Embedded newform 784.2.w.e.411.5

• Label: 784.2.w.e.411.5
• Level: $784$
• Weight: $2$
• Character: 784.411
• 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			411.5
Character	\(\chi\)	\(=\)	784.411
Dual form			784.2.w.e.227.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.655294 - 1.25323i) q^{2} +(-3.05239 - 0.817885i) q^{3} +(-1.14118 - 1.64247i) q^{4} +(0.213773 + 0.797811i) q^{5} +(-3.02521 + 3.28940i) q^{6} +(-2.80620 + 0.353863i) q^{8} +(6.05007 + 3.49301i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{2} + 8 q^{4} - 32 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.655294	−	1.25323i	0.463363	−	0.886169i
							\(3\)	−3.05239	−	0.817885i	−1.76230	−	0.472206i	−0.775118	−	0.631816i	\(-0.782310\pi\)
−0.987180	+	0.159610i	\(0.948976\pi\)
\(5\)	0.213773	+	0.797811i	0.0956021	+	0.356792i	0.997110	−	0.0759706i	\(-0.0242055\pi\)
							−0.901508	+	0.432763i	\(0.857539\pi\)
\(7\)		0			0
							\(11\)	2.73205	+	0.732051i	0.823744	+	0.220722i	0.645983	−	0.763352i	\(-0.276448\pi\)
0.177762	+	0.984074i	\(0.443114\pi\)
\(13\)	2.91203	+	2.91203i	0.807651	+	0.807651i	0.984278	−	0.176627i	\(-0.0565185\pi\)
							−0.176627	+	0.984278i	\(0.556519\pi\)
\(17\)	−2.30469	+	1.33061i	−0.558969	+	0.322721i	−0.752732	−	0.658328i	\(-0.771264\pi\)
							0.193763	+	0.981048i	\(0.437931\pi\)
\(19\)	0.117075	+	0.436928i	0.0268587	+	0.100238i	0.978054	−	0.208351i	\(-0.0668097\pi\)
							−0.951195	+	0.308590i	\(0.900143\pi\)
\(23\)	1.63915	−	2.83909i	0.341786	−	0.591991i	−0.642978	−	0.765884i	\(-0.722302\pi\)
							0.984764	+	0.173894i	\(0.0556349\pi\)
\(29\)	−2.04184	+	2.04184i	−0.379160	+	0.379160i	−0.870799	−	0.491639i	\(-0.836398\pi\)
							0.491639	+	0.870799i	\(0.336398\pi\)
\(31\)	−1.26156	−	2.18509i	−0.226583	−	0.392454i	0.730210	−	0.683223i	\(-0.239422\pi\)
							−0.956793	+	0.290769i	\(0.906089\pi\)
\(37\)	−2.33280	+	0.625071i	−0.383509	+	0.102761i	−0.445422	−	0.895321i	\(-0.646946\pi\)
							0.0619131	+	0.998082i	\(0.480280\pi\)
\(41\)	11.9895			1.87245			0.936224	−	0.351405i	\(-0.114296\pi\)
							0.936224	+	0.351405i	\(0.114296\pi\)
\(43\)	−3.27830	+	3.27830i	−0.499936	+	0.499936i	−0.911418	−	0.411482i	\(-0.865011\pi\)
							0.411482	+	0.911418i	\(0.365011\pi\)
\(47\)	4.98400	−	8.63254i	0.726991	−	1.25918i	−0.231159	−	0.972916i	\(-0.574252\pi\)
							0.958149	−	0.286269i	\(-0.0924150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.w.e.411.5		32
7.2	even	3		112.2.j.d.27.2	yes	16
7.3	odd	6	inner	784.2.w.e.619.5		32
7.4	even	3	inner	784.2.w.e.619.6		32
7.5	odd	6		112.2.j.d.27.1	✓	16
7.6	odd	2	inner	784.2.w.e.411.6		32
16.3	odd	4	inner	784.2.w.e.19.5		32
28.19	even	6		448.2.j.d.335.8		16
28.23	odd	6		448.2.j.d.335.1		16
56.5	odd	6		896.2.j.h.671.8		16
56.19	even	6		896.2.j.g.671.1		16
56.37	even	6		896.2.j.h.671.1		16
56.51	odd	6		896.2.j.g.671.8		16
112.3	even	12	inner	784.2.w.e.227.5		32
112.5	odd	12		896.2.j.g.223.8		16
112.19	even	12		112.2.j.d.83.2	yes	16
112.37	even	12		896.2.j.g.223.1		16
112.51	odd	12		112.2.j.d.83.1	yes	16
112.61	odd	12		448.2.j.d.111.1		16
112.67	odd	12	inner	784.2.w.e.227.6		32
112.75	even	12		896.2.j.h.223.1		16
112.83	even	4	inner	784.2.w.e.19.6		32
112.93	even	12		448.2.j.d.111.8		16
112.107	odd	12		896.2.j.h.223.8		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.d.27.1	✓	16	7.5	odd	6
112.2.j.d.27.2	yes	16	7.2	even	3
112.2.j.d.83.1	yes	16	112.51	odd	12
112.2.j.d.83.2	yes	16	112.19	even	12
448.2.j.d.111.1		16	112.61	odd	12
448.2.j.d.111.8		16	112.93	even	12
448.2.j.d.335.1		16	28.23	odd	6
448.2.j.d.335.8		16	28.19	even	6
784.2.w.e.19.5		32	16.3	odd	4	inner
784.2.w.e.19.6		32	112.83	even	4	inner
784.2.w.e.227.5		32	112.3	even	12	inner
784.2.w.e.227.6		32	112.67	odd	12	inner
784.2.w.e.411.5		32	1.1	even	1	trivial
784.2.w.e.411.6		32	7.6	odd	2	inner
784.2.w.e.619.5		32	7.3	odd	6	inner
784.2.w.e.619.6		32	7.4	even	3	inner
896.2.j.g.223.1		16	112.37	even	12
896.2.j.g.223.8		16	112.5	odd	12
896.2.j.g.671.1		16	56.19	even	6
896.2.j.g.671.8		16	56.51	odd	6
896.2.j.h.223.1		16	112.75	even	12
896.2.j.h.223.8		16	112.107	odd	12
896.2.j.h.671.1		16	56.37	even	6
896.2.j.h.671.8		16	56.5	odd	6