Embedded newform 49.2.g.a.4.1

• Label: 49.2.g.a.4.1
• Level: $49$
• Weight: $2$
• Character: 49.4
• Analytic conductor: $0.391$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.391266969904\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(4\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	49.4
Dual form			49.2.g.a.37.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.09732 - 1.42993i) q^{2} +(-1.40598 - 1.30456i) q^{3} +(1.62338 + 4.13631i) q^{4} +(-2.38971 + 0.737129i) q^{5} +(1.08336 + 4.74653i) q^{6} +(-1.40100 - 2.24437i) q^{7} +(1.38019 - 6.04701i) q^{8} +(0.0507148 + 0.676742i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 13 q^{2} - 14 q^{3} - 9 q^{4} - 14 q^{5} - 14 q^{7} - 20 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.09732	−	1.42993i	−1.48303	−	1.01111i	−0.989994	−	0.141107i	\(-0.954934\pi\)
							−0.493038	−	0.870008i	\(-0.664114\pi\)
\(3\)	−1.40598	−	1.30456i	−0.811741	−	0.753186i	0.160533	−	0.987031i	\(-0.448679\pi\)
							−0.972274	+	0.233845i	\(0.924869\pi\)
\(5\)	−2.38971	+	0.737129i	−1.06871	+	0.329654i	−0.778718	−	0.627374i	\(-0.784130\pi\)
							−0.289994	+	0.957028i	\(0.593653\pi\)
\(7\)	−1.40100	−	2.24437i	−0.529530	−	0.848291i
							\(11\)	0.229140	−	3.05767i	0.0690884	−	0.921921i	−0.849724	−	0.527227i	\(-0.823232\pi\)
0.918813	−	0.394694i	\(-0.129149\pi\)
\(13\)	−0.101149	+	0.0487107i	−0.0280536	+	0.0135099i	−0.447858	−	0.894105i	\(-0.647813\pi\)
							0.419804	+	0.907615i	\(0.362099\pi\)
\(17\)	−0.565579	+	0.0852474i	−0.137173	+	0.0206755i	−0.217270	−	0.976112i	\(-0.569715\pi\)
							0.0800967	+	0.996787i	\(0.474477\pi\)
\(19\)	−1.46037	−	2.52943i	−0.335031	−	0.580291i	0.648460	−	0.761249i	\(-0.275413\pi\)
							−0.983491	+	0.180958i	\(0.942080\pi\)
\(23\)	6.98498	+	1.05282i	1.45647	+	0.219527i	0.829058	−	0.559162i	\(-0.188877\pi\)
							0.627410	+	0.778689i	\(0.284115\pi\)
\(29\)	0.419642	+	0.526214i	0.0779255	+	0.0977155i	0.819266	−	0.573413i	\(-0.194381\pi\)
							−0.741341	+	0.671129i	\(0.765810\pi\)
\(31\)	2.54599	−	4.40979i	0.457274	−	0.792022i	−0.541542	−	0.840674i	\(-0.682159\pi\)
							0.998816	+	0.0486519i	\(0.0154925\pi\)
\(37\)	−0.977961	+	2.49180i	−0.160776	+	0.409650i	−0.988448	−	0.151562i	\(-0.951570\pi\)
							0.827672	+	0.561212i	\(0.189665\pi\)
\(41\)	0.963373	−	4.22081i	0.150454	−	0.659180i	−0.842300	−	0.539010i	\(-0.818799\pi\)
							0.992753	−	0.120171i	\(-0.0383442\pi\)
\(43\)	−2.61140	−	11.4413i	−0.398235	−	1.74478i	−0.634338	−	0.773056i	\(-0.718727\pi\)
							0.236103	−	0.971728i	\(-0.424130\pi\)
\(47\)	2.74905	+	1.87427i	0.400990	+	0.273391i	0.746978	−	0.664848i	\(-0.231504\pi\)
							−0.345988	+	0.938239i	\(0.612456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.2.g.a.4.1	✓	48
3.2	odd	2		441.2.bb.d.298.4		48
4.3	odd	2		784.2.bg.c.641.4		48
7.2	even	3		343.2.g.i.275.1		48
7.3	odd	6		343.2.e.c.148.8		48
7.4	even	3		343.2.e.d.148.8		48
7.5	odd	6		343.2.g.h.275.1		48
7.6	odd	2		343.2.g.g.263.1		48
49.12	odd	42		343.2.g.g.30.1		48
49.17	odd	42		343.2.e.c.197.8		48
49.20	odd	14		343.2.g.h.116.1		48
49.24	odd	42		2401.2.a.i.1.2		24
49.25	even	21		2401.2.a.h.1.2		24
49.29	even	7		343.2.g.i.116.1		48
49.32	even	21		343.2.e.d.197.8		48
49.37	even	21	inner	49.2.g.a.37.1	yes	48
147.86	odd	42		441.2.bb.d.37.4		48
196.135	odd	42		784.2.bg.c.625.4		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.2.g.a.4.1	✓	48	1.1	even	1	trivial
49.2.g.a.37.1	yes	48	49.37	even	21	inner
343.2.e.c.148.8		48	7.3	odd	6
343.2.e.c.197.8		48	49.17	odd	42
343.2.e.d.148.8		48	7.4	even	3
343.2.e.d.197.8		48	49.32	even	21
343.2.g.g.30.1		48	49.12	odd	42
343.2.g.g.263.1		48	7.6	odd	2
343.2.g.h.116.1		48	49.20	odd	14
343.2.g.h.275.1		48	7.5	odd	6
343.2.g.i.116.1		48	49.29	even	7
343.2.g.i.275.1		48	7.2	even	3
441.2.bb.d.37.4		48	147.86	odd	42
441.2.bb.d.298.4		48	3.2	odd	2
784.2.bg.c.625.4		48	196.135	odd	42
784.2.bg.c.641.4		48	4.3	odd	2
2401.2.a.h.1.2		24	49.25	even	21
2401.2.a.i.1.2		24	49.24	odd	42