Embedded newform 392.2.z.a.53.15

• Label: 392.2.z.a.53.15
• Level: $392$
• Weight: $2$
• Character: 392.53
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			53.15
Character	\(\chi\)	\(=\)	392.53
Dual form			392.2.z.a.37.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.824960 - 1.14867i) q^{2} +(-1.48945 + 1.60524i) q^{3} +(-0.638881 + 1.89521i) q^{4} +(1.25029 + 4.05335i) q^{5} +(3.07263 + 0.386620i) q^{6} +(2.55536 + 0.685667i) q^{7} +(2.70402 - 0.829613i) q^{8} +(-0.134159 - 1.79023i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.824960	−	1.14867i	−0.583335	−	0.812232i
							\(3\)	−1.48945	+	1.60524i	−0.859933	+	0.926787i	−0.997973	−	0.0636398i	\(-0.979729\pi\)
0.138040	+	0.990427i	\(0.455920\pi\)
\(5\)	1.25029	+	4.05335i	0.559148	+	1.81271i	0.582290	+	0.812981i	\(0.302157\pi\)
							−0.0231418	+	0.999732i	\(0.507367\pi\)
\(7\)	2.55536	+	0.685667i	0.965835	+	0.259158i
							\(11\)	−5.31750	−	0.398492i	−1.60329	−	0.120150i	−0.757196	−	0.653188i	\(-0.773431\pi\)
−0.846091	+	0.533038i	\(0.821050\pi\)
\(13\)	1.33482	+	2.77179i	0.370213	+	0.768756i	0.999968	−	0.00803866i	\(-0.00255881\pi\)
							−0.629755	+	0.776794i	\(0.716845\pi\)
\(17\)	0.221290	−	0.0333541i	0.0536707	−	0.00808955i	−0.122152	−	0.992511i	\(-0.538980\pi\)
							0.175823	+	0.984422i	\(0.443741\pi\)
\(19\)	4.52324	−	2.61149i	1.03770	−	0.599118i	0.118521	−	0.992952i	\(-0.462185\pi\)
							0.919182	+	0.393834i	\(0.128852\pi\)
\(23\)	2.76583	+	0.416882i	0.576716	+	0.0869259i	0.430922	−	0.902389i	\(-0.358188\pi\)
							0.145794	+	0.989315i	\(0.453426\pi\)
\(29\)	1.75874	−	1.40254i	0.326589	−	0.260446i	−0.446444	−	0.894811i	\(-0.647310\pi\)
							0.773033	+	0.634365i	\(0.218739\pi\)
\(31\)	−0.285976	+	0.495326i	−0.0513629	+	0.0889631i	−0.890564	−	0.454858i	\(-0.849690\pi\)
							0.839201	+	0.543822i	\(0.183023\pi\)
\(37\)	−2.64765	−	1.03913i	−0.435271	−	0.170831i	0.137580	−	0.990491i	\(-0.456068\pi\)
							−0.572851	+	0.819659i	\(0.694163\pi\)
\(41\)	−0.686642	+	3.00838i	−0.107235	+	0.469829i	0.892585	+	0.450879i	\(0.148889\pi\)
							−0.999820	+	0.0189500i	\(0.993968\pi\)
\(43\)	7.43650	−	1.69733i	1.13406	−	0.258841i	0.386014	−	0.922493i	\(-0.373852\pi\)
							0.748041	+	0.663652i	\(0.230994\pi\)
\(47\)	−9.37630	−	6.39265i	−1.36767	−	0.932464i	−0.999997	−	0.00251019i	\(-0.999201\pi\)
							−0.367677	−	0.929954i	\(-0.619847\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.53.15	yes	648
8.5	even	2	inner	392.2.z.a.53.6	yes	648
49.37	even	21	inner	392.2.z.a.37.6	✓	648
392.37	even	42	inner	392.2.z.a.37.15	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.37.6	✓	648	49.37	even	21	inner
392.2.z.a.37.15	yes	648	392.37	even	42	inner
392.2.z.a.53.6	yes	648	8.5	even	2	inner
392.2.z.a.53.15	yes	648	1.1	even	1	trivial