Embedded newform 392.2.z.a.109.27

• Label: 392.2.z.a.109.27
• Level: $392$
• Weight: $2$
• Character: 392.109
• 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			109.27
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.157461 + 1.40542i) q^{2} +(-0.121277 - 0.804621i) q^{3} +(-1.95041 - 0.442599i) q^{4} +(-1.19401 - 0.468613i) q^{5} +(1.14993 - 0.0437485i) q^{6} +(-1.79864 + 1.94034i) q^{7} +(0.929152 - 2.67146i) q^{8} +(2.23401 - 0.689101i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.157461	+	1.40542i	−0.111342	+	0.993782i
							\(3\)	−0.121277	−	0.804621i	−0.0700194	−	0.464548i	−0.996173	−	0.0874020i	\(-0.972144\pi\)
0.926154	−	0.377146i	\(-0.123095\pi\)
\(5\)	−1.19401	−	0.468613i	−0.533976	−	0.209570i	0.0830035	−	0.996549i	\(-0.473549\pi\)
							−0.616980	+	0.786979i	\(0.711644\pi\)
\(7\)	−1.79864	+	1.94034i	−0.679821	+	0.733378i
							\(11\)	1.35408	−	4.38982i	0.408270	−	1.32358i	−0.485179	−	0.874415i	\(-0.661246\pi\)
0.893449	−	0.449165i	\(-0.148278\pi\)
\(13\)	1.99747	−	0.455909i	0.553998	−	0.126447i	0.0636489	−	0.997972i	\(-0.479726\pi\)
							0.490350	+	0.871526i	\(0.336869\pi\)
\(17\)	2.29264	−	1.56309i	0.556046	−	0.379106i	−0.252448	−	0.967611i	\(-0.581236\pi\)
							0.808494	+	0.588505i	\(0.200283\pi\)
\(19\)	1.16705	−	0.673799i	0.267741	−	0.154580i	−0.360120	−	0.932906i	\(-0.617264\pi\)
							0.627860	+	0.778326i	\(0.283931\pi\)
\(23\)	1.28552	+	0.876450i	0.268049	+	0.182752i	0.689884	−	0.723920i	\(-0.257662\pi\)
							−0.421835	+	0.906672i	\(0.638614\pi\)
\(29\)	4.13982	−	8.59642i	0.768745	−	1.59632i	−0.0335856	−	0.999436i	\(-0.510693\pi\)
							0.802331	−	0.596880i	\(-0.203593\pi\)
\(31\)	1.74546	−	3.02323i	0.313494	−	0.542988i	−0.665622	−	0.746289i	\(-0.731834\pi\)
							0.979116	+	0.203301i	\(0.0651669\pi\)
\(37\)	−1.22121	+	0.0915172i	−0.200766	+	0.0150453i	−0.174733	−	0.984616i	\(-0.555906\pi\)
							−0.0260325	+	0.999661i	\(0.508287\pi\)
\(41\)	6.82269	+	8.55538i	1.06552	+	1.33613i	0.938906	+	0.344174i	\(0.111841\pi\)
							0.126619	+	0.991951i	\(0.459587\pi\)
\(43\)	−3.64488	−	2.90670i	−0.555839	−	0.443267i	0.304848	−	0.952401i	\(-0.401394\pi\)
							−0.860687	+	0.509134i	\(0.829966\pi\)
\(47\)	−9.72908	+	9.02726i	−1.41913	+	1.31676i	−0.538774	+	0.842450i	\(0.681112\pi\)
							−0.880357	+	0.474311i	\(0.842697\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.27	yes	648
8.5	even	2	inner	392.2.z.a.109.1	✓	648
49.9	even	21	inner	392.2.z.a.205.1	yes	648
392.205	even	42	inner	392.2.z.a.205.27	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.1	✓	648	8.5	even	2	inner
392.2.z.a.109.27	yes	648	1.1	even	1	trivial
392.2.z.a.205.1	yes	648	49.9	even	21	inner
392.2.z.a.205.27	yes	648	392.205	even	42	inner