Embedded newform 784.2.x.k.557.3

• Label: 784.2.x.k.557.3
• 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			557.3
Root			\(0.0165007 + 1.41412i\) of defining polynomial
Character	\(\chi\)	\(=\)	784.557
Dual form			784.2.x.k.373.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.533564 + 1.30970i) q^{2} +(-1.65049 + 0.442248i) q^{3} +(-1.43062 + 1.39762i) q^{4} +(3.54317 + 0.949390i) q^{5} +(-1.45986 - 1.92568i) q^{6} +(-2.59378 - 1.12796i) q^{8} +(-0.0695300 + 0.0401432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 4 q^{4} + 4 q^{5} - 32 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.533564	+	1.30970i	0.377287	+	0.926096i
							\(3\)	−1.65049	+	0.442248i	−0.952913	+	0.255332i	−0.701598	−	0.712573i	\(-0.747530\pi\)
−0.251315	+	0.967905i	\(0.580863\pi\)
\(5\)	3.54317	+	0.949390i	1.58456	+	0.424580i	0.940332	−	0.340258i	\(-0.110514\pi\)
							0.644223	+	0.764838i	\(0.277181\pi\)
\(7\)		0			0
							\(11\)	−0.395412	−	1.47570i	−0.119221	−	0.444940i	0.880347	−	0.474331i	\(-0.157310\pi\)
−0.999568	+	0.0293909i	\(0.990643\pi\)
\(13\)	2.97932	+	2.97932i	0.826315	+	0.826315i	0.987005	−	0.160690i	\(-0.0513720\pi\)
							−0.160690	+	0.987005i	\(0.551372\pi\)
\(17\)	−3.59378	+	6.22461i	−0.871620	+	1.50969i	−0.0113004	+	0.999936i	\(0.503597\pi\)
							−0.860320	+	0.509755i	\(0.829736\pi\)
\(19\)	−0.826331	+	3.08391i	−0.189573	+	0.707497i	0.804032	+	0.594586i	\(0.202684\pi\)
							−0.993605	+	0.112911i	\(0.963983\pi\)
\(23\)	−3.02830	+	1.74839i	−0.631444	+	0.364564i	−0.781311	−	0.624142i	\(-0.785449\pi\)
							0.149867	+	0.988706i	\(0.452115\pi\)
\(29\)	0.851361	+	0.851361i	0.158094	+	0.158094i	0.781721	−	0.623628i	\(-0.214342\pi\)
							−0.623628	+	0.781721i	\(0.714342\pi\)
\(31\)	1.97932	−	3.42828i	0.355496	−	0.615738i	−0.631706	−	0.775208i	\(-0.717645\pi\)
							0.987203	+	0.159470i	\(0.0509785\pi\)
\(37\)	−8.10278	−	2.17113i	−1.33209	−	0.356932i	−0.478595	−	0.878036i	\(-0.658854\pi\)
							−0.853494	+	0.521103i	\(0.825521\pi\)
\(41\)			2.67573i			0.417879i	0.977929	+	0.208939i	\(0.0670011\pi\)
							−0.977929	+	0.208939i	\(0.932999\pi\)
\(43\)	−4.25592	+	4.25592i	−0.649021	+	0.649021i	−0.952757	−	0.303735i	\(-0.901766\pi\)
							0.303735	+	0.952757i	\(0.401766\pi\)
\(47\)	−1.17729	−	2.03913i	−0.171726	−	0.297438i	0.767298	−	0.641291i	\(-0.221601\pi\)
							−0.939023	+	0.343854i	\(0.888268\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.k.557.3		16
7.2	even	3	inner	784.2.x.k.765.4		16
7.3	odd	6		784.2.m.g.589.1		8
7.4	even	3		112.2.m.c.29.1	✓	8
7.5	odd	6		784.2.x.j.765.4		16
7.6	odd	2		784.2.x.j.557.3		16
16.5	even	4	inner	784.2.x.k.165.4		16
28.11	odd	6		448.2.m.c.337.2		8
56.11	odd	6		896.2.m.f.673.3		8
56.53	even	6		896.2.m.e.673.2		8
112.5	odd	12		784.2.x.j.373.3		16
112.11	odd	12		448.2.m.c.113.2		8
112.37	even	12	inner	784.2.x.k.373.3		16
112.53	even	12		112.2.m.c.85.1	yes	8
112.67	odd	12		896.2.m.f.225.3		8
112.69	odd	4		784.2.x.j.165.4		16
112.101	odd	12		784.2.m.g.197.1		8
112.109	even	12		896.2.m.e.225.2		8
224.11	odd	24		7168.2.a.bd.1.3		8
224.53	even	24		7168.2.a.bc.1.6		8
224.123	odd	24		7168.2.a.bd.1.6		8
224.165	even	24		7168.2.a.bc.1.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.c.29.1	✓	8	7.4	even	3
112.2.m.c.85.1	yes	8	112.53	even	12
448.2.m.c.113.2		8	112.11	odd	12
448.2.m.c.337.2		8	28.11	odd	6
784.2.m.g.197.1		8	112.101	odd	12
784.2.m.g.589.1		8	7.3	odd	6
784.2.x.j.165.4		16	112.69	odd	4
784.2.x.j.373.3		16	112.5	odd	12
784.2.x.j.557.3		16	7.6	odd	2
784.2.x.j.765.4		16	7.5	odd	6
784.2.x.k.165.4		16	16.5	even	4	inner
784.2.x.k.373.3		16	112.37	even	12	inner
784.2.x.k.557.3		16	1.1	even	1	trivial
784.2.x.k.765.4		16	7.2	even	3	inner
896.2.m.e.225.2		8	112.109	even	12
896.2.m.e.673.2		8	56.53	even	6
896.2.m.f.225.3		8	112.67	odd	12
896.2.m.f.673.3		8	56.11	odd	6
7168.2.a.bc.1.3		8	224.165	even	24
7168.2.a.bc.1.6		8	224.53	even	24
7168.2.a.bd.1.3		8	224.11	odd	24
7168.2.a.bd.1.6		8	224.123	odd	24