Embedded newform 784.2.x.j.557.2

• Label: 784.2.x.j.557.2
• Level: $784$
• Weight: $2$
• Character: 784.557
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $16$
• 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:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 2*x^15 + 4*x^14 + 4*x^13 - 13*x^12 + 32*x^11 - 4*x^10 - 34*x^9 + 121*x^8 - 68*x^7 - 16*x^6 + 256*x^5 - 208*x^4 + 128*x^3 + 256*x^2 - 256*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.2
Root			\(0.772089 + 1.18485i\) of defining polynomial
Character	\(\chi\)	\(=\)	784.557
Dual form			784.2.x.j.373.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.146726 - 1.40658i) q^{2} +(-0.977085 + 0.261809i) q^{3} +(-1.95694 + 0.412764i) q^{4} +(1.18533 + 0.317608i) q^{5} +(0.511620 + 1.33594i) q^{6} +(0.867721 + 2.69204i) q^{8} +(-1.71192 + 0.988380i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 4 q^{4} - 4 q^{5} + 32 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.146726	−	1.40658i	−0.103751	−	0.994603i
							\(3\)	−0.977085	+	0.261809i	−0.564120	+	0.151156i	−0.529599	−	0.848248i	\(-0.677657\pi\)
−0.0345217	+	0.999404i	\(0.510991\pi\)
\(5\)	1.18533	+	0.317608i	0.530095	+	0.142039i	0.513932	−	0.857831i	\(-0.328188\pi\)
							0.0161629	+	0.999869i	\(0.494855\pi\)
\(7\)		0			0
							\(11\)	−1.08957	−	4.06633i	−0.328518	−	1.22604i	−0.910728	−	0.413007i	\(-0.864479\pi\)
0.582210	−	0.813038i	\(-0.302188\pi\)
\(13\)	2.02017	+	2.02017i	0.560293	+	0.560293i	0.929391	−	0.369098i	\(-0.120333\pi\)
							−0.369098	+	0.929391i	\(0.620333\pi\)
\(17\)	0.132279	−	0.229115i	0.0320825	−	0.0555685i	−0.849538	−	0.527527i	\(-0.823119\pi\)
							0.881621	+	0.471958i	\(0.156453\pi\)
\(19\)	1.66161	−	6.20120i	0.381199	−	1.42265i	−0.462874	−	0.886424i	\(-0.653182\pi\)
							0.844073	−	0.536228i	\(-0.180151\pi\)
\(23\)	−1.33906	+	0.773104i	−0.279212	+	0.161203i	−0.633067	−	0.774097i	\(-0.718204\pi\)
							0.353854	+	0.935301i	\(0.384871\pi\)
\(29\)	−0.328129	−	0.328129i	−0.0609320	−	0.0609320i	0.675984	−	0.736916i	\(-0.263719\pi\)
							−0.736916	+	0.675984i	\(0.763719\pi\)
\(31\)	3.02017	−	5.23108i	0.542438	−	0.939530i	−0.456326	−	0.889813i	\(-0.650835\pi\)
							0.998763	−	0.0497168i	\(-0.0158319\pi\)
\(37\)	−9.08220	−	2.43357i	−1.49310	−	0.400076i	−0.582320	−	0.812959i	\(-0.697855\pi\)
							−0.910784	+	0.412883i	\(0.864522\pi\)
\(41\)		−	11.0327i		−	1.72302i	−0.507741	−	0.861510i	\(-0.669519\pi\)
							0.507741	−	0.861510i	\(-0.330481\pi\)
\(43\)	3.38407	−	3.38407i	0.516066	−	0.516066i	−0.400312	−	0.916379i	\(-0.631098\pi\)
							0.916379	+	0.400312i	\(0.131098\pi\)
\(47\)	1.56283	+	2.70690i	0.227962	+	0.394842i	0.957204	−	0.289414i	\(-0.0934603\pi\)
							−0.729242	+	0.684256i	\(0.760127\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.j.557.2		16
7.2	even	3	inner	784.2.x.j.765.2		16
7.3	odd	6		112.2.m.c.29.4	✓	8
7.4	even	3		784.2.m.g.589.4		8
7.5	odd	6		784.2.x.k.765.2		16
7.6	odd	2		784.2.x.k.557.2		16
16.5	even	4	inner	784.2.x.j.165.2		16
28.3	even	6		448.2.m.c.337.3		8
56.3	even	6		896.2.m.f.673.2		8
56.45	odd	6		896.2.m.e.673.3		8
112.3	even	12		896.2.m.f.225.2		8
112.5	odd	12		784.2.x.k.373.2		16
112.37	even	12	inner	784.2.x.j.373.2		16
112.45	odd	12		896.2.m.e.225.3		8
112.53	even	12		784.2.m.g.197.4		8
112.59	even	12		448.2.m.c.113.3		8
112.69	odd	4		784.2.x.k.165.2		16
112.101	odd	12		112.2.m.c.85.4	yes	8
224.59	even	24		7168.2.a.bd.1.4		8
224.101	odd	24		7168.2.a.bc.1.5		8
224.171	even	24		7168.2.a.bd.1.5		8
224.213	odd	24		7168.2.a.bc.1.4		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.c.29.4	✓	8	7.3	odd	6
112.2.m.c.85.4	yes	8	112.101	odd	12
448.2.m.c.113.3		8	112.59	even	12
448.2.m.c.337.3		8	28.3	even	6
784.2.m.g.197.4		8	112.53	even	12
784.2.m.g.589.4		8	7.4	even	3
784.2.x.j.165.2		16	16.5	even	4	inner
784.2.x.j.373.2		16	112.37	even	12	inner
784.2.x.j.557.2		16	1.1	even	1	trivial
784.2.x.j.765.2		16	7.2	even	3	inner
784.2.x.k.165.2		16	112.69	odd	4
784.2.x.k.373.2		16	112.5	odd	12
784.2.x.k.557.2		16	7.6	odd	2
784.2.x.k.765.2		16	7.5	odd	6
896.2.m.e.225.3		8	112.45	odd	12
896.2.m.e.673.3		8	56.45	odd	6
896.2.m.f.225.2		8	112.3	even	12
896.2.m.f.673.2		8	56.3	even	6
7168.2.a.bc.1.4		8	224.213	odd	24
7168.2.a.bc.1.5		8	224.101	odd	24
7168.2.a.bd.1.4		8	224.59	even	24
7168.2.a.bd.1.5		8	224.171	even	24