LMFDB - Embedded newform 1140.2.bg.a.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1140,2,Mod(49,1140)] 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("1140.49"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1140, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1140.bg (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$9.10294583043$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1140.49
Dual form			1140.2.bg.a.349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.866025 - 0.500000i) q^{3} +(-1.86603 + 1.23205i) q^{5} +(0.500000 + 0.866025i) q^{9} -4.00000 q^{11} +(3.46410 - 2.00000i) q^{13} +(2.23205 - 0.133975i) q^{15} +(2.59808 + 1.50000i) q^{17} +(-3.50000 + 2.59808i) q^{19} +(1.96410 - 4.59808i) q^{25} -1.00000i q^{27} +(-4.00000 - 6.92820i) q^{29} +9.00000 q^{31} +(3.46410 + 2.00000i) q^{33} -10.0000i q^{37} -4.00000 q^{39} +(-2.00000 - 1.00000i) q^{45} +(-2.59808 + 1.50000i) q^{47} +7.00000 q^{49} +(-1.50000 - 2.59808i) q^{51} +(9.52628 - 5.50000i) q^{53} +(7.46410 - 4.92820i) q^{55} +(4.33013 - 0.500000i) q^{57} +(4.00000 - 6.92820i) q^{59} +(-3.00000 - 5.19615i) q^{61} +(-4.00000 + 8.00000i) q^{65} +(6.92820 - 4.00000i) q^{67} +(-3.00000 + 5.19615i) q^{71} +(-8.66025 - 5.00000i) q^{73} +(-4.00000 + 3.00000i) q^{75} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} -15.0000i q^{83} +(-6.69615 + 0.401924i) q^{85} +8.00000i q^{87} +(-1.00000 - 1.73205i) q^{89} +(-7.79423 - 4.50000i) q^{93} +(3.33013 - 9.16025i) q^{95} +(8.66025 + 5.00000i) q^{97} +(-2.00000 - 3.46410i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 2 q^{9} - 16 q^{11} + 2 q^{15} - 14 q^{19} - 6 q^{25} - 16 q^{29} + 36 q^{31} - 16 q^{39} - 8 q^{45} + 28 q^{49} - 6 q^{51} + 16 q^{55} + 16 q^{59} - 12 q^{61} - 16 q^{65} - 12 q^{71} - 16 q^{75}+ \cdots - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 + 2 * q^9 - 16 * q^11 + 2 * q^15 - 14 * q^19 - 6 * q^25 - 16 * q^29 + 36 * q^31 - 16 * q^39 - 8 * q^45 + 28 * q^49 - 6 * q^51 + 16 * q^55 + 16 * q^59 - 12 * q^61 - 16 * q^65 - 12 * q^71 - 16 * q^75 - 16 * q^79 - 2 * q^81 - 6 * q^85 - 4 * q^89 - 4 * q^95 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1140\mathbb{Z}\right)^\times$.

$n$	$457$	$571$	$761$	$781$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−0.866025	−	0.500000i	−0.500000	−	0.288675i
$3$	−0.866025	−	0.500000i	−0.500000	−	0.288675i
$4$		0			0
$5$	−1.86603	+	1.23205i	−0.834512	+	0.550990i
$5$	−1.86603	+	1.23205i	−0.834512	+	0.550990i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	0.500000	+	0.866025i	0.166667	+	0.288675i
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	3.46410	−	2.00000i	0.960769	−	0.554700i	0.0643593	−	0.997927i	$-0.479500\pi$
$13$	3.46410	−	2.00000i	0.960769	−	0.554700i	0.896410	+	0.443227i	$0.146166\pi$
$14$		0			0
$15$	2.23205	−	0.133975i	0.576313	−	0.0345921i
$16$		0			0
$17$	2.59808	+	1.50000i	0.630126	+	0.363803i	0.780801	−	0.624780i	$-0.214811\pi$
$17$	2.59808	+	1.50000i	0.630126	+	0.363803i	−0.150675	+	0.988583i	$0.548145\pi$
$18$		0			0
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	1.96410	−	4.59808i	0.392820	−	0.919615i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−4.00000	−	6.92820i	−0.742781	−	1.28654i	−0.951224	−	0.308500i	$-0.900173\pi$
$29$	−4.00000	−	6.92820i	−0.742781	−	1.28654i	0.208443	−	0.978035i	$-0.433160\pi$
$30$		0			0
$31$	9.00000			1.61645			0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000			1.61645			0.808224	+	0.588875i	$0.200429\pi$
$32$		0			0
$33$	3.46410	+	2.00000i	0.603023	+	0.348155i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	10.0000i		−	1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$		−	10.0000i		−	1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$		0			0
$39$	−4.00000			−0.640513
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$	−2.00000	−	1.00000i	−0.298142	−	0.149071i
$46$		0			0
$47$	−2.59808	+	1.50000i	−0.378968	+	0.218797i	−0.677369	−	0.735643i	$-0.736880\pi$
$47$	−2.59808	+	1.50000i	−0.378968	+	0.218797i	0.298401	+	0.954441i	$0.403547\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−1.50000	−	2.59808i	−0.210042	−	0.363803i
$52$		0			0
$53$	9.52628	−	5.50000i	1.30854	−	0.755483i	0.326683	−	0.945134i	$-0.394069\pi$
$53$	9.52628	−	5.50000i	1.30854	−	0.755483i	0.981852	+	0.189651i	$0.0607356\pi$
$54$		0			0
$55$	7.46410	−	4.92820i	1.00646	−	0.664519i
$56$		0			0
$57$	4.33013	−	0.500000i	0.573539	−	0.0662266i
$58$		0			0
$59$	4.00000	−	6.92820i	0.520756	−	0.901975i	−0.478953	−	0.877841i	$-0.658984\pi$
$59$	4.00000	−	6.92820i	0.520756	−	0.901975i	0.999709	−	0.0241347i	$-0.00768307\pi$
$60$		0			0
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	0.607535	−	0.794293i	$-0.292159\pi$
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	−0.991645	+	0.128994i	$0.958825\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	+	8.00000i	−0.496139	+	0.992278i
$66$		0			0
$67$	6.92820	−	4.00000i	0.846415	−	0.488678i	−0.0130248	−	0.999915i	$-0.504146\pi$
$67$	6.92820	−	4.00000i	0.846415	−	0.488678i	0.859440	+	0.511237i	$0.170813\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	−0.987294	−	0.158901i	$-0.949205\pi$
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	0.631260	+	0.775571i	$0.282538\pi$
$72$		0			0
$73$	−8.66025	−	5.00000i	−1.01361	−	0.585206i	−0.101361	−	0.994850i	$-0.532320\pi$
$73$	−8.66025	−	5.00000i	−1.01361	−	0.585206i	−0.912245	+	0.409644i	$0.865653\pi$
$74$		0			0
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		−	15.0000i		−	1.64646i	−0.567705	−	0.823232i	$-0.692169\pi$
$83$		−	15.0000i		−	1.64646i	0.567705	−	0.823232i	$-0.307831\pi$
$84$		0			0
$85$	−6.69615	+	0.401924i	−0.726300	+	0.0435948i
$86$		0			0
$87$			8.00000i			0.857690i
$88$		0			0
$89$	−1.00000	−	1.73205i	−0.106000	−	0.183597i	0.808146	−	0.588982i	$-0.200471\pi$
$89$	−1.00000	−	1.73205i	−0.106000	−	0.183597i	−0.914146	+	0.405385i	$0.867138\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−7.79423	−	4.50000i	−0.808224	−	0.466628i
$94$		0			0
$95$	3.33013	−	9.16025i	0.341664	−	0.939822i
$96$		0			0
$97$	8.66025	+	5.00000i	0.879316	+	0.507673i	0.870433	−	0.492287i	$-0.163839\pi$
$97$	8.66025	+	5.00000i	0.879316	+	0.507673i	0.00888289	+	0.999961i	$0.497172\pi$
$98$		0			0
$99$	−2.00000	−	3.46410i	−0.201008	−	0.348155i

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	1140.2.bg.a.49.1	✓	4
3.2	odd	2		3420.2.bj.b.1189.2		4
5.4	even	2	inner	1140.2.bg.a.49.2	yes	4
15.14	odd	2		3420.2.bj.b.1189.1		4
19.7	even	3	inner	1140.2.bg.a.349.2	yes	4
57.26	odd	6		3420.2.bj.b.2629.1		4
95.64	even	6	inner	1140.2.bg.a.349.1	yes	4
285.254	odd	6		3420.2.bj.b.2629.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.bg.a.49.1	✓	4	1.1	even	1	trivial
1140.2.bg.a.49.2	yes	4	5.4	even	2	inner
1140.2.bg.a.349.1	yes	4	95.64	even	6	inner
1140.2.bg.a.349.2	yes	4	19.7	even	3	inner
3420.2.bj.b.1189.1		4	15.14	odd	2
3420.2.bj.b.1189.2		4	3.2	odd	2
3420.2.bj.b.2629.1		4	57.26	odd	6
3420.2.bj.b.2629.2		4	285.254	odd	6

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1140.bg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{3} +(-1.86603 + 1.23205i) q^{5} +(0.500000 + 0.866025i) q^{9} -4.00000 q^{11} +(3.46410 - 2.00000i) q^{13} +(2.23205 - 0.133975i) q^{15} +(2.59808 + 1.50000i) q^{17} +(-3.50000 + 2.59808i) q^{19} +(1.96410 - 4.59808i) q^{25} -1.00000i q^{27} +(-4.00000 - 6.92820i) q^{29} +9.00000 q^{31} +(3.46410 + 2.00000i) q^{33} -10.0000i q^{37} -4.00000 q^{39} +(-2.00000 - 1.00000i) q^{45} +(-2.59808 + 1.50000i) q^{47} +7.00000 q^{49} +(-1.50000 - 2.59808i) q^{51} +(9.52628 - 5.50000i) q^{53} +(7.46410 - 4.92820i) q^{55} +(4.33013 - 0.500000i) q^{57} +(4.00000 - 6.92820i) q^{59} +(-3.00000 - 5.19615i) q^{61} +(-4.00000 + 8.00000i) q^{65} +(6.92820 - 4.00000i) q^{67} +(-3.00000 + 5.19615i) q^{71} +(-8.66025 - 5.00000i) q^{73} +(-4.00000 + 3.00000i) q^{75} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} -15.0000i q^{83} +(-6.69615 + 0.401924i) q^{85} +8.00000i q^{87} +(-1.00000 - 1.73205i) q^{89} +(-7.79423 - 4.50000i) q^{93} +(3.33013 - 9.16025i) q^{95} +(8.66025 + 5.00000i) q^{97} +(-2.00000 - 3.46410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 2 q^{9} - 16 q^{11} + 2 q^{15} - 14 q^{19} - 6 q^{25} - 16 q^{29} + 36 q^{31} - 16 q^{39} - 8 q^{45} + 28 q^{49} - 6 q^{51} + 16 q^{55} + 16 q^{59} - 12 q^{61} - 16 q^{65} - 12 q^{71} - 16 q^{75}+ \cdots - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 + 2 * q^9 - 16 * q^11 + 2 * q^15 - 14 * q^19 - 6 * q^25 - 16 * q^29 + 36 * q^31 - 16 * q^39 - 8 * q^45 + 28 * q^49 - 6 * q^51 + 16 * q^55 + 16 * q^59 - 12 * q^61 - 16 * q^65 - 12 * q^71 - 16 * q^75 - 16 * q^79 - 2 * q^81 - 6 * q^85 - 4 * q^89 - 4 * q^95 - 8 * q^99

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