Embedded newform 784.2.x.j.165.3

• Label: 784.2.x.j.165.3
• 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			165.3
Root			\(0.840224 - 1.13755i\) of defining polynomial
Character	\(\chi\)	\(=\)	784.165
Dual form			784.2.x.j.765.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.105414 - 1.41028i) q^{2} +(0.719263 + 2.68432i) q^{3} +(-1.97778 - 0.297327i) q^{4} +(0.229791 - 0.857592i) q^{5} +(3.86147 - 0.731395i) q^{6} +(-0.627801 + 2.75787i) q^{8} +(-4.09018 + 2.36147i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 4 q^{4} - 4 q^{5} + 32 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.105414	−	1.41028i	0.0745392	−	0.997218i
							\(3\)	0.719263	+	2.68432i	0.415266	+	1.54980i	0.784301	+	0.620380i	\(0.213022\pi\)
−0.369035	+	0.929415i	\(0.620312\pi\)
\(5\)	0.229791	−	0.857592i	0.102766	−	0.383527i	−0.895316	−	0.445431i	\(-0.853051\pi\)
							0.998082	+	0.0619040i	\(0.0197173\pi\)
\(7\)		0			0
							\(11\)	−5.08562	+	1.36269i	−1.53337	+	0.410866i	−0.924118	−	0.382108i	\(-0.875198\pi\)
−0.609256	+	0.792974i	\(0.708532\pi\)
\(13\)	−2.22066	+	2.22066i	−0.615901	+	0.615901i	−0.944477	−	0.328576i	\(-0.893431\pi\)
							0.328576	+	0.944477i	\(0.393431\pi\)
\(17\)	1.62780	−	2.81943i	0.394800	−	0.683813i	−0.598276	−	0.801290i	\(-0.704147\pi\)
							0.993076	+	0.117477i	\(0.0374807\pi\)
\(19\)	−2.50664	−	0.671653i	−0.575063	−	0.154088i	−0.0404454	−	0.999182i	\(-0.512878\pi\)
							−0.534618	+	0.845094i	\(0.679544\pi\)
\(23\)	−6.62774	+	3.82653i	−1.38198	+	0.797887i	−0.992394	−	0.123103i	\(-0.960715\pi\)
							−0.389586	+	0.920990i	\(0.627382\pi\)
\(29\)	−1.53721	+	1.53721i	−0.285453	+	0.285453i	−0.835279	−	0.549826i	\(-0.814694\pi\)
							0.549826	+	0.835279i	\(0.314694\pi\)
\(31\)	−1.22066	+	2.11425i	−0.219238	+	0.379731i	−0.954575	−	0.297970i	\(-0.903690\pi\)
							0.735338	+	0.677701i	\(0.237024\pi\)
\(37\)	−0.461245	+	1.72139i	−0.0758283	+	0.282995i	−0.993420	−	0.114530i	\(-0.963464\pi\)
							0.917592	+	0.397524i	\(0.130131\pi\)
\(41\)		−	7.77589i		−	1.21439i	−0.794553	−	0.607195i	\(-0.792295\pi\)
							0.794553	−	0.607195i	\(-0.207705\pi\)
\(43\)	−7.51575	−	7.51575i	−1.14614	−	1.14614i	−0.987305	−	0.158836i	\(-0.949226\pi\)
							−0.158836	−	0.987305i	\(-0.550774\pi\)
\(47\)	5.55792	+	9.62661i	0.810707	+	1.40418i	0.912370	+	0.409366i	\(0.134250\pi\)
							−0.101664	+	0.994819i	\(0.532417\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.j.165.3		16
7.2	even	3	inner	784.2.x.j.373.1		16
7.3	odd	6		112.2.m.c.85.3	yes	8
7.4	even	3		784.2.m.g.197.3		8
7.5	odd	6		784.2.x.k.373.1		16
7.6	odd	2		784.2.x.k.165.3		16
16.13	even	4	inner	784.2.x.j.557.1		16
28.3	even	6		448.2.m.c.113.4		8
56.3	even	6		896.2.m.f.225.1		8
56.45	odd	6		896.2.m.e.225.4		8
112.3	even	12		448.2.m.c.337.4		8
112.13	odd	4		784.2.x.k.557.1		16
112.45	odd	12		112.2.m.c.29.3	✓	8
112.59	even	12		896.2.m.f.673.1		8
112.61	odd	12		784.2.x.k.765.3		16
112.93	even	12	inner	784.2.x.j.765.3		16
112.101	odd	12		896.2.m.e.673.4		8
112.109	even	12		784.2.m.g.589.3		8
224.3	even	24		7168.2.a.bd.1.8		8
224.45	odd	24		7168.2.a.bc.1.8		8
224.115	even	24		7168.2.a.bd.1.1		8
224.157	odd	24		7168.2.a.bc.1.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.c.29.3	✓	8	112.45	odd	12
112.2.m.c.85.3	yes	8	7.3	odd	6
448.2.m.c.113.4		8	28.3	even	6
448.2.m.c.337.4		8	112.3	even	12
784.2.m.g.197.3		8	7.4	even	3
784.2.m.g.589.3		8	112.109	even	12
784.2.x.j.165.3		16	1.1	even	1	trivial
784.2.x.j.373.1		16	7.2	even	3	inner
784.2.x.j.557.1		16	16.13	even	4	inner
784.2.x.j.765.3		16	112.93	even	12	inner
784.2.x.k.165.3		16	7.6	odd	2
784.2.x.k.373.1		16	7.5	odd	6
784.2.x.k.557.1		16	112.13	odd	4
784.2.x.k.765.3		16	112.61	odd	12
896.2.m.e.225.4		8	56.45	odd	6
896.2.m.e.673.4		8	112.101	odd	12
896.2.m.f.225.1		8	56.3	even	6
896.2.m.f.673.1		8	112.59	even	12
7168.2.a.bc.1.1		8	224.157	odd	24
7168.2.a.bc.1.8		8	224.45	odd	24
7168.2.a.bd.1.1		8	224.115	even	24
7168.2.a.bd.1.8		8	224.3	even	24