Embedded newform 49.2.e.b.8.1

• Label: 49.2.e.b.8.1
• Level: $49$
• Weight: $2$
• Character: 49.8
• Analytic conductor: $0.391$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.391266969904\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{7})\)
Coefficient field:	\(\Q(\zeta_{21})\)
Defining polynomial:	\( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 7 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			8.1
Root			\(-0.988831 + 0.149042i\) of defining polynomial
Character	\(\chi\)	\(=\)	49.8
Dual form			49.2.e.b.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19158 - 1.49419i) q^{2} +(1.55929 - 0.750915i) q^{3} +(-0.367711 + 1.61105i) q^{4} +(-1.91647 + 0.922924i) q^{5} +(-2.98003 - 1.43511i) q^{6} +(1.91879 - 1.82161i) q^{7} +(-0.598393 + 0.288171i) q^{8} +(-0.00295512 + 0.00370560i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} + 2 q^{4} - 7 q^{5} - 7 q^{6} - 7 q^{7} + 6 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19158	−	1.49419i	−0.842574	−	1.05655i	−0.997641	−	0.0686506i	\(-0.978131\pi\)
							0.155067	−	0.987904i	\(-0.450441\pi\)
\(3\)	1.55929	−	0.750915i	0.900257	−	0.433541i	0.0742752	−	0.997238i	\(-0.476336\pi\)
							0.825982	+	0.563697i	\(0.190621\pi\)
\(5\)	−1.91647	+	0.922924i	−0.857072	+	0.412744i	−0.810198	−	0.586157i	\(-0.800640\pi\)
							−0.0468741	+	0.998901i	\(0.514926\pi\)
\(7\)	1.91879	−	1.82161i	0.725233	−	0.688503i
							\(11\)	2.91390	+	3.65392i	0.878575	+	1.10170i	0.994108	+	0.108395i	\(0.0345710\pi\)
−0.115533	+	0.993304i	\(0.536858\pi\)
\(13\)	0.609562	+	0.764367i	0.169062	+	0.211997i	0.859144	−	0.511734i	\(-0.170997\pi\)
							−0.690082	+	0.723732i	\(0.742425\pi\)
\(17\)	−0.463787	−	2.03198i	−0.112485	−	0.492828i	−0.999516	−	0.0311182i	\(-0.990093\pi\)
							0.887031	−	0.461710i	\(-0.152764\pi\)
\(19\)	−5.29116			−1.21387			−0.606937	−	0.794750i	\(-0.707602\pi\)
							−0.606937	+	0.794750i	\(0.707602\pi\)
\(23\)	0.841040	−	3.68484i	0.175369	−	0.768342i	−0.808361	−	0.588687i	\(-0.799645\pi\)
							0.983730	−	0.179654i	\(-0.0574979\pi\)
\(29\)	−1.23460	−	5.40913i	−0.229259	−	1.00445i	−0.950246	−	0.311501i	\(-0.899168\pi\)
							0.720987	−	0.692949i	\(-0.243689\pi\)
\(31\)	−10.1646			−1.82562			−0.912811	−	0.408383i	\(-0.866093\pi\)
							−0.912811	+	0.408383i	\(0.866093\pi\)
\(37\)	0.384244	+	1.68348i	0.0631693	+	0.276763i	0.996642	−	0.0818866i	\(-0.0260945\pi\)
							−0.933472	+	0.358649i	\(0.883237\pi\)
\(41\)	3.19850	−	1.54031i	0.499521	−	0.240557i	−0.167119	−	0.985937i	\(-0.553446\pi\)
							0.666640	+	0.745380i	\(0.267732\pi\)
\(43\)	9.05500	+	4.36066i	1.38087	+	0.664994i	0.969186	−	0.246331i	\(-0.0792249\pi\)
							0.411688	+	0.911325i	\(0.364939\pi\)
\(47\)	−2.69886	−	3.38427i	−0.393670	−	0.493646i	0.545014	−	0.838427i	\(-0.316525\pi\)
							−0.938683	+	0.344781i	\(0.887953\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.2.e.b.8.1	✓	12
3.2	odd	2		441.2.u.b.253.2		12
4.3	odd	2		784.2.u.b.449.1		12
7.2	even	3		343.2.g.b.214.1		12
7.3	odd	6		343.2.g.a.79.1		12
7.4	even	3		343.2.g.d.79.1		12
7.5	odd	6		343.2.g.c.214.1		12
7.6	odd	2		343.2.e.b.50.1		12
49.6	odd	14		343.2.e.b.295.1		12
49.10	odd	42		343.2.g.c.226.1		12
49.16	even	21		343.2.g.d.165.1		12
49.22	even	7		2401.2.a.c.1.1		6
49.27	odd	14		2401.2.a.d.1.1		6
49.33	odd	42		343.2.g.a.165.1		12
49.39	even	21		343.2.g.b.226.1		12
49.43	even	7	inner	49.2.e.b.43.1	yes	12
147.92	odd	14		441.2.u.b.190.2		12
196.43	odd	14		784.2.u.b.337.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.2.e.b.8.1	✓	12	1.1	even	1	trivial
49.2.e.b.43.1	yes	12	49.43	even	7	inner
343.2.e.b.50.1		12	7.6	odd	2
343.2.e.b.295.1		12	49.6	odd	14
343.2.g.a.79.1		12	7.3	odd	6
343.2.g.a.165.1		12	49.33	odd	42
343.2.g.b.214.1		12	7.2	even	3
343.2.g.b.226.1		12	49.39	even	21
343.2.g.c.214.1		12	7.5	odd	6
343.2.g.c.226.1		12	49.10	odd	42
343.2.g.d.79.1		12	7.4	even	3
343.2.g.d.165.1		12	49.16	even	21
441.2.u.b.190.2		12	147.92	odd	14
441.2.u.b.253.2		12	3.2	odd	2
784.2.u.b.337.1		12	196.43	odd	14
784.2.u.b.449.1		12	4.3	odd	2
2401.2.a.c.1.1		6	49.22	even	7
2401.2.a.d.1.1		6	49.27	odd	14