Embedded newform 784.2.x.k.165.4

• Label: 784.2.x.k.165.4
• 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			165.4
Root			\(1.21641 + 0.721349i\) of defining polynomial
Character	\(\chi\)	\(=\)	784.165
Dual form			784.2.x.k.765.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.867450 + 1.11693i) q^{2} +(0.442248 + 1.65049i) q^{3} +(-0.495063 + 1.93776i) q^{4} +(-0.949390 + 3.54317i) q^{5} +(-1.45986 + 1.92568i) q^{6} +(-2.59378 + 1.12796i) q^{8} +(0.0695300 - 0.0401432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 4 q^{4} + 4 q^{5} - 32 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.867450	+	1.11693i	0.613379	+	0.789788i
							\(3\)	0.442248	+	1.65049i	0.255332	+	0.952913i	0.967905	+	0.251315i	\(0.0808630\pi\)
−0.712573	+	0.701598i	\(0.752470\pi\)
\(5\)	−0.949390	+	3.54317i	−0.424580	+	1.58456i	0.340258	+	0.940332i	\(0.389486\pi\)
							−0.764838	+	0.644223i	\(0.777181\pi\)
\(7\)		0			0
							\(11\)	1.47570	−	0.395412i	0.444940	−	0.119221i	−0.0293909	−	0.999568i	\(-0.509357\pi\)
0.474331	+	0.880347i	\(0.342690\pi\)
\(13\)	2.97932	−	2.97932i	0.826315	−	0.826315i	−0.160690	−	0.987005i	\(-0.551372\pi\)
							0.987005	+	0.160690i	\(0.0513720\pi\)
\(17\)	−3.59378	+	6.22461i	−0.871620	+	1.50969i	−0.0113004	+	0.999936i	\(0.503597\pi\)
							−0.860320	+	0.509755i	\(0.829736\pi\)
\(19\)	3.08391	+	0.826331i	0.707497	+	0.189573i	0.594586	−	0.804032i	\(-0.297316\pi\)
							0.112911	+	0.993605i	\(0.463983\pi\)
\(23\)	3.02830	−	1.74839i	0.631444	−	0.364564i	−0.149867	−	0.988706i	\(-0.547885\pi\)
							0.781311	+	0.624142i	\(0.214551\pi\)
\(29\)	0.851361	−	0.851361i	0.158094	−	0.158094i	−0.623628	−	0.781721i	\(-0.714342\pi\)
							0.781721	+	0.623628i	\(0.214342\pi\)
\(31\)	1.97932	−	3.42828i	0.355496	−	0.615738i	−0.631706	−	0.775208i	\(-0.717645\pi\)
							0.987203	+	0.159470i	\(0.0509785\pi\)
\(37\)	2.17113	−	8.10278i	0.356932	−	1.33209i	−0.521103	−	0.853494i	\(-0.674479\pi\)
							0.878036	−	0.478595i	\(-0.158854\pi\)
\(41\)		−	2.67573i		−	0.417879i	−0.977929	−	0.208939i	\(-0.932999\pi\)
							0.977929	−	0.208939i	\(-0.0670011\pi\)
\(43\)	−4.25592	−	4.25592i	−0.649021	−	0.649021i	0.303735	−	0.952757i	\(-0.401766\pi\)
							−0.952757	+	0.303735i	\(0.901766\pi\)
\(47\)	−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.165.4		16
7.2	even	3	inner	784.2.x.k.373.3		16
7.3	odd	6		784.2.m.g.197.1		8
7.4	even	3		112.2.m.c.85.1	yes	8
7.5	odd	6		784.2.x.j.373.3		16
7.6	odd	2		784.2.x.j.165.4		16
16.13	even	4	inner	784.2.x.k.557.3		16
28.11	odd	6		448.2.m.c.113.2		8
56.11	odd	6		896.2.m.f.225.3		8
56.53	even	6		896.2.m.e.225.2		8
112.11	odd	12		896.2.m.f.673.3		8
112.13	odd	4		784.2.x.j.557.3		16
112.45	odd	12		784.2.m.g.589.1		8
112.53	even	12		896.2.m.e.673.2		8
112.61	odd	12		784.2.x.j.765.4		16
112.67	odd	12		448.2.m.c.337.2		8
112.93	even	12	inner	784.2.x.k.765.4		16
112.109	even	12		112.2.m.c.29.1	✓	8
224.67	odd	24		7168.2.a.bd.1.3		8
224.109	even	24		7168.2.a.bc.1.3		8
224.179	odd	24		7168.2.a.bd.1.6		8
224.221	even	24		7168.2.a.bc.1.6		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.c.29.1	✓	8	112.109	even	12
112.2.m.c.85.1	yes	8	7.4	even	3
448.2.m.c.113.2		8	28.11	odd	6
448.2.m.c.337.2		8	112.67	odd	12
784.2.m.g.197.1		8	7.3	odd	6
784.2.m.g.589.1		8	112.45	odd	12
784.2.x.j.165.4		16	7.6	odd	2
784.2.x.j.373.3		16	7.5	odd	6
784.2.x.j.557.3		16	112.13	odd	4
784.2.x.j.765.4		16	112.61	odd	12
784.2.x.k.165.4		16	1.1	even	1	trivial
784.2.x.k.373.3		16	7.2	even	3	inner
784.2.x.k.557.3		16	16.13	even	4	inner
784.2.x.k.765.4		16	112.93	even	12	inner
896.2.m.e.225.2		8	56.53	even	6
896.2.m.e.673.2		8	112.53	even	12
896.2.m.f.225.3		8	56.11	odd	6
896.2.m.f.673.3		8	112.11	odd	12
7168.2.a.bc.1.3		8	224.109	even	24
7168.2.a.bc.1.6		8	224.221	even	24
7168.2.a.bd.1.3		8	224.67	odd	24
7168.2.a.bd.1.6		8	224.179	odd	24