Embedded newform 784.2.x.k.373.1

• Label: 784.2.x.k.373.1
• Level: $784$
• Weight: $2$
• Character: 784.373
• 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			373.1
Root			\(-1.40526 - 0.158880i\) of defining polynomial
Character	\(\chi\)	\(=\)	784.373
Dual form			784.2.x.k.557.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27404 + 0.613848i) q^{2} +(2.68432 + 0.719263i) q^{3} +(1.24638 - 1.56414i) q^{4} +(0.857592 - 0.229791i) q^{5} +(-3.86147 + 0.731395i) q^{6} +(-0.627801 + 2.75787i) q^{8} +(4.09018 + 2.36147i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 4 q^{4} + 4 q^{5} - 32 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27404	+	0.613848i	−0.900886	+	0.434056i
							\(3\)	2.68432	+	0.719263i	1.54980	+	0.415266i	0.929415	−	0.369035i	\(-0.120312\pi\)
0.620380	+	0.784301i	\(0.286978\pi\)
\(5\)	0.857592	−	0.229791i	0.383527	−	0.102766i	−0.0619040	−	0.998082i	\(-0.519717\pi\)
							0.445431	+	0.895316i	\(0.353051\pi\)
\(7\)		0			0
							\(11\)	1.36269	−	5.08562i	0.410866	−	1.53337i	−0.382108	−	0.924118i	\(-0.624802\pi\)
0.792974	−	0.609256i	\(-0.208532\pi\)
\(13\)	2.22066	−	2.22066i	0.615901	−	0.615901i	−0.328576	−	0.944477i	\(-0.606569\pi\)
							0.944477	+	0.328576i	\(0.106569\pi\)
\(17\)	−1.62780	−	2.81943i	−0.394800	−	0.683813i	0.598276	−	0.801290i	\(-0.295853\pi\)
							−0.993076	+	0.117477i	\(0.962519\pi\)
\(19\)	−0.671653	−	2.50664i	−0.154088	−	0.575063i	−0.999182	−	0.0404454i	\(-0.987122\pi\)
							0.845094	−	0.534618i	\(-0.179544\pi\)
\(23\)	6.62774	+	3.82653i	1.38198	+	0.797887i	0.992394	−	0.123103i	\(-0.0392847\pi\)
							0.389586	+	0.920990i	\(0.372618\pi\)
\(29\)	−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	\(-0.0963097\pi\)
							−0.735338	+	0.677701i	\(0.762976\pi\)
\(37\)	1.72139	−	0.461245i	0.282995	−	0.0758283i	−0.114530	−	0.993420i	\(-0.536536\pi\)
							0.397524	+	0.917592i	\(0.369869\pi\)
\(41\)			7.77589i			1.21439i	0.794553	+	0.607195i	\(0.207705\pi\)
							−0.794553	+	0.607195i	\(0.792295\pi\)
\(43\)	−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	0.101664	+	0.994819i	\(0.467583\pi\)
							−0.912370	+	0.409366i	\(0.865750\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.k.373.1		16
7.2	even	3		112.2.m.c.85.3	yes	8
7.3	odd	6		784.2.x.j.165.3		16
7.4	even	3	inner	784.2.x.k.165.3		16
7.5	odd	6		784.2.m.g.197.3		8
7.6	odd	2		784.2.x.j.373.1		16
16.13	even	4	inner	784.2.x.k.765.3		16
28.23	odd	6		448.2.m.c.113.4		8
56.37	even	6		896.2.m.e.225.4		8
56.51	odd	6		896.2.m.f.225.1		8
112.13	odd	4		784.2.x.j.765.3		16
112.37	even	12		896.2.m.e.673.4		8
112.45	odd	12		784.2.x.j.557.1		16
112.51	odd	12		448.2.m.c.337.4		8
112.61	odd	12		784.2.m.g.589.3		8
112.93	even	12		112.2.m.c.29.3	✓	8
112.107	odd	12		896.2.m.f.673.1		8
112.109	even	12	inner	784.2.x.k.557.1		16
224.51	odd	24		7168.2.a.bd.1.1		8
224.93	even	24		7168.2.a.bc.1.1		8
224.163	odd	24		7168.2.a.bd.1.8		8
224.205	even	24		7168.2.a.bc.1.8		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.c.29.3	✓	8	112.93	even	12
112.2.m.c.85.3	yes	8	7.2	even	3
448.2.m.c.113.4		8	28.23	odd	6
448.2.m.c.337.4		8	112.51	odd	12
784.2.m.g.197.3		8	7.5	odd	6
784.2.m.g.589.3		8	112.61	odd	12
784.2.x.j.165.3		16	7.3	odd	6
784.2.x.j.373.1		16	7.6	odd	2
784.2.x.j.557.1		16	112.45	odd	12
784.2.x.j.765.3		16	112.13	odd	4
784.2.x.k.165.3		16	7.4	even	3	inner
784.2.x.k.373.1		16	1.1	even	1	trivial
784.2.x.k.557.1		16	112.109	even	12	inner
784.2.x.k.765.3		16	16.13	even	4	inner
896.2.m.e.225.4		8	56.37	even	6
896.2.m.e.673.4		8	112.37	even	12
896.2.m.f.225.1		8	56.51	odd	6
896.2.m.f.673.1		8	112.107	odd	12
7168.2.a.bc.1.1		8	224.93	even	24
7168.2.a.bc.1.8		8	224.205	even	24
7168.2.a.bd.1.1		8	224.51	odd	24
7168.2.a.bd.1.8		8	224.163	odd	24