Embedded newform 784.2.w.e.19.1

• Label: 784.2.w.e.19.1
• 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			19.1
Character	\(\chi\)	\(=\)	784.19
Dual form			784.2.w.e.619.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41028 + 0.105414i) q^{2} +(-0.551303 + 2.05749i) q^{3} +(1.97778 - 0.297327i) q^{4} +(-3.47044 + 0.929901i) q^{5} +(0.560603 - 2.95975i) q^{6} +(-2.75787 + 0.627801i) q^{8} +(-1.33126 - 0.768605i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{2} + 8 q^{4} - 32 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41028	+	0.105414i	−0.997218	+	0.0745392i
							\(3\)	−0.551303	+	2.05749i	−0.318295	+	1.18789i	0.602587	+	0.798053i	\(0.294137\pi\)
−0.920882	+	0.389841i	\(0.872530\pi\)
\(5\)	−3.47044	+	0.929901i	−1.55203	+	0.415864i	−0.930129	−	0.367233i	\(-0.880305\pi\)
							−0.621898	+	0.783098i	\(0.713638\pi\)
\(7\)		0			0
							\(11\)	−0.732051	+	2.73205i	−0.220722	+	0.823744i	0.763352	+	0.645983i	\(0.223552\pi\)
−0.984074	+	0.177762i	\(0.943114\pi\)
\(13\)	−1.17573	+	1.17573i	−0.326090	+	0.326090i	−0.851098	−	0.525008i	\(-0.824062\pi\)
							0.525008	+	0.851098i	\(0.324062\pi\)
\(17\)	−5.31204	+	3.06691i	−1.28836	+	0.743834i	−0.978361	−	0.206903i	\(-0.933661\pi\)
							−0.309997	+	0.950737i	\(0.600328\pi\)
\(19\)	1.33846	−	0.358639i	0.307063	−	0.0822774i	−0.101997	−	0.994785i	\(-0.532523\pi\)
							0.409060	+	0.912507i	\(0.365857\pi\)
\(23\)	−0.103594	+	0.179430i	−0.0216009	+	0.0374138i	−0.876624	−	0.481176i	\(-0.840210\pi\)
							0.855023	+	0.518590i	\(0.173543\pi\)
\(29\)	−3.46733	−	3.46733i	−0.643868	−	0.643868i	0.307636	−	0.951504i	\(-0.400462\pi\)
							−0.951504	+	0.307636i	\(0.900462\pi\)
\(31\)	−2.87099	−	4.97270i	−0.515645	−	0.893124i	−0.999835	−	0.0181610i	\(-0.994219\pi\)
							0.484190	−	0.874963i	\(-0.339114\pi\)
\(37\)	−0.0935566	−	0.349158i	−0.0153806	−	0.0574012i	0.957809	−	0.287405i	\(-0.0927927\pi\)
							−0.973190	+	0.230004i	\(0.926126\pi\)
\(41\)	6.75794			1.05541			0.527707	−	0.849427i	\(-0.323052\pi\)
							0.527707	+	0.849427i	\(0.323052\pi\)
\(43\)	0.207188	+	0.207188i	0.0315959	+	0.0315959i	0.722728	−	0.691132i	\(-0.242888\pi\)
							−0.691132	+	0.722728i	\(0.742888\pi\)
\(47\)	5.46606	−	9.46749i	0.797307	−	1.38098i	−0.124058	−	0.992275i	\(-0.539591\pi\)
							0.921364	−	0.388700i	\(-0.127076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.w.e.19.1		32
7.2	even	3		112.2.j.d.83.7	yes	16
7.3	odd	6	inner	784.2.w.e.227.3		32
7.4	even	3	inner	784.2.w.e.227.4		32
7.5	odd	6		112.2.j.d.83.8	yes	16
7.6	odd	2	inner	784.2.w.e.19.2		32
16.11	odd	4	inner	784.2.w.e.411.3		32
28.19	even	6		448.2.j.d.111.2		16
28.23	odd	6		448.2.j.d.111.7		16
56.5	odd	6		896.2.j.h.223.2		16
56.19	even	6		896.2.j.g.223.7		16
56.37	even	6		896.2.j.h.223.7		16
56.51	odd	6		896.2.j.g.223.2		16
112.5	odd	12		448.2.j.d.335.7		16
112.11	odd	12	inner	784.2.w.e.619.2		32
112.19	even	12		896.2.j.h.671.7		16
112.27	even	4	inner	784.2.w.e.411.4		32
112.37	even	12		448.2.j.d.335.2		16
112.51	odd	12		896.2.j.h.671.2		16
112.59	even	12	inner	784.2.w.e.619.1		32
112.61	odd	12		896.2.j.g.671.2		16
112.75	even	12		112.2.j.d.27.7	✓	16
112.93	even	12		896.2.j.g.671.7		16
112.107	odd	12		112.2.j.d.27.8	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.d.27.7	✓	16	112.75	even	12
112.2.j.d.27.8	yes	16	112.107	odd	12
112.2.j.d.83.7	yes	16	7.2	even	3
112.2.j.d.83.8	yes	16	7.5	odd	6
448.2.j.d.111.2		16	28.19	even	6
448.2.j.d.111.7		16	28.23	odd	6
448.2.j.d.335.2		16	112.37	even	12
448.2.j.d.335.7		16	112.5	odd	12
784.2.w.e.19.1		32	1.1	even	1	trivial
784.2.w.e.19.2		32	7.6	odd	2	inner
784.2.w.e.227.3		32	7.3	odd	6	inner
784.2.w.e.227.4		32	7.4	even	3	inner
784.2.w.e.411.3		32	16.11	odd	4	inner
784.2.w.e.411.4		32	112.27	even	4	inner
784.2.w.e.619.1		32	112.59	even	12	inner
784.2.w.e.619.2		32	112.11	odd	12	inner
896.2.j.g.223.2		16	56.51	odd	6
896.2.j.g.223.7		16	56.19	even	6
896.2.j.g.671.2		16	112.61	odd	12
896.2.j.g.671.7		16	112.93	even	12
896.2.j.h.223.2		16	56.5	odd	6
896.2.j.h.223.7		16	56.37	even	6
896.2.j.h.671.2		16	112.51	odd	12
896.2.j.h.671.7		16	112.19	even	12