LMFDB - Embedded newform 3375.1.g.a.568.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3375,1,Mod(568,3375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3375, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("3375.568");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3375 = 3^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3375.g (of order $4$, degree $2$, 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:	$1.68434441764$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.6879707136000000000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 21x^{12} + 86x^{8} + 36x^{4} + 1 $ x^16 + 21x^12 + 86x^8 + 36*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{30}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{30} - \cdots)$

Embedding invariants

Embedding label			568.8
Root			$0.575212 + 0.575212i$ of defining polynomial
Character	$\chi$	$=$	3375.568
Dual form			3375.1.g.a.1432.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.40647 - 1.40647i) q^{2} -2.95630i q^{4} +(-2.75146 - 2.75146i) q^{8} +O(q^{10})$	$q+(1.40647 - 1.40647i) q^{2} -2.95630i q^{4} +(-2.75146 - 2.75146i) q^{8} -4.78339 q^{16} +(1.05097 - 1.05097i) q^{17} +0.209057i q^{19} +(-0.294032 - 0.294032i) q^{23} -1.33826 q^{31} +(-3.97621 + 3.97621i) q^{32} -2.95630i q^{34} +(0.294032 + 0.294032i) q^{38} -0.827091 q^{46} +(0.831254 - 0.831254i) q^{47} +1.00000i q^{49} +(0.575212 + 0.575212i) q^{53} +1.82709 q^{61} +(-1.88222 + 1.88222i) q^{62} +6.40142i q^{64} +(-3.10696 - 3.10696i) q^{68} +0.618034 q^{76} +1.82709i q^{79} +(-0.575212 - 0.575212i) q^{83} +(-0.869244 + 0.869244i) q^{92} -2.33826i q^{94} +(1.40647 + 1.40647i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{16} - 4 q^{31} + 12 q^{46} + 4 q^{61} - 8 q^{76}+O(q^{100}) $ 16 * q - 16 * q^16 - 4 * q^31 + 12 * q^46 + 4 * q^61 - 8 * q^76

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3375\mathbb{Z}\right)^\times$.

$n$	$1001$	$2377$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.629320	−	0.777146i	$-0.283333\pi$
$2$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.777146	−	0.629320i	$-0.216667\pi$
$3$		0			0
$3$		0			0
$4$		−	2.95630i		−	2.95630i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	−2.75146	−	2.75146i	−2.75146	−	2.75146i
$9$		0			0
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$	−4.78339			−4.78339
$17$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.0523360	−	0.998630i	$-0.483333\pi$
$17$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.998630	−	0.0523360i	$-0.0166667\pi$
$18$		0			0
$19$			0.209057i			0.209057i	0.994522	+	0.104528i	$0.0333333\pi$
$19$			0.209057i			0.209057i	−0.994522	+	0.104528i	$0.966667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.294032	−	0.294032i	−0.294032	−	0.294032i	0.544639	−	0.838671i	$-0.316667\pi$
$23$	−0.294032	−	0.294032i	−0.294032	−	0.294032i	−0.838671	+	0.544639i	$0.816667\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−1.33826			−1.33826			−0.669131	−	0.743145i	$-0.733333\pi$
$31$	−1.33826			−1.33826			−0.669131	+	0.743145i	$0.733333\pi$
$32$	−3.97621	+	3.97621i	−3.97621	+	3.97621i
$33$		0			0
$34$		−	2.95630i		−	2.95630i
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	0.294032	+	0.294032i	0.294032	+	0.294032i
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$	−0.827091			−0.827091
$47$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$47$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$48$		0			0
$49$			1.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.575212	+	0.575212i	0.575212	+	0.575212i	0.933580	−	0.358368i	$-0.116667\pi$
$53$	0.575212	+	0.575212i	0.575212	+	0.575212i	−0.358368	+	0.933580i	$0.616667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$61$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$62$	−1.88222	+	1.88222i	−1.88222	+	1.88222i
$63$		0			0
$64$			6.40142i			6.40142i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	−3.10696	−	3.10696i	−3.10696	−	3.10696i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$		0			0
$76$	0.618034			0.618034
$77$		0			0
$78$		0			0
$79$			1.82709i			1.82709i	0.406737	+	0.913545i	$0.366667\pi$
$79$			1.82709i			1.82709i	−0.406737	+	0.913545i	$0.633333\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.575212	−	0.575212i	−0.575212	−	0.575212i	0.358368	−	0.933580i	$-0.383333\pi$
$83$	−0.575212	−	0.575212i	−0.575212	−	0.575212i	−0.933580	+	0.358368i	$0.883333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−0.869244	+	0.869244i	−0.869244	+	0.869244i
$93$		0			0
$94$		−	2.33826i		−	2.33826i
$95$		0			0
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	1.40647	+	1.40647i	1.40647	+	1.40647i
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$	1.61803			1.61803
$107$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$107$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$108$		0			0
$109$			1.33826i			1.33826i	0.743145	+	0.669131i	$0.233333\pi$
$109$			1.33826i			1.33826i	−0.743145	+	0.669131i	$0.766667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.831254	+	0.831254i	0.831254	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$113$	0.831254	+	0.831254i	0.831254	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−1.00000
$122$	2.56974	−	2.56974i	2.56974	−	2.56974i
$123$		0			0
$124$			3.95630i			3.95630i
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	5.02717	+	5.02717i	5.02717	+	5.02717i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−5.78339			−5.78339
$137$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$137$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$138$		0			0
$139$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$139$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$140$		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			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$151$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$152$	0.575212	−	0.575212i	0.575212	−	0.575212i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$	2.56974	+	2.56974i	2.56974	+	2.56974i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$	−1.61803			−1.61803
$167$	0.575212	−	0.575212i	0.575212	−	0.575212i	−0.358368	−	0.933580i	$-0.616667\pi$
$167$	0.575212	−	0.575212i	0.575212	−	0.575212i	0.933580	+	0.358368i	$0.116667\pi$
$168$		0			0
$169$		−	1.00000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$173$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	0.209057			0.209057			0.104528	−	0.994522i	$-0.466667\pi$
$181$	0.209057			0.209057			0.104528	+	0.994522i	$0.466667\pi$
$182$		0			0
$183$		0			0
$184$			1.61803i			1.61803i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−2.45743	−	2.45743i	−2.45743	−	2.45743i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	2.95630			2.95630
$197$	0.294032	−	0.294032i	0.294032	−	0.294032i	−0.544639	−	0.838671i	$-0.683333\pi$
$197$	0.294032	−	0.294032i	0.294032	−	0.294032i	0.838671	+	0.544639i	$0.183333\pi$
$198$		0			0
$199$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$199$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$211$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$212$	1.70050	−	1.70050i	1.70050	−	1.70050i
$213$		0			0
$214$		−	3.78339i		−	3.78339i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.88222	+	1.88222i	1.88222	+	1.88222i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		0			0
$226$	2.33826			2.33826
$227$	−1.05097	+	1.05097i	−1.05097	+	1.05097i	−0.0523360	+	0.998630i	$0.516667\pi$
$227$	−1.05097	+	1.05097i	−1.05097	+	1.05097i	−0.998630	+	0.0523360i	$0.983333\pi$
$228$		0			0
$229$			1.95630i			1.95630i	0.207912	+	0.978148i	$0.433333\pi$
$229$			1.95630i			1.95630i	−0.207912	+	0.978148i	$0.566667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$241$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$242$	−1.40647	+	1.40647i	−1.40647	+	1.40647i
$243$		0			0
$244$		−	5.40142i		−	5.40142i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	3.68218	+	3.68218i	3.68218	+	3.68218i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	7.73968			7.73968
$257$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	−0.933580	−	0.358368i	$-0.883333\pi$
$257$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	0.358368	+	0.933580i	$0.383333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$263$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$271$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$272$	−5.02717	+	5.02717i	−5.02717	+	5.02717i
$273$		0			0
$274$		−	3.44512i		−	3.44512i
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	−0.869244	−	0.869244i	−0.869244	−	0.869244i
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	1.20906i		−	1.20906i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.294032	+	0.294032i	0.294032	+	0.294032i	0.838671	−	0.544639i	$-0.183333\pi$
$293$	0.294032	+	0.294032i	0.294032	+	0.294032i	−0.544639	+	0.838671i	$0.683333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	2.27571	−	2.27571i	2.27571	−	2.27571i
$303$		0			0
$304$		−	1.00000i		−	1.00000i
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$	5.40142			5.40142
$317$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$317$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.219712	+	0.219712i	0.219712	+	0.219712i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$331$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$332$	−1.70050	+	1.70050i	−1.70050	+	1.70050i
$333$		0			0
$334$		−	1.61803i		−	1.61803i
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	−1.40647	−	1.40647i	−1.40647	−	1.40647i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−3.44512			−3.44512
$347$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$347$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$348$		0			0
$349$			1.95630i			1.95630i	0.207912	+	0.978148i	$0.433333\pi$
$349$			1.95630i			1.95630i	−0.207912	+	0.978148i	$0.566667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$353$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	0.956295			0.956295
$362$	0.294032	−	0.294032i	0.294032	−	0.294032i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$	1.40647	+	1.40647i	1.40647	+	1.40647i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$	−4.57433			−4.57433
$377$		0			0
$378$		0			0
$379$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$379$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.05097	−	1.05097i	−1.05097	−	1.05097i	−0.998630	−	0.0523360i	$-0.983333\pi$
$383$	−1.05097	−	1.05097i	−1.05097	−	1.05097i	−0.0523360	−	0.998630i	$-0.516667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	−0.618034			−0.618034
$392$	2.75146	−	2.75146i	2.75146	−	2.75146i
$393$		0			0
$394$		−	0.827091i		−	0.827091i
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	0.869244	+	0.869244i	0.869244	+	0.869244i
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	1.82709i		−	1.82709i	−0.406737	−	0.913545i	$-0.633333\pi$
$409$		−	1.82709i		−	1.82709i	0.406737	−	0.913545i	$-0.366667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	0.209057			0.209057			0.104528	−	0.994522i	$-0.466667\pi$
$421$	0.209057			0.209057			0.104528	+	0.994522i	$0.466667\pi$
$422$	−2.75146	+	2.75146i	−2.75146	+	2.75146i
$423$		0			0
$424$		−	3.16535i		−	3.16535i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−3.97621	−	3.97621i	−3.97621	−	3.97621i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$	3.95630			3.95630
$437$	0.0614693	−	0.0614693i	0.0614693	−	0.0614693i
$438$		0			0
$439$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$439$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.40647	+	1.40647i	1.40647	+	1.40647i	0.777146	+	0.629320i	$0.216667\pi$
$443$	1.40647	+	1.40647i	1.40647	+	1.40647i	0.629320	+	0.777146i	$0.283333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$	2.45743	−	2.45743i	2.45743	−	2.45743i
$453$		0			0
$454$			2.95630i			2.95630i
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	2.75146	+	2.75146i	2.75146	+	2.75146i
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.629320	−	0.777146i	$-0.283333\pi$
$467$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.777146	−	0.629320i	$-0.216667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$	1.88222	−	1.88222i	1.88222	−	1.88222i
$483$		0			0
$484$			2.95630i			2.95630i
$485$		0			0
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	−5.02717	−	5.02717i	−5.02717	−	5.02717i
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	6.40142			6.40142
$497$		0			0
$498$		0			0
$499$			1.82709i			1.82709i	0.406737	+	0.913545i	$0.366667\pi$
$499$			1.82709i			1.82709i	−0.406737	+	0.913545i	$0.633333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$503$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	5.85843	−	5.85843i	5.85843	−	5.85843i
$513$		0			0
$514$			1.61803i			1.61803i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$	3.78339			3.78339
$527$	−1.40647	+	1.40647i	−1.40647	+	1.40647i
$528$		0			0
$529$		−	0.827091i		−	0.827091i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$541$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$542$	−2.75146	+	2.75146i	−2.75146	+	2.75146i
$543$		0			0
$544$			8.35772i			8.35772i
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	−3.62071	−	3.62071i	−3.62071	−	3.62071i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.82709			−1.82709
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$563$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−1.70050	−	1.70050i	−1.70050	−	1.70050i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.827091			0.827091
$587$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	−0.838671	−	0.544639i	$-0.816667\pi$
$587$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	0.544639	+	0.838671i	$0.316667\pi$
$588$		0			0
$589$		−	0.279773i		−	0.279773i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.777146	−	0.629320i	$-0.783333\pi$
$593$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.629320	−	0.777146i	$-0.716667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	1.95630			1.95630			0.978148	−	0.207912i	$-0.0666667\pi$
$601$	1.95630			1.95630			0.978148	+	0.207912i	$0.0666667\pi$
$602$		0			0
$603$		0			0
$604$		−	4.78339i		−	4.78339i
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$	−0.831254	−	0.831254i	−0.831254	−	0.831254i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	−0.838671	−	0.544639i	$-0.816667\pi$
$617$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	0.544639	+	0.838671i	$0.316667\pi$
$618$		0			0
$619$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$619$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$632$	5.02717	−	5.02717i	5.02717	−	5.02717i
$633$		0			0
$634$			3.44512i			3.44512i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$	0.618034			0.618034
$647$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$647$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.40647	+	1.40647i	1.40647	+	1.40647i	0.777146	+	0.629320i	$0.216667\pi$
$653$	1.40647	+	1.40647i	1.40647	+	1.40647i	0.629320	+	0.777146i	$0.283333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$661$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$662$	−2.27571	+	2.27571i	−2.27571	+	2.27571i
$663$		0			0
$664$			3.16535i			3.16535i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.70050	−	1.70050i	−1.70050	−	1.70050i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$	−2.95630			−2.95630
$677$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$677$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.294032	+	0.294032i	0.294032	+	0.294032i	0.838671	−	0.544639i	$-0.183333\pi$
$683$	0.294032	+	0.294032i	0.294032	+	0.294032i	−0.544639	+	0.838671i	$0.683333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.33826			−1.33826			−0.669131	−	0.743145i	$-0.733333\pi$
$691$	−1.33826			−1.33826			−0.669131	+	0.743145i	$0.733333\pi$
$692$	−3.62071	+	3.62071i	−3.62071	+	3.62071i
$693$		0			0
$694$			2.33826i			2.33826i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	2.75146	+	2.75146i	2.75146	+	2.75146i
$699$		0			0
$700$		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
$706$	−3.78339			−3.78339
$707$		0			0
$708$		0			0
$709$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$709$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.393491	+	0.393491i	0.393491	+	0.393491i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	1.34500	−	1.34500i	1.34500	−	1.34500i
$723$		0			0
$724$		−	0.618034i		−	0.618034i
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$	2.33826			2.33826
$737$		0			0
$738$		0			0
$739$		−	1.33826i		−	1.33826i	−0.743145	−	0.669131i	$-0.766667\pi$
$739$		−	1.33826i		−	1.33826i	0.743145	−	0.669131i	$-0.233333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.82709			−1.82709			−0.913545	−	0.406737i	$-0.866667\pi$
$751$	−1.82709			−1.82709			−0.913545	+	0.406737i	$0.866667\pi$
$752$	−3.97621	+	3.97621i	−3.97621	+	3.97621i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	1.40647	+	1.40647i	1.40647	+	1.40647i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−2.95630			−2.95630
$767$		0			0
$768$		0			0
$769$		−	0.209057i		−	0.209057i	−0.994522	−	0.104528i	$-0.966667\pi$
$769$		−	0.209057i		−	0.209057i	0.994522	−	0.104528i	$-0.0333333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.777146	−	0.629320i	$-0.783333\pi$
$773$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.629320	−	0.777146i	$-0.716667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.869244	+	0.869244i	−0.869244	+	0.869244i
$783$		0			0
$784$		−	4.78339i		−	4.78339i
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	−0.869244	−	0.869244i	−0.869244	−	0.869244i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.82709			1.82709
$797$	−1.05097	+	1.05097i	−1.05097	+	1.05097i	−0.0523360	+	0.998630i	$0.516667\pi$
$797$	−1.05097	+	1.05097i	−1.05097	+	1.05097i	−0.998630	+	0.0523360i	$0.983333\pi$
$798$		0			0
$799$		−	1.74724i		−	1.74724i
$800$		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		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$811$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−2.56974	−	2.56974i	−2.56974	−	2.56974i
$819$		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			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	−0.933580	−	0.358368i	$-0.883333\pi$
$827$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	0.358368	+	0.933580i	$0.383333\pi$
$828$		0			0
$829$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$829$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.05097	+	1.05097i	1.05097	+	1.05097i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$	0.294032	−	0.294032i	0.294032	−	0.294032i
$843$		0			0
$844$			5.78339i			5.78339i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−2.75146	−	2.75146i	−2.75146	−	2.75146i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$	−7.40142			−7.40142
$857$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.629320	+	0.777146i	$0.716667\pi$
$857$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.777146	+	0.629320i	$0.783333\pi$
$858$		0			0
$859$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$859$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$863$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	3.68218	−	3.68218i	3.68218	−	3.68218i
$873$		0			0
$874$		−	0.172909i		−	0.172909i
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$	−1.40647	−	1.40647i	−1.40647	−	1.40647i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$	3.95630			3.95630
$887$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.629320	+	0.777146i	$0.716667\pi$
$887$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.777146	+	0.629320i	$0.783333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.173779	+	0.173779i	0.173779	+	0.173779i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	1.20906			1.20906
$902$		0			0
$903$		0			0
$904$		−	4.57433i		−	4.57433i
$905$		0			0
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$	3.10696	+	3.10696i	3.10696	+	3.10696i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	5.78339			5.78339
$917$		0			0
$918$		0			0
$919$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$919$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	−0.209057			−0.209057
$932$		0			0
$933$		0			0
$934$		−	3.95630i		−	3.95630i
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	−0.838671	−	0.544639i	$-0.816667\pi$
$947$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	0.544639	+	0.838671i	$0.316667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$953$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.790943			0.790943
$962$		0			0
$963$		0			0
$964$		−	3.95630i		−	3.95630i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	2.75146	+	2.75146i	2.75146	+	2.75146i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−8.73968			−8.73968
$977$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$977$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.777146	−	0.629320i	$-0.783333\pi$
$983$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.629320	−	0.777146i	$-0.716667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$991$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$992$	5.32121	−	5.32121i	5.32121	−	5.32121i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$	2.56974	+	2.56974i	2.56974	+	2.56974i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3375.1.g.a.568.8	yes	16
3.2	odd	2	inner	3375.1.g.a.568.1	✓	16
5.2	odd	4	inner	3375.1.g.a.1432.8	yes	16
5.3	odd	4	inner	3375.1.g.a.1432.1	yes	16
5.4	even	2	inner	3375.1.g.a.568.1	✓	16
15.2	even	4	inner	3375.1.g.a.1432.1	yes	16
15.8	even	4	inner	3375.1.g.a.1432.8	yes	16
15.14	odd	2	CM	3375.1.g.a.568.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3375.1.g.a.568.1	✓	16	3.2	odd	2	inner
3375.1.g.a.568.1	✓	16	5.4	even	2	inner
3375.1.g.a.568.8	yes	16	1.1	even	1	trivial
3375.1.g.a.568.8	yes	16	15.14	odd	2	CM
3375.1.g.a.1432.1	yes	16	5.3	odd	4	inner
3375.1.g.a.1432.1	yes	16	15.2	even	4	inner
3375.1.g.a.1432.8	yes	16	5.2	odd	4	inner
3375.1.g.a.1432.8	yes	16	15.8	even	4	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:	\( N \)	\(=\)	\( 3375 = 3^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3375.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.40647 - 1.40647i) q^{2} -2.95630i q^{4} +(-2.75146 - 2.75146i) q^{8} +O(q^{10})\)	\(q+(1.40647 - 1.40647i) q^{2} -2.95630i q^{4} +(-2.75146 - 2.75146i) q^{8} -4.78339 q^{16} +(1.05097 - 1.05097i) q^{17} +0.209057i q^{19} +(-0.294032 - 0.294032i) q^{23} -1.33826 q^{31} +(-3.97621 + 3.97621i) q^{32} -2.95630i q^{34} +(0.294032 + 0.294032i) q^{38} -0.827091 q^{46} +(0.831254 - 0.831254i) q^{47} +1.00000i q^{49} +(0.575212 + 0.575212i) q^{53} +1.82709 q^{61} +(-1.88222 + 1.88222i) q^{62} +6.40142i q^{64} +(-3.10696 - 3.10696i) q^{68} +0.618034 q^{76} +1.82709i q^{79} +(-0.575212 - 0.575212i) q^{83} +(-0.869244 + 0.869244i) q^{92} -2.33826i q^{94} +(1.40647 + 1.40647i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{16} - 4 q^{31} + 12 q^{46} + 4 q^{61} - 8 q^{76}+O(q^{100}) \) 16 * q - 16 * q^16 - 4 * q^31 + 12 * q^46 + 4 * q^61 - 8 * q^76

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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