Embedded newform 784.2.x.k.765.1

• Label: 784.2.x.k.765.1
• Level: $784$
• Weight: $2$
• Character: 784.765
• 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			765.1
Root			\(-1.33068 + 0.478848i\) of defining polynomial
Character	\(\chi\)	\(=\)	784.765
Dual form			784.2.x.k.165.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19412 - 0.757684i) q^{2} +(0.538823 - 2.01092i) q^{3} +(0.851831 + 1.80953i) q^{4} +(0.129523 + 0.483385i) q^{5} +(-2.16706 + 1.99301i) q^{6} +(0.353863 - 2.80620i) q^{8} +(-1.15537 - 0.667056i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 4 q^{4} + 4 q^{5} - 32 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19412	−	0.757684i	−0.844368	−	0.535763i
							\(3\)	0.538823	−	2.01092i	0.311090	−	1.16100i	−0.616486	−	0.787366i	\(-0.711444\pi\)
0.927575	−	0.373636i	\(-0.121889\pi\)
\(5\)	0.129523	+	0.483385i	0.0579243	+	0.216177i	0.988821	−	0.149105i	\(-0.0476393\pi\)
							−0.930897	+	0.365282i	\(0.880973\pi\)
\(7\)		0			0
							\(11\)	−0.456405	−	0.122293i	−0.137611	−	0.0368728i	0.189356	−	0.981909i	\(-0.439360\pi\)
−0.326967	+	0.945036i	\(0.606027\pi\)
\(13\)	−3.17982	−	3.17982i	−0.881923	−	0.881923i	0.111807	−	0.993730i	\(-0.464336\pi\)
							−0.993730	+	0.111807i	\(0.964336\pi\)
\(17\)	−0.646137	−	1.11914i	−0.156711	−	0.271432i	0.776970	−	0.629538i	\(-0.216756\pi\)
							−0.933681	+	0.358106i	\(0.883423\pi\)
\(19\)	−3.59560	+	0.963437i	−0.824886	+	0.221028i	−0.646482	−	0.762930i	\(-0.723760\pi\)
							−0.178405	+	0.983957i	\(0.557094\pi\)
\(23\)	2.26039	+	1.30504i	0.471324	+	0.272119i	0.716794	−	0.697285i	\(-0.245609\pi\)
							−0.245470	+	0.969404i	\(0.578942\pi\)
\(29\)	−6.98602	−	6.98602i	−1.29727	−	1.29727i	−0.930188	−	0.367084i	\(-0.880356\pi\)
							−0.367084	−	0.930188i	\(-0.619644\pi\)
\(31\)	−4.17982	−	7.23966i	−0.750717	−	1.30028i	−0.947475	−	0.319829i	\(-0.896375\pi\)
							0.196758	−	0.980452i	\(-0.436959\pi\)
\(37\)	−1.21525	−	4.53539i	−0.199786	−	0.745613i	−0.990976	−	0.134042i	\(-0.957204\pi\)
							0.791189	−	0.611571i	\(-0.209462\pi\)
\(41\)		−	9.93254i		−	1.55120i	−0.631223	−	0.775601i	\(-0.717447\pi\)
							0.631223	−	0.775601i	\(-0.282553\pi\)
\(43\)	−7.61241	+	7.61241i	−1.16088	+	1.16088i	−0.176598	+	0.984283i	\(0.556509\pi\)
							−0.984283	+	0.176598i	\(0.943491\pi\)
\(47\)	2.29805	−	3.98033i	0.335205	−	0.580591i	−0.648319	−	0.761368i	\(-0.724528\pi\)
							0.983524	+	0.180777i	\(0.0578612\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.k.765.1		16
7.2	even	3		112.2.m.c.29.2	✓	8
7.3	odd	6		784.2.x.j.557.4		16
7.4	even	3	inner	784.2.x.k.557.4		16
7.5	odd	6		784.2.m.g.589.2		8
7.6	odd	2		784.2.x.j.765.1		16
16.5	even	4	inner	784.2.x.k.373.4		16
28.23	odd	6		448.2.m.c.337.1		8
56.37	even	6		896.2.m.e.673.1		8
56.51	odd	6		896.2.m.f.673.4		8
112.5	odd	12		784.2.m.g.197.2		8
112.37	even	12		112.2.m.c.85.2	yes	8
112.51	odd	12		896.2.m.f.225.4		8
112.53	even	12	inner	784.2.x.k.165.1		16
112.69	odd	4		784.2.x.j.373.4		16
112.93	even	12		896.2.m.e.225.1		8
112.101	odd	12		784.2.x.j.165.1		16
112.107	odd	12		448.2.m.c.113.1		8
224.37	even	24		7168.2.a.bc.1.2		8
224.107	odd	24		7168.2.a.bd.1.2		8
224.149	even	24		7168.2.a.bc.1.7		8
224.219	odd	24		7168.2.a.bd.1.7		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.c.29.2	✓	8	7.2	even	3
112.2.m.c.85.2	yes	8	112.37	even	12
448.2.m.c.113.1		8	112.107	odd	12
448.2.m.c.337.1		8	28.23	odd	6
784.2.m.g.197.2		8	112.5	odd	12
784.2.m.g.589.2		8	7.5	odd	6
784.2.x.j.165.1		16	112.101	odd	12
784.2.x.j.373.4		16	112.69	odd	4
784.2.x.j.557.4		16	7.3	odd	6
784.2.x.j.765.1		16	7.6	odd	2
784.2.x.k.165.1		16	112.53	even	12	inner
784.2.x.k.373.4		16	16.5	even	4	inner
784.2.x.k.557.4		16	7.4	even	3	inner
784.2.x.k.765.1		16	1.1	even	1	trivial
896.2.m.e.225.1		8	112.93	even	12
896.2.m.e.673.1		8	56.37	even	6
896.2.m.f.225.4		8	112.51	odd	12
896.2.m.f.673.4		8	56.51	odd	6
7168.2.a.bc.1.2		8	224.37	even	24
7168.2.a.bc.1.7		8	224.149	even	24
7168.2.a.bd.1.2		8	224.107	odd	24
7168.2.a.bd.1.7		8	224.219	odd	24