LMFDB - Embedded newform 2592.2.i.j.1729.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(865,2592)] 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("2592.865"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-1,0,-2,0,0,0,-2,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1729.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2592.1729
Dual form			2592.2.i.j.865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +(-0.500000 + 0.866025i) q^{13} +3.00000 q^{17} +2.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.00000 + 3.46410i) q^{25} +(-0.500000 - 0.866025i) q^{29} +(4.00000 - 6.92820i) q^{31} +2.00000 q^{35} +1.00000 q^{37} +(1.00000 - 1.73205i) q^{41} +(-5.00000 - 8.66025i) q^{43} +(-2.00000 - 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} -10.0000 q^{53} +2.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-4.50000 - 7.79423i) q^{61} +(-0.500000 - 0.866025i) q^{65} +(7.00000 - 12.1244i) q^{67} +10.0000 q^{71} -9.00000 q^{73} +(-2.00000 + 3.46410i) q^{77} +(-5.00000 - 8.66025i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-1.50000 + 2.59808i) q^{85} +11.0000 q^{89} +2.00000 q^{91} +(-1.00000 + 1.73205i) q^{95} +(1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 2 q^{7} - 2 q^{11} - q^{13} + 6 q^{17} + 4 q^{19} - 6 q^{23} + 4 q^{25} - q^{29} + 8 q^{31} + 4 q^{35} + 2 q^{37} + 2 q^{41} - 10 q^{43} - 4 q^{47} + 3 q^{49} - 20 q^{53} + 4 q^{55} - 4 q^{59}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q - q^5 - 2 * q^7 - 2 * q^11 - q^13 + 6 * q^17 + 4 * q^19 - 6 * q^23 + 4 * q^25 - q^29 + 8 * q^31 + 4 * q^35 + 2 * q^37 + 2 * q^41 - 10 * q^43 - 4 * q^47 + 3 * q^49 - 20 * q^53 + 4 * q^55 - 4 * q^59 - 9 * q^61 - q^65 + 14 * q^67 + 20 * q^71 - 18 * q^73 - 4 * q^77 - 10 * q^79 - 12 * q^83 - 3 * q^85 + 22 * q^89 + 4 * q^91 - 2 * q^95 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. 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
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	−0.955901	−	0.293691i	$-0.905116\pi$
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	0.732294	+	0.680989i	$0.238450\pi$
$6$		0			0
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	$-0.290043\pi$
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	−0.990766	+	0.135583i	$0.956709\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	−0.926995	−	0.375073i	$-0.877618\pi$
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	0.788320	+	0.615265i	$0.210951\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	0.815861	−	0.578249i	$-0.196264\pi$
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	−0.908708	+	0.417432i	$0.862930\pi$
$30$		0			0
$31$	4.00000	−	6.92820i	0.718421	−	1.24434i	−0.243204	−	0.969975i	$-0.578198\pi$
$31$	4.00000	−	6.92820i	0.718421	−	1.24434i	0.961625	−	0.274367i	$-0.0884683\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000			0.338062
$36$		0			0
$37$	1.00000			0.164399			0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000			0.164399			0.0821995	+	0.996616i	$0.473806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.00000	−	1.73205i	0.156174	−	0.270501i	−0.777312	−	0.629115i	$-0.783417\pi$
$41$	1.00000	−	1.73205i	0.156174	−	0.270501i	0.933486	+	0.358614i	$0.116751\pi$
$42$		0			0
$43$	−5.00000	−	8.66025i	−0.762493	−	1.32068i	−0.941562	−	0.336840i	$-0.890642\pi$
$43$	−5.00000	−	8.66025i	−0.762493	−	1.32068i	0.179069	−	0.983836i	$-0.442691\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	0.682489	−	0.730896i	$-0.260898\pi$
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	−0.974219	+	0.225605i	$0.927564\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	−0.966342	−	0.257260i	$-0.917180\pi$
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	0.705965	+	0.708247i	$0.250514\pi$
$60$		0			0
$61$	−4.50000	−	7.79423i	−0.576166	−	0.997949i	−0.995914	−	0.0903080i	$-0.971215\pi$
$61$	−4.50000	−	7.79423i	−0.576166	−	0.997949i	0.419748	−	0.907641i	$-0.362118\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.500000	−	0.866025i	−0.0620174	−	0.107417i
$66$		0			0
$67$	7.00000	−	12.1244i	0.855186	−	1.48123i	−0.0212861	−	0.999773i	$-0.506776\pi$
$67$	7.00000	−	12.1244i	0.855186	−	1.48123i	0.876472	−	0.481452i	$-0.159891\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.0000			1.18678			0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000			1.18678			0.593391	+	0.804914i	$0.297789\pi$
$72$		0			0
$73$	−9.00000			−1.05337			−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000			−1.05337			−0.526685	+	0.850060i	$0.676565\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00000	+	3.46410i	−0.227921	+	0.394771i
$78$		0			0
$79$	−5.00000	−	8.66025i	−0.562544	−	0.974355i	−0.997274	−	0.0737937i	$-0.976489\pi$
$79$	−5.00000	−	8.66025i	−0.562544	−	0.974355i	0.434730	−	0.900561i	$-0.356844\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	$-0.937822\pi$
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	0.322396	−	0.946605i	$-0.395512\pi$
$84$		0			0
$85$	−1.50000	+	2.59808i	−0.162698	+	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.0000			1.16600			0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000			1.16600			0.582999	+	0.812473i	$0.301879\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000	+	1.73205i	−0.102598	+	0.177705i
$96$		0			0
$97$	1.00000	+	1.73205i	0.101535	+	0.175863i	0.912317	−	0.409484i	$-0.134291\pi$
$97$	1.00000	+	1.73205i	0.101535	+	0.175863i	−0.810782	+	0.585348i	$0.800958\pi$
$98$		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	2592.2.i.j.1729.1		2
3.2	odd	2		2592.2.i.n.1729.1		2
4.3	odd	2		2592.2.i.k.1729.1		2
9.2	odd	6		2592.2.i.n.865.1		2
9.4	even	3		2592.2.a.f.1.1	yes	1
9.5	odd	6		2592.2.a.d.1.1	yes	1
9.7	even	3	inner	2592.2.i.j.865.1		2
12.11	even	2		2592.2.i.o.1729.1		2
36.7	odd	6		2592.2.i.k.865.1		2
36.11	even	6		2592.2.i.o.865.1		2
36.23	even	6		2592.2.a.c.1.1	✓	1
36.31	odd	6		2592.2.a.e.1.1	yes	1
72.5	odd	6		5184.2.a.w.1.1		1
72.13	even	6		5184.2.a.m.1.1		1
72.59	even	6		5184.2.a.t.1.1		1
72.67	odd	6		5184.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.a.c.1.1	✓	1	36.23	even	6
2592.2.a.d.1.1	yes	1	9.5	odd	6
2592.2.a.e.1.1	yes	1	36.31	odd	6
2592.2.a.f.1.1	yes	1	9.4	even	3
2592.2.i.j.865.1		2	9.7	even	3	inner
2592.2.i.j.1729.1		2	1.1	even	1	trivial
2592.2.i.k.865.1		2	36.7	odd	6
2592.2.i.k.1729.1		2	4.3	odd	2
2592.2.i.n.865.1		2	9.2	odd	6
2592.2.i.n.1729.1		2	3.2	odd	2
2592.2.i.o.865.1		2	36.11	even	6
2592.2.i.o.1729.1		2	12.11	even	2
5184.2.a.j.1.1		1	72.67	odd	6
5184.2.a.m.1.1		1	72.13	even	6
5184.2.a.t.1.1		1	72.59	even	6
5184.2.a.w.1.1		1	72.5	odd	6

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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +(-0.500000 + 0.866025i) q^{13} +3.00000 q^{17} +2.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.00000 + 3.46410i) q^{25} +(-0.500000 - 0.866025i) q^{29} +(4.00000 - 6.92820i) q^{31} +2.00000 q^{35} +1.00000 q^{37} +(1.00000 - 1.73205i) q^{41} +(-5.00000 - 8.66025i) q^{43} +(-2.00000 - 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} -10.0000 q^{53} +2.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-4.50000 - 7.79423i) q^{61} +(-0.500000 - 0.866025i) q^{65} +(7.00000 - 12.1244i) q^{67} +10.0000 q^{71} -9.00000 q^{73} +(-2.00000 + 3.46410i) q^{77} +(-5.00000 - 8.66025i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-1.50000 + 2.59808i) q^{85} +11.0000 q^{89} +2.00000 q^{91} +(-1.00000 + 1.73205i) q^{95} +(1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 2 q^{7} - 2 q^{11} - q^{13} + 6 q^{17} + 4 q^{19} - 6 q^{23} + 4 q^{25} - q^{29} + 8 q^{31} + 4 q^{35} + 2 q^{37} + 2 q^{41} - 10 q^{43} - 4 q^{47} + 3 q^{49} - 20 q^{53} + 4 q^{55} - 4 q^{59}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q - q^5 - 2 * q^7 - 2 * q^11 - q^13 + 6 * q^17 + 4 * q^19 - 6 * q^23 + 4 * q^25 - q^29 + 8 * q^31 + 4 * q^35 + 2 * q^37 + 2 * q^41 - 10 * q^43 - 4 * q^47 + 3 * q^49 - 20 * q^53 + 4 * q^55 - 4 * q^59 - 9 * q^61 - q^65 + 14 * q^67 + 20 * q^71 - 18 * q^73 - 4 * q^77 - 10 * q^79 - 12 * q^83 - 3 * q^85 + 22 * q^89 + 4 * q^91 - 2 * q^95 + 2 * q^97

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