Embedded newform 784.2.x.k.165.2

• Label: 784.2.x.k.165.2
• Level: $784$
• Weight: $2$
• Character: 784.165
• 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			165.2
Root			\(0.640069 + 1.26108i\) of defining polynomial
Character	\(\chi\)	\(=\)	784.165
Dual form			784.2.x.k.765.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.14477 - 0.830359i) q^{2} +(-0.261809 - 0.977085i) q^{3} +(0.621007 + 1.90114i) q^{4} +(0.317608 - 1.18533i) q^{5} +(-0.511620 + 1.33594i) q^{6} +(0.867721 - 2.69204i) q^{8} +(1.71192 - 0.988380i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 4 q^{4} + 4 q^{5} - 32 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.14477	−	0.830359i	−0.809476	−	0.587153i
							\(3\)	−0.261809	−	0.977085i	−0.151156	−	0.564120i	−0.999404	−	0.0345217i	\(-0.989009\pi\)
0.848248	−	0.529599i	\(-0.177657\pi\)
\(5\)	0.317608	−	1.18533i	0.142039	−	0.530095i	−0.857831	−	0.513932i	\(-0.828188\pi\)
							0.999869	−	0.0161629i	\(-0.00514502\pi\)
\(7\)		0			0
							\(11\)	4.06633	−	1.08957i	1.22604	−	0.328518i	0.413007	−	0.910728i	\(-0.364479\pi\)
0.813038	+	0.582210i	\(0.197812\pi\)
\(13\)	−2.02017	+	2.02017i	−0.560293	+	0.560293i	−0.929391	−	0.369098i	\(-0.879667\pi\)
							0.369098	+	0.929391i	\(0.379667\pi\)
\(17\)	−0.132279	+	0.229115i	−0.0320825	+	0.0555685i	−0.881621	−	0.471958i	\(-0.843547\pi\)
							0.849538	+	0.527527i	\(0.176881\pi\)
\(19\)	6.20120	+	1.66161i	1.42265	+	0.381199i	0.886424	−	0.462874i	\(-0.153182\pi\)
							0.536228	+	0.844073i	\(0.319849\pi\)
\(23\)	1.33906	−	0.773104i	0.279212	−	0.161203i	−0.353854	−	0.935301i	\(-0.615129\pi\)
							0.633067	+	0.774097i	\(0.281796\pi\)
\(29\)	−0.328129	+	0.328129i	−0.0609320	+	0.0609320i	−0.736916	−	0.675984i	\(-0.763719\pi\)
							0.675984	+	0.736916i	\(0.263719\pi\)
\(31\)	−3.02017	+	5.23108i	−0.542438	+	0.939530i	0.456326	+	0.889813i	\(0.349165\pi\)
							−0.998763	+	0.0497168i	\(0.984168\pi\)
\(37\)	2.43357	−	9.08220i	0.400076	−	1.49310i	−0.412883	−	0.910784i	\(-0.635478\pi\)
							0.812959	−	0.582320i	\(-0.197855\pi\)
\(41\)		−	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	0.916379	−	0.400312i	\(-0.131098\pi\)
							−0.400312	+	0.916379i	\(0.631098\pi\)
\(47\)	−1.56283	−	2.70690i	−0.227962	−	0.394842i	0.729242	−	0.684256i	\(-0.239873\pi\)
							−0.957204	+	0.289414i	\(0.906540\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.k.165.2		16
7.2	even	3	inner	784.2.x.k.373.2		16
7.3	odd	6		784.2.m.g.197.4		8
7.4	even	3		112.2.m.c.85.4	yes	8
7.5	odd	6		784.2.x.j.373.2		16
7.6	odd	2		784.2.x.j.165.2		16
16.13	even	4	inner	784.2.x.k.557.2		16
28.11	odd	6		448.2.m.c.113.3		8
56.11	odd	6		896.2.m.f.225.2		8
56.53	even	6		896.2.m.e.225.3		8
112.11	odd	12		896.2.m.f.673.2		8
112.13	odd	4		784.2.x.j.557.2		16
112.45	odd	12		784.2.m.g.589.4		8
112.53	even	12		896.2.m.e.673.3		8
112.61	odd	12		784.2.x.j.765.2		16
112.67	odd	12		448.2.m.c.337.3		8
112.93	even	12	inner	784.2.x.k.765.2		16
112.109	even	12		112.2.m.c.29.4	✓	8
224.67	odd	24		7168.2.a.bd.1.5		8
224.109	even	24		7168.2.a.bc.1.5		8
224.179	odd	24		7168.2.a.bd.1.4		8
224.221	even	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	112.109	even	12
112.2.m.c.85.4	yes	8	7.4	even	3
448.2.m.c.113.3		8	28.11	odd	6
448.2.m.c.337.3		8	112.67	odd	12
784.2.m.g.197.4		8	7.3	odd	6
784.2.m.g.589.4		8	112.45	odd	12
784.2.x.j.165.2		16	7.6	odd	2
784.2.x.j.373.2		16	7.5	odd	6
784.2.x.j.557.2		16	112.13	odd	4
784.2.x.j.765.2		16	112.61	odd	12
784.2.x.k.165.2		16	1.1	even	1	trivial
784.2.x.k.373.2		16	7.2	even	3	inner
784.2.x.k.557.2		16	16.13	even	4	inner
784.2.x.k.765.2		16	112.93	even	12	inner
896.2.m.e.225.3		8	56.53	even	6
896.2.m.e.673.3		8	112.53	even	12
896.2.m.f.225.2		8	56.11	odd	6
896.2.m.f.673.2		8	112.11	odd	12
7168.2.a.bc.1.4		8	224.221	even	24
7168.2.a.bc.1.5		8	224.109	even	24
7168.2.a.bd.1.4		8	224.179	odd	24
7168.2.a.bd.1.5		8	224.67	odd	24