Embedded newform 784.2.x.l.557.2

• Label: 784.2.x.l.557.2
• Level: $784$
• Weight: $2$
• Character: 784.557
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26027151847\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.2
Character	\(\chi\)	\(=\)	784.557
Dual form			784.2.x.l.373.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02649 - 0.972785i) q^{2} +(-0.862219 + 0.231031i) q^{3} +(0.107378 + 1.99712i) q^{4} +(3.20004 + 0.857449i) q^{5} +(1.10981 + 0.601602i) q^{6} +(1.83254 - 2.15448i) q^{8} +(-1.90803 + 1.10160i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} - 4 q^{3} + 6 q^{4} - 4 q^{5} + 8 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(687\)	\(689\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02649	−	0.972785i	−0.725840	−	0.687863i
							\(3\)	−0.862219	+	0.231031i	−0.497802	+	0.133386i	−0.498980	−	0.866614i	\(-0.666292\pi\)
0.00117766	+	0.999999i	\(0.499625\pi\)
\(5\)	3.20004	+	0.857449i	1.43110	+	0.383463i	0.889409	−	0.457113i	\(-0.151116\pi\)
							0.541694	+	0.840576i	\(0.317783\pi\)
\(7\)		0			0
							\(11\)	−0.799070	−	2.98217i	−0.240929	−	0.899158i	−0.975386	−	0.220503i	\(-0.929230\pi\)
0.734458	−	0.678654i	\(-0.237437\pi\)
\(13\)	−4.03390	−	4.03390i	−1.11880	−	1.11880i	−0.991918	−	0.126884i	\(-0.959502\pi\)
							−0.126884	−	0.991918i	\(-0.540498\pi\)
\(17\)	−0.173966	+	0.301317i	−0.0421929	+	0.0730802i	−0.886351	−	0.463015i	\(-0.846768\pi\)
							0.844158	+	0.536095i	\(0.180101\pi\)
\(19\)	1.56048	−	5.82380i	0.357999	−	1.33607i	−0.518669	−	0.854975i	\(-0.673572\pi\)
							0.876668	−	0.481096i	\(-0.159761\pi\)
\(23\)	5.40216	−	3.11894i	1.12643	−	0.650344i	0.183394	−	0.983040i	\(-0.441292\pi\)
							0.943034	+	0.332696i	\(0.107958\pi\)
\(29\)	1.21961	+	1.21961i	0.226476	+	0.226476i	0.811219	−	0.584743i	\(-0.198805\pi\)
							−0.584743	+	0.811219i	\(0.698805\pi\)
\(31\)	0.631188	−	1.09325i	0.113365	−	0.196353i	−0.803760	−	0.594953i	\(-0.797170\pi\)
							0.917125	+	0.398600i	\(0.130504\pi\)
\(37\)	8.77373	+	2.35091i	1.44239	+	0.386488i	0.893371	−	0.449321i	\(-0.148334\pi\)
							0.549021	+	0.835808i	\(0.315001\pi\)
\(41\)		−	2.68519i		−	0.419356i	−0.977770	−	0.209678i	\(-0.932758\pi\)
							0.977770	−	0.209678i	\(-0.0672416\pi\)
\(43\)	4.05478	−	4.05478i	0.618348	−	0.618348i	−0.326760	−	0.945107i	\(-0.605957\pi\)
							0.945107	+	0.326760i	\(0.105957\pi\)
\(47\)	−2.32249	−	4.02267i	−0.338770	−	0.586767i	0.645432	−	0.763818i	\(-0.276677\pi\)
							−0.984202	+	0.177051i	\(0.943344\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.l.557.2		24
7.2	even	3	inner	784.2.x.l.765.3		24
7.3	odd	6		784.2.m.h.589.6		12
7.4	even	3		112.2.m.d.29.6	✓	12
7.5	odd	6		784.2.x.m.765.3		24
7.6	odd	2		784.2.x.m.557.2		24
16.5	even	4	inner	784.2.x.l.165.3		24
28.11	odd	6		448.2.m.d.337.3		12
56.11	odd	6		896.2.m.h.673.4		12
56.53	even	6		896.2.m.g.673.3		12
112.5	odd	12		784.2.x.m.373.2		24
112.11	odd	12		448.2.m.d.113.3		12
112.37	even	12	inner	784.2.x.l.373.2		24
112.53	even	12		112.2.m.d.85.6	yes	12
112.67	odd	12		896.2.m.h.225.4		12
112.69	odd	4		784.2.x.m.165.3		24
112.101	odd	12		784.2.m.h.197.6		12
112.109	even	12		896.2.m.g.225.3		12
224.11	odd	24		7168.2.a.bi.1.4		12
224.53	even	24		7168.2.a.bj.1.9		12
224.123	odd	24		7168.2.a.bi.1.9		12
224.165	even	24		7168.2.a.bj.1.4		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.d.29.6	✓	12	7.4	even	3
112.2.m.d.85.6	yes	12	112.53	even	12
448.2.m.d.113.3		12	112.11	odd	12
448.2.m.d.337.3		12	28.11	odd	6
784.2.m.h.197.6		12	112.101	odd	12
784.2.m.h.589.6		12	7.3	odd	6
784.2.x.l.165.3		24	16.5	even	4	inner
784.2.x.l.373.2		24	112.37	even	12	inner
784.2.x.l.557.2		24	1.1	even	1	trivial
784.2.x.l.765.3		24	7.2	even	3	inner
784.2.x.m.165.3		24	112.69	odd	4
784.2.x.m.373.2		24	112.5	odd	12
784.2.x.m.557.2		24	7.6	odd	2
784.2.x.m.765.3		24	7.5	odd	6
896.2.m.g.225.3		12	112.109	even	12
896.2.m.g.673.3		12	56.53	even	6
896.2.m.h.225.4		12	112.67	odd	12
896.2.m.h.673.4		12	56.11	odd	6
7168.2.a.bi.1.4		12	224.11	odd	24
7168.2.a.bi.1.9		12	224.123	odd	24
7168.2.a.bj.1.4		12	224.165	even	24
7168.2.a.bj.1.9		12	224.53	even	24