017	+	0.587785i	$0.200000\pi$
$710$		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		0.0224381	−	0.999748i	$-0.492857\pi$
$719$		0			0		−0.0224381	+	0.999748i	$0.507143\pi$
$720$		0			0
$721$	0.212755	+	1.17238i	0.212755	+	1.17238i
$722$		0			0
$723$		0			0
$724$	−1.53483	−	0.656016i	−1.53483	−	0.656016i
$725$		0			0
$726$		0			0
$727$	−0.507074	−	1.44913i	−0.507074	−	1.44913i	−0.858449	−	0.512899i	$-0.828571\pi$
$727$	−0.507074	−	1.44913i	−0.507074	−	1.44913i	0.351375	−	0.936235i	$-0.385714\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.53853	+	1.22694i	1.53853	+	1.22694i	0.880596	+	0.473869i	$0.157143\pi$
$733$	1.53853	+	1.22694i	1.53853	+	1.22694i	0.657939	+	0.753071i	$0.271429\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.699921	+	1.86493i	−0.699921	+	1.86493i	−0.266037	+	0.963963i	$0.585714\pi$
$739$	−0.699921	+	1.86493i	−0.699921	+	1.86493i	−0.433884	+	0.900969i	$0.642857\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.550897	−	0.834573i	$-0.685714\pi$
$743$		0			0		0.550897	+	0.834573i	$0.314286\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.632555	+	0.757678i	0.632555	+	0.757678i	0.983930	−	0.178557i	$-0.0571429\pi$
$751$	0.632555	+	0.757678i	0.632555	+	0.757678i	−0.351375	+	0.936235i	$0.614286\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.16493	−	0.157801i	1.16493	−	0.157801i	0.473869	−	0.880596i	$-0.342857\pi$
$757$	1.16493	−	0.157801i	1.16493	−	0.157801i	0.691063	+	0.722795i	$0.257143\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.550897	−	0.834573i	$-0.314286\pi$
$761$		0			0		−0.550897	+	0.834573i	$0.685714\pi$
$762$		0			0
$763$	3.44256	+	0.466327i	3.44256	+	0.466327i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.448096	+	0.879438i	−0.448096	+	0.879438i	0.550897	+	0.834573i	$0.314286\pi$
$769$	−0.448096	+	0.879438i	−0.448096	+	0.879438i	−0.998993	+	0.0448648i	$0.985714\pi$
$770$		0			0
$771$		0			0
$772$	0.222521	−	0.0250721i	0.222521	−	0.0250721i
$773$		0			0		0.795625	−	0.605790i	$-0.207143\pi$
$773$		0			0		−0.795625	+	0.605790i	$0.792857\pi$
$774$		0			0
$775$	0.635928	−	0.462029i	0.635928	−	0.462029i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	2.54365	+	0.954649i	2.54365	+	0.954649i
$785$		0			0
$786$		0			0
$787$	1.21709	−	1.27298i	1.21709	−	1.27298i	0.266037	−	0.963963i	$-0.414286\pi$
$787$	1.21709	−	1.27298i	1.21709	−	1.27298i	0.951057	−	0.309017i	$-0.100000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.18899	−	1.17794i	−2.18899	−	1.17794i
$794$		0			0
$795$		0			0
$796$	1.50463	−	0.0675728i	1.50463	−	0.0675728i
$797$		0			0		0.351375	−	0.936235i	$-0.385714\pi$
$797$		0			0		−0.351375	+	0.936235i	$0.614286\pi$
$798$		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		−0.900969	−	0.433884i	$-0.857143\pi$
$809$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$810$		0			0
$811$	−0.194046	+	0.185527i	−0.194046	+	0.185527i	−0.781831	−	0.623490i	$-0.785714\pi$
$811$	−0.194046	+	0.185527i	−0.194046	+	0.185527i	0.587785	+	0.809017i	$0.300000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.10850	−	0.150156i	−1.10850	−	0.150156i
$818$		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
$823$	0.879438	+	1.54989i	0.879438	+	1.54989i	0.834573	+	0.550897i	$0.185714\pi$
$823$	0.879438	+	1.54989i	0.879438	+	1.54989i	0.0448648	+	0.998993i	$0.485714\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.178557	−	0.983930i	$-0.557143\pi$
$827$		0			0		0.178557	+	0.983930i	$0.442857\pi$
$828$		0			0
$829$	−1.76007	+	0.278768i	−1.76007	+	0.278768i	−0.951057	−	0.309017i	$-0.900000\pi$
$829$	−1.76007	+	0.278768i	−1.76007	+	0.278768i	−0.809017	+	0.587785i	$0.800000\pi$
$830$		0			0
$831$		0			0
$832$	−0.802047	+	1.76576i	−0.802047	+	1.76576i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.858449	−	0.512899i	$-0.828571\pi$
$839$		0			0		0.858449	+	0.512899i	$0.171429\pi$
$840$		0			0
$841$			1.00000i			1.00000i
$842$		0			0
$843$		0			0
$844$	1.78394	−	0.810306i	1.78394	−	0.810306i
$845$		0			0
$846$		0			0
$847$	1.26846	+	1.45187i	1.26846	+	1.45187i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.66243	+	0.149621i	−1.66243	+	0.149621i	−0.880596	−	0.473869i	$-0.842857\pi$
$853$	−1.66243	+	0.149621i	−1.66243	+	0.149621i	−0.781831	+	0.623490i	$0.785714\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.983930	−	0.178557i	$-0.0571429\pi$
$857$		0			0		−0.983930	+	0.178557i	$0.942857\pi$
$858$		0			0
$859$	−0.919528	−	0.606975i	−0.919528	−	0.606975i		−	1.00000i	$-0.5\pi$
$859$	−0.919528	−	0.606975i	−0.919528	−	0.606975i	−0.919528	+	0.393025i	$0.871429\pi$
$860$		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
$868$	0.834855	+	1.26475i	0.834855	+	1.26475i
$869$		0			0
$870$		0			0
$871$	−2.73993	+	2.73993i	−2.73993	+	2.73993i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.612441	−	1.88490i	0.612441	−	1.88490i	0.178557	−	0.983930i	$-0.442857\pi$
$877$	0.612441	−	1.88490i	0.612441	−	1.88490i	0.433884	−	0.900969i	$-0.357143\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.928115	−	0.372294i	$-0.878571\pi$
$881$		0			0		0.928115	+	0.372294i	$0.121429\pi$
$882$		0			0
$883$	0.974865	+	1.71807i	0.974865	+	1.71807i	0.623490	+	0.781831i	$0.285714\pi$
$883$	0.974865	+	1.71807i	0.974865	+	1.71807i	0.351375	+	0.936235i	$0.385714\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.936235	−	0.351375i	$-0.885714\pi$
$887$		0			0		0.936235	+	0.351375i	$0.114286\pi$
$888$		0			0
$889$	0.236159	−	3.50271i	0.236159	−	3.50271i
$890$		0			0
$891$		0			0
$892$	0.406727	+	1.35446i	0.406727	+	1.35446i
$893$		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
$907$	−1.27885	+	1.53181i	−1.27885	+	1.53181i	−0.587785	+	0.809017i	$0.700000\pi$
$907$	−1.27885	+	1.53181i	−1.27885	+	1.53181i	−0.691063	+	0.722795i	$0.742857\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.605790	−	0.795625i	$-0.707143\pi$
$911$		0			0		0.605790	+	0.795625i	$0.292857\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.350072	+	1.26846i	−0.350072	+	1.26846i
$917$		0			0
$918$		0			0
$919$	0.602949	+	0.829888i	0.602949	+	0.829888i	0.995974	−	0.0896393i	$-0.0285714\pi$
$919$	0.602949	+	0.829888i	0.602949	+	0.829888i	−0.393025	+	0.919528i	$0.628571\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.25234	+	1.37012i	1.25234	+	1.37012i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.244340	−	0.969690i	$-0.578571\pi$
$929$		0			0		0.244340	+	0.969690i	$0.421429\pi$
$930$		0			0
$931$	−0.0402689	−	1.79422i	−0.0402689	−	1.79422i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.542920	+	0.189976i	0.542920	+	0.189976i	0.587785	−	0.809017i	$-0.300000\pi$
$937$	0.542920	+	0.189976i	0.542920	+	0.189976i	−0.0448648	+	0.998993i	$0.514286\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.834573	−	0.550897i	$-0.185714\pi$
$941$		0			0		−0.834573	+	0.550897i	$0.814286\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.657939	−	0.753071i	$-0.728571\pi$
$947$		0			0		0.657939	+	0.753071i	$0.271429\pi$
$948$		0			0
$949$	0.544154	−	1.67473i	0.544154	−	1.67473i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$953$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.0171440	−	0.381741i	0.0171440	−	0.381741i
$962$		0			0
$963$		0			0
$964$	0.261589	−	1.14610i	0.261589	−	1.14610i
$965$		0			0
$966$		0			0
$967$	−1.20645	+	0.796371i	−1.20645	+	0.796371i	−0.983930	−	0.178557i	$-0.942857\pi$
$967$	−1.20645	+	0.796371i	−1.20645	+	0.796371i	−0.222521	+	0.974928i	$0.571429\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.928115	−	0.372294i	$-0.121429\pi$
$971$		0			0		−0.928115	+	0.372294i	$0.878571\pi$
$972$		0			0
$973$	−0.979278	−	2.60928i	−0.979278	−	2.60928i
$974$		0			0
$975$		0			0
$976$	1.11479	+	0.632555i	1.11479	+	0.632555i
$977$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$977$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.822002	−	0.569484i	$-0.807143\pi$
$983$		0			0		0.822002	+	0.569484i	$0.192857\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.27978	+	0.0574751i	1.27978	+	0.0574751i
$989$		0			0
$990$		0			0
$991$	1.50385	+	0.724216i	1.50385	+	0.724216i	0.990950	−	0.134233i	$-0.0428571\pi$
$991$	1.50385	+	0.724216i	1.50385	+	0.724216i	0.512899	+	0.858449i	$0.328571\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.170504	+	1.89446i	−0.170504	+	1.89446i	0.222521	+	0.974928i	$0.428571\pi$
$997$	−0.170504	+	1.89446i	−0.170504	+	1.89446i	−0.393025	+	0.919528i	$0.628571\pi$
$998$		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.685.1	✓	48
3.2	odd	2	CM	3789.1.du.a.685.1	✓	48
421.362	odd	140	inner	3789.1.du.a.2467.1	yes	48
1263.362	even	140	inner	3789.1.du.a.2467.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3789.1.du.a.685.1	✓	48	1.1	even	1	trivial
3789.1.du.a.685.1	✓	48	3.2	odd	2	CM
3789.1.du.a.2467.1	yes	48	421.362	odd	140	inner
3789.1.du.a.2467.1	yes	48	1263.362	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)

\(f(q)\)	\(=\)	\(q+(-0.657939 + 0.753071i) q^{4} +(-0.988832 - 1.65503i) q^{7} +O(q^{10})\)	\(q+(-0.657939 + 0.753071i) q^{4} +(-0.988832 - 1.65503i) q^{7} +(0.303386 + 1.91550i) q^{13} +(-0.134233 - 0.990950i) q^{16} +(-0.574515 + 0.325991i) q^{19} +(0.134233 - 0.990950i) q^{25} +(1.89694 + 0.344244i) q^{28} +(0.543210 + 0.568153i) q^{31} +(-1.18961 + 1.42493i) q^{37} +(1.34735 + 1.02587i) q^{43} +(-1.28745 + 2.39249i) q^{49} +(-1.64212 - 1.03181i) q^{52} +(-0.776472 + 1.01979i) q^{61} +(0.834573 + 0.550897i) q^{64} +(1.17439 + 1.61640i) q^{67} +(-0.809017 - 0.412215i) q^{73} +(0.132501 - 0.647133i) q^{76} +(-0.0943324 + 0.696390i) q^{79} +(2.87021 - 2.39622i) q^{91} +(0.344244 - 0.0950054i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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

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