Embedded newform 392.2.z.a.205.11

• Label: 392.2.z.a.205.11
• Level: $392$
• Weight: $2$
• Character: 392.205
• 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			205.11
Character	\(\chi\)	\(=\)	392.205
Dual form			392.2.z.a.109.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19093 - 0.762688i) q^{2} +(-0.209889 + 1.39252i) q^{3} +(0.836613 + 1.81661i) q^{4} +(1.73139 - 0.679519i) q^{5} +(1.31202 - 1.49831i) q^{6} +(-0.592319 + 2.57860i) q^{7} +(0.389164 - 2.80153i) q^{8} +(0.971651 + 0.299715i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19093	−	0.762688i	−0.842112	−	0.539302i
							\(3\)	−0.209889	+	1.39252i	−0.121180	+	0.803974i	0.842844	+	0.538158i	\(0.180879\pi\)
−0.964024	+	0.265816i	\(0.914359\pi\)
\(5\)	1.73139	−	0.679519i	0.774300	−	0.303890i	0.0549049	−	0.998492i	\(-0.482514\pi\)
							0.719395	+	0.694601i	\(0.244419\pi\)
\(7\)	−0.592319	+	2.57860i	−0.223875	+	0.974618i
							\(11\)	−0.224646	−	0.728285i	−0.0677333	−	0.219586i	0.915378	−	0.402596i	\(-0.131892\pi\)
−0.983111	+	0.183010i	\(0.941416\pi\)
\(13\)	1.74405	+	0.398067i	0.483712	+	0.110404i	0.457421	−	0.889250i	\(-0.348773\pi\)
							0.0262907	+	0.999654i	\(0.491630\pi\)
\(17\)	4.21955	+	2.87684i	1.02339	+	0.697737i	0.953986	−	0.299851i	\(-0.0969371\pi\)
							0.0694063	+	0.997588i	\(0.477890\pi\)
\(19\)	−7.23191	−	4.17535i	−1.65911	−	0.957890i	−0.973125	−	0.230279i	\(-0.926036\pi\)
							−0.685990	−	0.727611i	\(-0.740630\pi\)
\(23\)	−0.115232	+	0.0785636i	−0.0240275	+	0.0163816i	−0.575274	−	0.817961i	\(-0.695105\pi\)
							0.551247	+	0.834342i	\(0.314152\pi\)
\(29\)	2.76656	+	5.74483i	0.513738	+	1.06679i	0.982979	+	0.183718i	\(0.0588133\pi\)
							−0.469241	+	0.883070i	\(0.655472\pi\)
\(31\)	3.93613	+	6.81758i	0.706951	+	1.22447i	0.965983	+	0.258606i	\(0.0832630\pi\)
							−0.259032	+	0.965869i	\(0.583404\pi\)
\(37\)	4.03411	+	0.302315i	0.663204	+	0.0497003i	0.402080	−	0.915605i	\(-0.368288\pi\)
							0.261125	+	0.965305i	\(0.415907\pi\)
\(41\)	2.31407	−	2.90175i	0.361397	−	0.453177i	−0.567578	−	0.823319i	\(-0.692120\pi\)
							0.928975	+	0.370142i	\(0.120691\pi\)
\(43\)	6.71348	−	5.35382i	1.02380	−	0.816450i	0.0406320	−	0.999174i	\(-0.487063\pi\)
							0.983164	+	0.182724i	\(0.0584914\pi\)
\(47\)	4.15655	+	3.85672i	0.606295	+	0.562560i	0.922435	−	0.386153i	\(-0.126196\pi\)
							−0.316140	+	0.948713i	\(0.602387\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.205.11	yes	648
8.5	even	2	inner	392.2.z.a.205.16	yes	648
49.11	even	21	inner	392.2.z.a.109.16	yes	648
392.109	even	42	inner	392.2.z.a.109.11	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.11	✓	648	392.109	even	42	inner
392.2.z.a.109.16	yes	648	49.11	even	21	inner
392.2.z.a.205.11	yes	648	1.1	even	1	trivial
392.2.z.a.205.16	yes	648	8.5	even	2	inner