LMFDB - Embedded newform 648.2.i.i.217.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,2,Mod(217,648)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("648.217"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.17430605098$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.2.i.i.433.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.86603 - 3.23205i) q^{5} +(-1.73205 + 3.00000i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(1.23205 + 2.13397i) q^{13} +2.26795 q^{17} -7.46410 q^{19} +(2.46410 + 4.26795i) q^{23} +(-4.46410 + 7.73205i) q^{25} +(-2.13397 + 3.69615i) q^{29} +(5.46410 + 9.46410i) q^{31} +12.9282 q^{35} -0.464102 q^{37} +(-3.46410 - 6.00000i) q^{41} +(2.26795 - 3.92820i) q^{43} +(-3.46410 + 6.00000i) q^{47} +(-2.50000 - 4.33013i) q^{49} -10.9282 q^{53} +7.46410 q^{55} +(-4.00000 - 6.92820i) q^{59} +(-5.23205 + 9.06218i) q^{61} +(4.59808 - 7.96410i) q^{65} +(0.267949 + 0.464102i) q^{67} -2.00000 q^{71} +1.00000 q^{73} +(-3.46410 - 6.00000i) q^{77} +(-0.267949 + 0.464102i) q^{79} +(1.46410 - 2.53590i) q^{83} +(-4.23205 - 7.33013i) q^{85} -5.19615 q^{89} -8.53590 q^{91} +(13.9282 + 24.1244i) q^{95} +(5.92820 - 10.2679i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{11} - 2 q^{13} + 16 q^{17} - 16 q^{19} - 4 q^{23} - 4 q^{25} - 12 q^{29} + 8 q^{31} + 24 q^{35} + 12 q^{37} + 16 q^{43} - 10 q^{49} - 16 q^{53} + 16 q^{55} - 16 q^{59} - 14 q^{61} + 8 q^{65}+ \cdots - 4 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^11 - 2 * q^13 + 16 * q^17 - 16 * q^19 - 4 * q^23 - 4 * q^25 - 12 * q^29 + 8 * q^31 + 24 * q^35 + 12 * q^37 + 16 * q^43 - 10 * q^49 - 16 * q^53 + 16 * q^55 - 16 * q^59 - 14 * q^61 + 8 * q^65 + 8 * q^67 - 8 * q^71 + 4 * q^73 - 8 * q^79 - 8 * q^83 - 10 * q^85 - 48 * q^91 + 28 * q^95 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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			0
$3$		0			0
$4$		0			0
$5$	−1.86603	−	3.23205i	−0.834512	−	1.44542i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−1.86603	−	3.23205i	−0.834512	−	1.44542i	0.0599153	−	0.998203i	$-0.480917\pi$
$6$		0			0
$7$	−1.73205	+	3.00000i	−0.654654	+	1.13389i	0.327327	+	0.944911i	$0.393852\pi$
$7$	−1.73205	+	3.00000i	−0.654654	+	1.13389i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	1.23205	+	2.13397i	0.341709	+	0.591858i	0.984750	−	0.173974i	$-0.0556608\pi$
$13$	1.23205	+	2.13397i	0.341709	+	0.591858i	−0.643041	+	0.765832i	$0.722327\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.26795			0.550058			0.275029	−	0.961436i	$-0.411312\pi$
$17$	2.26795			0.550058			0.275029	+	0.961436i	$0.411312\pi$
$18$		0			0
$19$	−7.46410			−1.71238			−0.856191	−	0.516659i	$-0.827175\pi$
$19$	−7.46410			−1.71238			−0.856191	+	0.516659i	$0.827175\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.46410	+	4.26795i	0.513801	+	0.889929i	0.999872	+	0.0160097i	$0.00509626\pi$
$23$	2.46410	+	4.26795i	0.513801	+	0.889929i	−0.486071	+	0.873919i	$0.661570\pi$
$24$		0			0
$25$	−4.46410	+	7.73205i	−0.892820	+	1.54641i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.13397	+	3.69615i	−0.396269	+	0.686358i	−0.993262	−	0.115888i	$-0.963029\pi$
$29$	−2.13397	+	3.69615i	−0.396269	+	0.686358i	0.596993	+	0.802246i	$0.296362\pi$
$30$		0			0
$31$	5.46410	+	9.46410i	0.981382	+	1.69980i	0.657027	+	0.753867i	$0.271814\pi$
$31$	5.46410	+	9.46410i	0.981382	+	1.69980i	0.324355	+	0.945935i	$0.394853\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	12.9282			2.18527
$36$		0			0
$37$	−0.464102			−0.0762978			−0.0381489	−	0.999272i	$-0.512146\pi$
$37$	−0.464102			−0.0762978			−0.0381489	+	0.999272i	$0.512146\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.46410	−	6.00000i	−0.541002	−	0.937043i	−0.998847	−	0.0480106i	$-0.984712\pi$
$41$	−3.46410	−	6.00000i	−0.541002	−	0.937043i	0.457845	−	0.889032i	$-0.348621\pi$
$42$		0			0
$43$	2.26795	−	3.92820i	0.345859	−	0.599045i	−0.639650	−	0.768666i	$-0.720921\pi$
$43$	2.26795	−	3.92820i	0.345859	−	0.599045i	0.985509	+	0.169621i	$0.0542542\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.46410	+	6.00000i	−0.505291	+	0.875190i	0.494690	+	0.869069i	$0.335282\pi$
$47$	−3.46410	+	6.00000i	−0.505291	+	0.875190i	−0.999981	+	0.00612051i	$0.998052\pi$
$48$		0			0
$49$	−2.50000	−	4.33013i	−0.357143	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.9282			−1.50110			−0.750552	−	0.660811i	$-0.770212\pi$
$53$	−10.9282			−1.50110			−0.750552	+	0.660811i	$0.770212\pi$
$54$		0			0
$55$	7.46410			1.00646
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	−0.999709	−	0.0241347i	$-0.992317\pi$
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	0.478953	−	0.877841i	$-0.341016\pi$
$60$		0			0
$61$	−5.23205	+	9.06218i	−0.669895	+	1.16029i	0.308038	+	0.951374i	$0.400328\pi$
$61$	−5.23205	+	9.06218i	−0.669895	+	1.16029i	−0.977933	+	0.208919i	$0.933006\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.59808	−	7.96410i	0.570321	−	0.987825i
$66$		0			0
$67$	0.267949	+	0.464102i	0.0327352	+	0.0566990i	0.881929	−	0.471383i	$-0.156245\pi$
$67$	0.267949	+	0.464102i	0.0327352	+	0.0566990i	−0.849194	+	0.528082i	$0.822912\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	1.00000			0.117041			0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000			0.117041			0.0585206	+	0.998286i	$0.481362\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.46410	−	6.00000i	−0.394771	−	0.683763i
$78$		0			0
$79$	−0.267949	+	0.464102i	−0.0301466	+	0.0522155i	−0.880705	−	0.473665i	$-0.842931\pi$
$79$	−0.267949	+	0.464102i	−0.0301466	+	0.0522155i	0.850558	+	0.525880i	$0.176264\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.46410	−	2.53590i	0.160706	−	0.278351i	−0.774416	−	0.632677i	$-0.781956\pi$
$83$	1.46410	−	2.53590i	0.160706	−	0.278351i	0.935122	+	0.354326i	$0.115290\pi$
$84$		0			0
$85$	−4.23205	−	7.33013i	−0.459030	−	0.795064i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.19615			−0.550791			−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615			−0.550791			−0.275396	+	0.961331i	$0.588809\pi$
$90$		0			0
$91$	−8.53590			−0.894805
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.9282	+	24.1244i	1.42900	+	2.47511i
$96$		0			0
$97$	5.92820	−	10.2679i	0.601918	−	1.04255i	−0.390613	−	0.920555i	$-0.627737\pi$
$97$	5.92820	−	10.2679i	0.601918	−	1.04255i	0.992530	−	0.121997i	$-0.0389299\pi$
$98$		0			0
$99$		0			0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.i.i.217.1		4
3.2	odd	2		648.2.i.j.217.2		4
4.3	odd	2		1296.2.i.r.865.1		4
9.2	odd	6		648.2.a.e.1.1	✓	2
9.4	even	3	inner	648.2.i.i.433.1		4
9.5	odd	6		648.2.i.j.433.2		4
9.7	even	3		648.2.a.h.1.2	yes	2
12.11	even	2		1296.2.i.t.865.2		4
36.7	odd	6		1296.2.a.q.1.2		2
36.11	even	6		1296.2.a.m.1.1		2
36.23	even	6		1296.2.i.t.433.2		4
36.31	odd	6		1296.2.i.r.433.1		4
72.11	even	6		5184.2.a.bz.1.2		2
72.29	odd	6		5184.2.a.cb.1.2		2
72.43	odd	6		5184.2.a.bi.1.1		2
72.61	even	6		5184.2.a.bg.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.a.e.1.1	✓	2	9.2	odd	6
648.2.a.h.1.2	yes	2	9.7	even	3
648.2.i.i.217.1		4	1.1	even	1	trivial
648.2.i.i.433.1		4	9.4	even	3	inner
648.2.i.j.217.2		4	3.2	odd	2
648.2.i.j.433.2		4	9.5	odd	6
1296.2.a.m.1.1		2	36.11	even	6
1296.2.a.q.1.2		2	36.7	odd	6
1296.2.i.r.433.1		4	36.31	odd	6
1296.2.i.r.865.1		4	4.3	odd	2
1296.2.i.t.433.2		4	36.23	even	6
1296.2.i.t.865.2		4	12.11	even	2
5184.2.a.bg.1.1		2	72.61	even	6
5184.2.a.bi.1.1		2	72.43	odd	6
5184.2.a.bz.1.2		2	72.11	even	6
5184.2.a.cb.1.2		2	72.29	odd	6

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Belyi maps

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.86603 - 3.23205i) q^{5} +(-1.73205 + 3.00000i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(1.23205 + 2.13397i) q^{13} +2.26795 q^{17} -7.46410 q^{19} +(2.46410 + 4.26795i) q^{23} +(-4.46410 + 7.73205i) q^{25} +(-2.13397 + 3.69615i) q^{29} +(5.46410 + 9.46410i) q^{31} +12.9282 q^{35} -0.464102 q^{37} +(-3.46410 - 6.00000i) q^{41} +(2.26795 - 3.92820i) q^{43} +(-3.46410 + 6.00000i) q^{47} +(-2.50000 - 4.33013i) q^{49} -10.9282 q^{53} +7.46410 q^{55} +(-4.00000 - 6.92820i) q^{59} +(-5.23205 + 9.06218i) q^{61} +(4.59808 - 7.96410i) q^{65} +(0.267949 + 0.464102i) q^{67} -2.00000 q^{71} +1.00000 q^{73} +(-3.46410 - 6.00000i) q^{77} +(-0.267949 + 0.464102i) q^{79} +(1.46410 - 2.53590i) q^{83} +(-4.23205 - 7.33013i) q^{85} -5.19615 q^{89} -8.53590 q^{91} +(13.9282 + 24.1244i) q^{95} +(5.92820 - 10.2679i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{11} - 2 q^{13} + 16 q^{17} - 16 q^{19} - 4 q^{23} - 4 q^{25} - 12 q^{29} + 8 q^{31} + 24 q^{35} + 12 q^{37} + 16 q^{43} - 10 q^{49} - 16 q^{53} + 16 q^{55} - 16 q^{59} - 14 q^{61} + 8 q^{65}+ \cdots - 4 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^11 - 2 * q^13 + 16 * q^17 - 16 * q^19 - 4 * q^23 - 4 * q^25 - 12 * q^29 + 8 * q^31 + 24 * q^35 + 12 * q^37 + 16 * q^43 - 10 * q^49 - 16 * q^53 + 16 * q^55 - 16 * q^59 - 14 * q^61 + 8 * q^65 + 8 * q^67 - 8 * q^71 + 4 * q^73 - 8 * q^79 - 8 * q^83 - 10 * q^85 - 48 * q^91 + 28 * q^95 - 4 * q^97

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