LMFDB - Embedded newform 3104.1.et.a.1263.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3104,1,Mod(335,3104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3104, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 24, 29]))

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

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

chi := DirichletCharacter("3104.335");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3104 = 2^{5} \cdot 97 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3104.et (of order $48$, degree $16$, not 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:	$1.54909779921$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 776)
Projective image:	$D_{48}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{48} + \cdots)$

Embedding invariants

Embedding label			1263.1
Root			$-0.608761 + 0.793353i$ of defining polynomial
Character	$\chi$	$=$	3104.1263
Dual form			3104.1.et.a.2959.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.207107 - 0.158919i) q^{3} +(-0.241181 + 0.900100i) q^{9} +(-0.0675653 - 0.513210i) q^{11} +(-0.128293 + 1.95737i) q^{17} +(-0.923880 + 1.38268i) q^{19} +(-0.608761 + 0.793353i) q^{25} +(0.192993 + 0.465926i) q^{27} +(-0.0955518 - 0.0955518i) q^{33} +(1.05217 + 0.357164i) q^{41} +(-0.410670 - 1.53264i) q^{43} +(0.991445 - 0.130526i) q^{49} +(0.284492 + 0.425773i) q^{51} +(0.0283924 + 0.433185i) q^{57} +(0.257264 - 0.293353i) q^{59} +(1.57469 + 1.05217i) q^{67} +(1.36603 - 0.366025i) q^{73} +0.261052i q^{75} +(-0.692993 - 0.400100i) q^{81} +(-1.50046 - 0.0983454i) q^{83} +(-1.12484 - 0.465926i) q^{89} +(-0.793353 + 0.608761i) q^{97} +(0.478235 + 0.0629609i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{3} - 8 q^{9} + 16 q^{27} + 8 q^{73} - 24 q^{81}+O(q^{100}) $ 16 * q - 8 * q^3 - 8 * q^9 + 16 * q^27 + 8 * q^73 - 24 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3104\mathbb{Z}\right)^\times$.

$n$	$389$	$2721$	$2911$
$\chi(n)$	$-1$	$e\left(\frac{17}{48}\right)$	$-1$

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
$2$		0			0
$3$	0.207107	−	0.158919i	0.207107	−	0.158919i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	0.207107	−	0.158919i	0.207107	−	0.158919i	0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$5$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$6$		0			0
$7$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$7$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$8$		0			0
$9$	−0.241181	+	0.900100i	−0.241181	+	0.900100i
$10$		0			0
$11$	−0.0675653	−	0.513210i	−0.0675653	−	0.513210i	−0.991445	−	0.130526i	$-0.958333\pi$
$11$	−0.0675653	−	0.513210i	−0.0675653	−	0.513210i	0.923880	−	0.382683i	$-0.125000\pi$
$12$		0			0
$13$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$13$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.128293	+	1.95737i	−0.128293	+	1.95737i	0.130526	+	0.991445i	$0.458333\pi$
$17$	−0.128293	+	1.95737i	−0.128293	+	1.95737i	−0.258819	+	0.965926i	$0.583333\pi$
$18$		0			0
$19$	−0.923880	+	1.38268i	−0.923880	+	1.38268i			1.00000i	$0.5\pi$
$19$	−0.923880	+	1.38268i	−0.923880	+	1.38268i	−0.923880	+	0.382683i	$0.875000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$23$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$24$		0			0
$25$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$26$		0			0
$27$	0.192993	+	0.465926i	0.192993	+	0.465926i
$28$		0			0
$29$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$29$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$30$		0			0
$31$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$31$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$32$		0			0
$33$	−0.0955518	−	0.0955518i	−0.0955518	−	0.0955518i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$37$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.05217	+	0.357164i	1.05217	+	0.357164i	0.793353	−	0.608761i	$-0.208333\pi$
$41$	1.05217	+	0.357164i	1.05217	+	0.357164i	0.258819	+	0.965926i	$0.416667\pi$
$42$		0			0
$43$	−0.410670	−	1.53264i	−0.410670	−	1.53264i	−0.793353	−	0.608761i	$-0.791667\pi$
$43$	−0.410670	−	1.53264i	−0.410670	−	1.53264i	0.382683	−	0.923880i	$-0.375000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$	0.991445	−	0.130526i	0.991445	−	0.130526i
$50$		0			0
$51$	0.284492	+	0.425773i	0.284492	+	0.425773i
$52$		0			0
$53$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$53$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.0283924	+	0.433185i	0.0283924	+	0.433185i
$58$		0			0
$59$	0.257264	−	0.293353i	0.257264	−	0.293353i	−0.608761	−	0.793353i	$-0.708333\pi$
$59$	0.257264	−	0.293353i	0.257264	−	0.293353i	0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.57469	+	1.05217i	1.57469	+	1.05217i	0.965926	+	0.258819i	$0.0833333\pi$
$67$	1.57469	+	1.05217i	1.57469	+	1.05217i	0.608761	+	0.793353i	$0.291667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$71$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$72$		0			0
$73$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$73$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$			0.261052i			0.261052i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$79$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$80$		0			0
$81$	−0.692993	−	0.400100i	−0.692993	−	0.400100i
$82$		0			0
$83$	−1.50046	−	0.0983454i	−1.50046	−	0.0983454i	−0.707107	−	0.707107i	$-0.750000\pi$
$83$	−1.50046	−	0.0983454i	−1.50046	−	0.0983454i	−0.793353	+	0.608761i	$0.791667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.12484	−	0.465926i	−1.12484	−	0.465926i	−0.258819	−	0.965926i	$-0.583333\pi$
$89$	−1.12484	−	0.465926i	−1.12484	−	0.465926i	−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$97$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$98$		0			0
$99$	0.478235	+	0.0629609i	0.478235	+	0.0629609i
$100$		0			0
$101$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$101$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.641502	+	1.88981i	0.641502	+	1.88981i	0.382683	+	0.923880i	$0.375000\pi$
$107$	0.641502	+	1.88981i	0.641502	+	1.88981i	0.258819	+	0.965926i	$0.416667\pi$
$108$		0			0
$109$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$109$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.05441	−	0.608761i	−1.05441	−	0.608761i	−0.130526	−	0.991445i	$-0.541667\pi$
$113$	−1.05441	−	0.608761i	−1.05441	−	0.608761i	−0.923880	+	0.382683i	$0.875000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.707107	−	0.189469i	0.707107	−	0.189469i
$122$		0			0
$123$	0.274672	−	0.0932386i	0.274672	−	0.0932386i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$127$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$128$		0			0
$129$	−0.328618	−	0.252157i	−0.328618	−	0.252157i
$130$		0			0
$131$	−0.389345	+	1.95737i	−0.389345	+	1.95737i	−0.130526	+	0.991445i	$0.541667\pi$
$131$	−0.389345	+	1.95737i	−0.389345	+	1.95737i	−0.258819	+	0.965926i	$0.583333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.117317	−	1.78990i	−0.117317	−	1.78990i	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−0.117317	−	1.78990i	−0.117317	−	1.78990i	0.382683	−	0.923880i	$-0.375000\pi$
$138$		0			0
$139$	1.49144	−	0.996552i	1.49144	−	0.996552i	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.49144	−	0.996552i	1.49144	−	0.996552i	0.991445	−	0.130526i	$-0.0416667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.184592	−	0.184592i	0.184592	−	0.184592i
$148$		0			0
$149$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$149$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$150$		0			0
$151$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$151$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$152$		0			0
$153$	−1.73089	−	0.587557i	−1.73089	−	0.587557i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$157$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.608761	+	0.793353i	0.608761	+	0.793353i	0.991445	−	0.130526i	$-0.0416667\pi$
$163$	0.608761	+	0.793353i	0.608761	+	0.793353i	−0.382683	+	0.923880i	$0.625000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$167$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$168$		0			0
$169$	0.608761	−	0.793353i	0.608761	−	0.793353i
$170$		0			0
$171$	−1.02173	−	1.16506i	−1.02173	−	1.16506i
$172$		0			0
$173$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$173$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.00666180	−	0.101640i	0.00666180	−	0.101640i
$178$		0			0
$179$	−0.349942	−	0.172572i	−0.349942	−	0.172572i	0.258819	−	0.965926i	$-0.416667\pi$
$179$	−0.349942	−	0.172572i	−0.349942	−	0.172572i	−0.608761	+	0.793353i	$0.708333\pi$
$180$		0			0
$181$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$181$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.01321	−	0.0664093i	1.01321	−	0.0664093i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$191$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$192$		0			0
$193$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$193$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$197$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$198$		0			0
$199$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$199$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$200$		0			0
$201$	0.493338	−	0.0323351i	0.493338	−	0.0323351i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.772029	+	0.380722i	0.772029	+	0.380722i
$210$		0			0
$211$	−0.0420463	+	0.641502i	−0.0420463	+	0.641502i	0.923880	+	0.382683i	$0.125000\pi$
$211$	−0.0420463	+	0.641502i	−0.0420463	+	0.641502i	−0.965926	+	0.258819i	$0.916667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	0.224745	−	0.292893i	0.224745	−	0.292893i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$223$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$224$		0			0
$225$	−0.567275	−	0.739288i	−0.567275	−	0.739288i
$226$		0			0
$227$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$227$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.583242	−	1.18270i	0.583242	−	1.18270i	−0.382683	−	0.923880i	$-0.625000\pi$
$233$	0.583242	−	1.18270i	0.583242	−	1.18270i	0.965926	−	0.258819i	$-0.0833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$239$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$240$		0			0
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$		0			0
$243$	−0.707107	+	0.0930924i	−0.707107	+	0.0930924i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.326384	+	0.218083i	−0.326384	+	0.218083i
$250$		0			0
$251$	−0.0726721	−	1.10876i	−0.0726721	−	1.10876i	−0.866025	−	0.500000i	$-0.833333\pi$
$251$	−0.0726721	−	1.10876i	−0.0726721	−	1.10876i	0.793353	−	0.608761i	$-0.208333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.0255190	−	0.128293i	0.0255190	−	0.128293i	−0.965926	−	0.258819i	$-0.916667\pi$
$257$	0.0255190	−	0.128293i	0.0255190	−	0.128293i	0.991445	+	0.130526i	$0.0416667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$263$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.307007	+	0.0822623i	−0.307007	+	0.0822623i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$271$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.448288	+	0.258819i	0.448288	+	0.258819i
$276$		0			0
$277$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$277$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.608761	+	1.79335i	0.608761	+	1.79335i	0.608761	+	0.793353i	$0.291667\pi$
$281$	0.608761	+	1.79335i	0.608761	+	1.79335i			1.00000i	$0.500000\pi$
$282$		0			0
$283$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$283$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.82340	−	0.371707i	−2.82340	−	0.371707i
$290$		0			0
$291$	−0.0675653	+	0.252157i	−0.0675653	+	0.252157i
$292$		0			0
$293$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$293$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	0.226078	−	0.130526i	0.226078	−	0.130526i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.71723	+	0.991445i	1.71723	+	0.991445i	0.923880	+	0.382683i	$0.125000\pi$
$307$	1.71723	+	0.991445i	1.71723	+	0.991445i	0.793353	+	0.608761i	$0.208333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$311$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$312$		0			0
$313$			1.58671i			1.58671i	0.608761	+	0.793353i	$0.291667\pi$
$313$			1.58671i			1.58671i	−0.608761	+	0.793353i	$0.708333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$317$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	0.433185	+	0.289445i	0.433185	+	0.289445i
$322$		0			0
$323$	−2.58790	−	1.98576i	−2.58790	−	1.98576i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.0578541	−	0.882683i	−0.0578541	−	0.882683i	−0.923880	−	0.382683i	$-0.875000\pi$
$331$	−0.0578541	−	0.882683i	−0.0578541	−	0.882683i	0.866025	−	0.500000i	$-0.166667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.732626	−	1.09645i	−0.732626	−	1.09645i	−0.991445	−	0.130526i	$-0.958333\pi$
$337$	−0.732626	−	1.09645i	−0.732626	−	1.09645i	0.258819	−	0.965926i	$-0.416667\pi$
$338$		0			0
$339$	−0.315118	+	0.0414861i	−0.315118	+	0.0414861i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.24871	−	0.423880i	−1.24871	−	0.423880i	−0.382683	−	0.923880i	$-0.625000\pi$
$347$	−1.24871	−	0.423880i	−1.24871	−	0.423880i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$349$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$359$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$360$		0			0
$361$	−0.675577	−	1.63099i	−0.675577	−	1.63099i
$362$		0			0
$363$	0.116337	−	0.151613i	0.116337	−	0.151613i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$367$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$368$		0			0
$369$	−0.575247	+	0.860919i	−0.575247	+	0.860919i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$373$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.448288	−	1.67303i	0.448288	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$379$	0.448288	−	1.67303i	0.448288	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$383$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.47858			1.47858
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	0.230427	+	0.467259i	0.230427	+	0.467259i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$397$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.793353	−	0.391239i	0.793353	−	0.391239i		−	1.00000i	$-0.5\pi$
$401$	0.793353	−	0.391239i	0.793353	−	0.391239i	0.793353	+	0.608761i	$0.208333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.13053	−	0.991445i	1.13053	−	0.991445i	0.130526	−	0.991445i	$-0.458333\pi$
$409$	1.13053	−	0.991445i	1.13053	−	0.991445i	1.00000			$0$
$410$		0			0
$411$	−0.308746	−	0.352058i	−0.308746	−	0.352058i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.150518	−	0.443411i	0.150518	−	0.443411i
$418$		0			0
$419$	1.20711	+	1.57313i	1.20711	+	1.57313i	0.707107	+	0.707107i	$0.250000\pi$
$419$	1.20711	+	1.57313i	1.20711	+	1.57313i	0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.47479	−	1.29335i	−1.47479	−	1.29335i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$431$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$432$		0			0
$433$	1.85747	−	0.369474i	1.85747	−	0.369474i	0.866025	−	0.500000i	$-0.166667\pi$
$433$	1.85747	−	0.369474i	1.85747	−	0.369474i	0.991445	+	0.130526i	$0.0416667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$439$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$440$		0			0
$441$	−0.121631	+	0.923880i	−0.121631	+	0.923880i
$442$		0			0
$443$	−0.735499	+	0.491445i	−0.735499	+	0.491445i	−0.866025	−	0.500000i	$-0.833333\pi$
$443$	−0.735499	+	0.491445i	−0.735499	+	0.491445i	0.130526	+	0.991445i	$0.458333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.991445	−	1.71723i	0.991445	−	1.71723i	0.382683	−	0.923880i	$-0.375000\pi$
$449$	0.991445	−	1.71723i	0.991445	−	1.71723i	0.608761	−	0.793353i	$-0.291667\pi$
$450$		0			0
$451$	0.112210	−	0.564117i	0.112210	−	0.564117i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.128293	+	0.0255190i	0.128293	+	0.0255190i	0.258819	−	0.965926i	$-0.416667\pi$
$457$	0.128293	+	0.0255190i	0.128293	+	0.0255190i	−0.130526	+	0.991445i	$0.541667\pi$
$458$		0			0
$459$	−0.936749	+	0.317983i	−0.936749	+	0.317983i
$460$		0			0
$461$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$461$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$467$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.758819	+	0.314313i	−0.758819	+	0.314313i
$474$		0			0
$475$	−0.534534	−	1.57469i	−0.534534	−	1.57469i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$487$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$488$		0			0
$489$	0.252157	+	0.0675653i	0.252157	+	0.0675653i
$490$		0			0
$491$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$491$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.65938	+	0.108761i	1.65938	+	0.108761i	0.866025	−	0.500000i	$-0.166667\pi$
$499$	1.65938	+	0.108761i	1.65938	+	0.108761i	0.793353	+	0.608761i	$0.208333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$503$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	0.261052i		−	0.261052i
$508$		0			0
$509$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$509$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−0.822530	−	0.163611i	−0.822530	−	0.163611i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$522$		0			0
$523$	0.869474	−	0.991445i	0.869474	−	0.991445i	−0.130526	−	0.991445i	$-0.541667\pi$
$523$	0.869474	−	0.991445i	0.869474	−	0.991445i	1.00000			$0$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$530$		0			0
$531$	0.202000	+	0.302314i	0.202000	+	0.302314i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.0999004	+	0.0198714i	−0.0999004	+	0.0198714i
$538$		0			0
$539$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$540$		0			0
$541$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$541$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.793353	−	1.37413i	−0.793353	−	1.37413i	−0.923880	−	0.382683i	$-0.875000\pi$
$547$	−0.793353	−	1.37413i	−0.793353	−	1.37413i	0.130526	−	0.991445i	$-0.458333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$557$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	0.199289	−	0.174772i	0.199289	−	0.174772i
$562$		0			0
$563$	−1.08979	+	1.63099i	−1.08979	+	1.63099i	−0.382683	+	0.923880i	$0.625000\pi$
$563$	−1.08979	+	1.63099i	−1.08979	+	1.63099i	−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.576581	+	0.284338i	−0.576581	+	0.284338i	−0.707107	−	0.707107i	$-0.750000\pi$
$569$	−0.576581	+	0.284338i	−0.576581	+	0.284338i	0.130526	+	0.991445i	$0.458333\pi$
$570$		0			0
$571$	0.258819	+	1.96593i	0.258819	+	1.96593i	0.258819	+	0.965926i	$0.416667\pi$
$571$	0.258819	+	1.96593i	0.258819	+	1.96593i			1.00000i	$0.500000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.284338	+	0.576581i	0.284338	+	0.576581i	0.991445	−	0.130526i	$-0.0416667\pi$
$577$	0.284338	+	0.576581i	0.284338	+	0.576581i	−0.707107	+	0.707107i	$0.750000\pi$
$578$		0			0
$579$	0.207107	−	0.158919i	0.207107	−	0.158919i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.391239	−	0.793353i	−0.391239	−	0.793353i	0.608761	−	0.793353i	$-0.291667\pi$
$587$	−0.391239	−	0.793353i	−0.391239	−	0.793353i	−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.130526	+	0.991445i	0.130526	+	0.991445i	0.923880	+	0.382683i	$0.125000\pi$
$593$	0.130526	+	0.991445i	0.130526	+	0.991445i	−0.793353	+	0.608761i	$0.791667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$599$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$600$		0			0
$601$	0.216773	−	0.324423i	0.216773	−	0.324423i	−0.707107	−	0.707107i	$-0.750000\pi$
$601$	0.216773	−	0.324423i	0.216773	−	0.324423i	0.923880	+	0.382683i	$0.125000\pi$
$602$		0			0
$603$	−1.32684	+	1.16361i	−1.32684	+	1.16361i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$607$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$613$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$617$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.130526	−	0.991445i	$-0.541667\pi$
$618$		0			0
$619$	−0.835400	−	0.732626i	−0.835400	−	0.732626i	0.130526	−	0.991445i	$-0.458333\pi$
$619$	−0.835400	−	0.732626i	−0.835400	−	0.732626i	−0.965926	+	0.258819i	$0.916667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$626$		0			0
$627$	0.220396	−	0.0438395i	0.220396	−	0.0438395i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$631$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$632$		0			0
$633$	0.0932386	+	0.139541i	0.0932386	+	0.139541i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.18270	−	1.34861i	1.18270	−	1.34861i	0.258819	−	0.965926i	$-0.416667\pi$
$641$	1.18270	−	1.34861i	1.18270	−	1.34861i	0.923880	−	0.382683i	$-0.125000\pi$
$642$		0			0
$643$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$643$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$647$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$648$		0			0
$649$	−0.167934	−	0.112210i	−0.167934	−	0.112210i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$653$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			1.31784i			1.31784i
$658$		0			0
$659$	−0.0255190	−	0.128293i	−0.0255190	−	0.128293i	0.965926	−	0.258819i	$-0.0833333\pi$
$659$	−0.0255190	−	0.128293i	−0.0255190	−	0.128293i	−0.991445	+	0.130526i	$0.958333\pi$
$660$		0			0
$661$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$661$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.71723	+	0.991445i	−1.71723	+	0.991445i	−0.793353	+	0.608761i	$0.791667\pi$
$673$	−1.71723	+	0.991445i	−1.71723	+	0.991445i	−0.923880	+	0.382683i	$0.875000\pi$
$674$		0			0
$675$	−0.487130	−	0.130526i	−0.487130	−	0.130526i
$676$		0			0
$677$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$677$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−0.448288	−	0.0590182i	−0.448288	−	0.0590182i
$682$		0			0
$683$	−1.91532	−	0.513210i	−1.91532	−	0.513210i	−0.991445	−	0.130526i	$-0.958333\pi$
$683$	−1.91532	−	0.513210i	−1.91532	−	0.513210i	−0.923880	−	0.382683i	$-0.875000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.707107	−	0.292893i	0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$691$	0.707107	−	0.292893i	0.707107	−	0.292893i	0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.834089	+	2.01367i	−0.834089	+	2.01367i
$698$		0			0
$699$	−0.0671594	−	0.337633i	−0.0671594	−	0.337633i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$709$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$710$		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		−0.0654031	−	0.997859i	$-0.520833\pi$
$719$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$727$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$728$		0			0
$729$	0.434174	−	0.434174i	0.434174	−	0.434174i
$730$		0			0
$731$	3.05263	−	0.607206i	3.05263	−	0.607206i
$732$		0			0
$733$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$733$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.433591	−	0.879235i	0.433591	−	0.879235i
$738$		0			0
$739$	1.34861	+	1.18270i	1.34861	+	1.18270i	0.965926	+	0.258819i	$0.0833333\pi$
$739$	1.34861	+	1.18270i	1.34861	+	1.18270i	0.382683	+	0.923880i	$0.375000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.450403	−	1.32684i	0.450403	−	1.32684i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$751$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$752$		0			0
$753$	−0.191254	−	0.218083i	−0.191254	−	0.218083i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$757$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.75928	+	0.867580i	1.75928	+	0.867580i	0.965926	+	0.258819i	$0.0833333\pi$
$761$	1.75928	+	0.867580i	1.75928	+	0.867580i	0.793353	+	0.608761i	$0.208333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.65938	−	0.108761i	1.65938	−	0.108761i	0.793353	−	0.608761i	$-0.208333\pi$
$769$	1.65938	−	0.108761i	1.65938	−	0.108761i	0.866025	+	0.500000i	$0.166667\pi$
$770$		0			0
$771$	−0.0151030	−	0.0306258i	−0.0151030	−	0.0306258i
$772$		0			0
$773$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$773$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.46593	+	1.12484i	−1.46593	+	1.12484i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−0.241181	−	1.83195i	−0.241181	−	1.83195i	−0.500000	−	0.866025i	$-0.666667\pi$
$787$	−0.241181	−	1.83195i	−0.241181	−	1.83195i	0.258819	−	0.965926i	$-0.416667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$797$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	0.690671	−	0.900100i	0.690671	−	0.900100i
$802$		0			0
$803$	−0.280144	−	0.676327i	−0.280144	−	0.676327i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$809$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$810$		0			0
$811$	0.608761	+	1.05441i	0.608761	+	1.05441i	0.991445	+	0.130526i	$0.0416667\pi$
$811$	0.608761	+	1.05441i	0.608761	+	1.05441i	−0.382683	+	0.923880i	$0.625000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.49857	+	0.848149i	2.49857	+	0.848149i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$821$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$	0.133975	−	0.0176381i	0.133975	−	0.0176381i
$826$		0			0
$827$	−0.357164	−	0.534534i	−0.357164	−	0.534534i	0.608761	−	0.793353i	$-0.291667\pi$
$827$	−0.357164	−	0.534534i	−0.357164	−	0.534534i	−0.965926	+	0.258819i	$0.916667\pi$
$828$		0			0
$829$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$829$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.128293	+	1.95737i	0.128293	+	1.95737i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$839$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$840$		0			0
$841$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$842$		0			0
$843$	0.411076	+	0.274672i	0.411076	+	0.274672i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−0.356604	+	0.0955518i	−0.356604	+	0.0955518i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$853$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.448288	+	0.258819i	0.448288	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$857$	0.448288	+	0.258819i	0.448288	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$858$		0			0
$859$	1.99144	+	0.130526i	1.99144	+	0.130526i	1.00000			$0$
$859$	1.99144	+	0.130526i	1.99144	+	0.130526i	0.991445	+	0.130526i	$0.0416667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$863$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.643816	+	0.371707i	−0.643816	+	0.371707i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.356604	−	0.860919i	−0.356604	−	0.860919i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$877$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.60021	−	0.662827i	−1.60021	−	0.662827i	−0.608761	−	0.793353i	$-0.708333\pi$
$881$	−1.60021	−	0.662827i	−1.60021	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$882$		0			0
$883$	−0.641502	−	1.88981i	−0.641502	−	1.88981i	−0.382683	−	0.923880i	$-0.625000\pi$
$883$	−0.641502	−	1.88981i	−0.641502	−	1.88981i	−0.258819	−	0.965926i	$-0.583333\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$887$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.158513	+	0.382683i	−0.158513	+	0.382683i
$892$		0			0
$893$		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
$907$	−0.293353	+	1.47479i	−0.293353	+	1.47479i	0.500000	+	0.866025i	$0.333333\pi$
$907$	−0.293353	+	1.47479i	−0.293353	+	1.47479i	−0.793353	+	0.608761i	$0.791667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$911$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$912$		0			0
$913$	0.0509073	+	0.776695i	0.0509073	+	0.776695i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$919$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$920$		0			0
$921$	0.513210	−	0.0675653i	0.513210	−	0.0675653i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.123864	−	0.0420463i	−0.123864	−	0.0420463i	0.258819	−	0.965926i	$-0.416667\pi$
$929$	−0.123864	−	0.0420463i	−0.123864	−	0.0420463i	−0.382683	+	0.923880i	$0.625000\pi$
$930$		0			0
$931$	−0.735499	+	1.49144i	−0.735499	+	1.49144i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$937$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$938$		0			0
$939$	0.252157	+	0.328618i	0.252157	+	0.328618i
$940$		0			0
$941$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$941$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.991445	+	1.13053i	0.991445	+	1.13053i	0.991445	+	0.130526i	$0.0416667\pi$
$947$	0.991445	+	1.13053i	0.991445	+	1.13053i			1.00000i	$0.500000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.0255190	+	0.389345i	−0.0255190	+	0.389345i	0.965926	+	0.258819i	$0.0833333\pi$
$953$	−0.0255190	+	0.389345i	−0.0255190	+	0.389345i	−0.991445	+	0.130526i	$0.958333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$962$		0			0
$963$	−1.85573	+	0.121631i	−1.85573	+	0.121631i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$967$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$968$		0			0
$969$	−0.851546			−0.851546
$970$		0			0
$971$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$971$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.991445	−	0.130526i	$-0.958333\pi$
$977$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.965926	+	0.258819i	$0.916667\pi$
$978$		0			0
$979$	−0.163117	+	0.608761i	−0.163117	+	0.608761i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$983$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$991$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$992$		0			0
$993$	−0.152257	−	0.173616i	−0.152257	−	0.173616i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$997$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3104.1.et.a.1263.1		16
4.3	odd	2		776.1.bp.a.99.1	✓	16
8.3	odd	2	CM	3104.1.et.a.1263.1		16
8.5	even	2		776.1.bp.a.99.1	✓	16
97.49	even	48	inner	3104.1.et.a.2959.1		16
388.243	odd	48		776.1.bp.a.243.1	yes	16
776.243	odd	48	inner	3104.1.et.a.2959.1		16
776.437	even	48		776.1.bp.a.243.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
776.1.bp.a.99.1	✓	16	4.3	odd	2
776.1.bp.a.99.1	✓	16	8.5	even	2
776.1.bp.a.243.1	yes	16	388.243	odd	48
776.1.bp.a.243.1	yes	16	776.437	even	48
3104.1.et.a.1263.1		16	1.1	even	1	trivial
3104.1.et.a.1263.1		16	8.3	odd	2	CM
3104.1.et.a.2959.1		16	97.49	even	48	inner
3104.1.et.a.2959.1		16	776.243	odd	48	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:	\( N \)	\(=\)	\( 3104 = 2^{5} \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3104.et (of order \(48\), degree \(16\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.207107 - 0.158919i) q^{3} +(-0.241181 + 0.900100i) q^{9} +(-0.0675653 - 0.513210i) q^{11} +(-0.128293 + 1.95737i) q^{17} +(-0.923880 + 1.38268i) q^{19} +(-0.608761 + 0.793353i) q^{25} +(0.192993 + 0.465926i) q^{27} +(-0.0955518 - 0.0955518i) q^{33} +(1.05217 + 0.357164i) q^{41} +(-0.410670 - 1.53264i) q^{43} +(0.991445 - 0.130526i) q^{49} +(0.284492 + 0.425773i) q^{51} +(0.0283924 + 0.433185i) q^{57} +(0.257264 - 0.293353i) q^{59} +(1.57469 + 1.05217i) q^{67} +(1.36603 - 0.366025i) q^{73} +0.261052i q^{75} +(-0.692993 - 0.400100i) q^{81} +(-1.50046 - 0.0983454i) q^{83} +(-1.12484 - 0.465926i) q^{89} +(-0.793353 + 0.608761i) q^{97} +(0.478235 + 0.0629609i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{3} - 8 q^{9} + 16 q^{27} + 8 q^{73} - 24 q^{81}+O(q^{100}) \) 16 * q - 8 * q^3 - 8 * q^9 + 16 * q^27 + 8 * q^73 - 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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