Embedded newform 784.2.w.e.19.7

• Label: 784.2.w.e.19.7
• Level: $784$
• Weight: $2$
• Character: 784.19
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.w (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26027151847\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			19.7
Character	\(\chi\)	\(=\)	784.19
Dual form			784.2.w.e.619.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.11693 + 0.867450i) q^{2} +(-0.379383 + 1.41588i) q^{3} +(0.495063 + 1.93776i) q^{4} +(2.30425 - 0.617422i) q^{5} +(-1.65195 + 1.25234i) q^{6} +(-1.12796 + 2.59378i) q^{8} +(0.737300 + 0.425680i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{2} + 8 q^{4} - 32 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(687\)	\(689\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.11693	+	0.867450i	0.789788	+	0.613379i
							\(3\)	−0.379383	+	1.41588i	−0.219037	+	0.817457i	0.765669	+	0.643234i	\(0.222408\pi\)
−0.984706	+	0.174223i	\(0.944259\pi\)
\(5\)	2.30425	−	0.617422i	1.03049	−	0.276120i	0.296325	−	0.955087i	\(-0.404239\pi\)
							0.734168	+	0.678968i	\(0.237572\pi\)
\(7\)		0			0
							\(11\)	−0.732051	+	2.73205i	−0.220722	+	0.823744i	0.763352	+	0.645983i	\(0.223552\pi\)
−0.984074	+	0.177762i	\(0.943114\pi\)
\(13\)	4.80976	−	4.80976i	1.33399	−	1.33399i	0.432219	−	0.901769i	\(-0.357731\pi\)
							0.901769	−	0.432219i	\(-0.142269\pi\)
\(17\)	−0.982280	+	0.567119i	−0.238238	+	0.137547i	−0.614367	−	0.789021i	\(-0.710588\pi\)
							0.376129	+	0.926567i	\(0.377255\pi\)
\(19\)	−1.66307	+	0.445620i	−0.381536	+	0.102232i	−0.444489	−	0.895784i	\(-0.646615\pi\)
							0.0629534	+	0.998016i	\(0.479948\pi\)
\(23\)	0.668102	−	1.15719i	0.139309	−	0.241290i	−0.787926	−	0.615770i	\(-0.788845\pi\)
							0.927235	+	0.374479i	\(0.122179\pi\)
\(29\)	5.26785	+	5.26785i	0.978215	+	0.978215i	0.999768	−	0.0215522i	\(-0.00686082\pi\)
							−0.0215522	+	0.999768i	\(0.506861\pi\)
\(31\)	−4.15942	−	7.20433i	−0.747055	−	1.29394i	−0.949229	−	0.314587i	\(-0.898134\pi\)
							0.202174	−	0.979350i	\(-0.435199\pi\)
\(37\)	−1.53276	−	5.72032i	−0.251983	−	0.940415i	−0.969744	−	0.244125i	\(-0.921499\pi\)
							0.717760	−	0.696290i	\(-0.245167\pi\)
\(41\)	−1.63570			−0.255453			−0.127726	−	0.991809i	\(-0.540768\pi\)
							−0.127726	+	0.991809i	\(0.540768\pi\)
\(43\)	−1.33620	−	1.33620i	−0.203769	−	0.203769i	0.597844	−	0.801613i	\(-0.296024\pi\)
							−0.801613	+	0.597844i	\(0.796024\pi\)
\(47\)	0.966727	−	1.67442i	0.141012	−	0.244239i	−0.786866	−	0.617124i	\(-0.788298\pi\)
							0.927878	+	0.372884i	\(0.121631\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.w.e.19.7		32
7.2	even	3		112.2.j.d.83.5	yes	16
7.3	odd	6	inner	784.2.w.e.227.1		32
7.4	even	3	inner	784.2.w.e.227.2		32
7.5	odd	6		112.2.j.d.83.6	yes	16
7.6	odd	2	inner	784.2.w.e.19.8		32
16.11	odd	4	inner	784.2.w.e.411.1		32
28.19	even	6		448.2.j.d.111.4		16
28.23	odd	6		448.2.j.d.111.5		16
56.5	odd	6		896.2.j.h.223.4		16
56.19	even	6		896.2.j.g.223.5		16
56.37	even	6		896.2.j.h.223.5		16
56.51	odd	6		896.2.j.g.223.4		16
112.5	odd	12		448.2.j.d.335.5		16
112.11	odd	12	inner	784.2.w.e.619.8		32
112.19	even	12		896.2.j.h.671.5		16
112.27	even	4	inner	784.2.w.e.411.2		32
112.37	even	12		448.2.j.d.335.4		16
112.51	odd	12		896.2.j.h.671.4		16
112.59	even	12	inner	784.2.w.e.619.7		32
112.61	odd	12		896.2.j.g.671.4		16
112.75	even	12		112.2.j.d.27.5	✓	16
112.93	even	12		896.2.j.g.671.5		16
112.107	odd	12		112.2.j.d.27.6	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.d.27.5	✓	16	112.75	even	12
112.2.j.d.27.6	yes	16	112.107	odd	12
112.2.j.d.83.5	yes	16	7.2	even	3
112.2.j.d.83.6	yes	16	7.5	odd	6
448.2.j.d.111.4		16	28.19	even	6
448.2.j.d.111.5		16	28.23	odd	6
448.2.j.d.335.4		16	112.37	even	12
448.2.j.d.335.5		16	112.5	odd	12
784.2.w.e.19.7		32	1.1	even	1	trivial
784.2.w.e.19.8		32	7.6	odd	2	inner
784.2.w.e.227.1		32	7.3	odd	6	inner
784.2.w.e.227.2		32	7.4	even	3	inner
784.2.w.e.411.1		32	16.11	odd	4	inner
784.2.w.e.411.2		32	112.27	even	4	inner
784.2.w.e.619.7		32	112.59	even	12	inner
784.2.w.e.619.8		32	112.11	odd	12	inner
896.2.j.g.223.4		16	56.51	odd	6
896.2.j.g.223.5		16	56.19	even	6
896.2.j.g.671.4		16	112.61	odd	12
896.2.j.g.671.5		16	112.93	even	12
896.2.j.h.223.4		16	56.5	odd	6
896.2.j.h.223.5		16	56.37	even	6
896.2.j.h.671.4		16	112.51	odd	12
896.2.j.h.671.5		16	112.19	even	12