Embedded newform 392.2.z.a.37.5

• Label: 392.2.z.a.37.5
• Level: $392$
• Weight: $2$
• Character: 392.37
• 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			37.5
Character	\(\chi\)	\(=\)	392.37
Dual form			392.2.z.a.53.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37458 - 0.332474i) q^{2} +(-0.288166 - 0.310569i) q^{3} +(1.77892 + 0.914022i) q^{4} +(-0.486553 + 1.57737i) q^{5} +(0.292850 + 0.522709i) q^{6} +(-2.63963 - 0.179924i) q^{7} +(-2.14138 - 1.84784i) q^{8} +(0.210777 - 2.81262i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37458	−	0.332474i	−0.971973	−	0.235094i
							\(3\)	−0.288166	−	0.310569i	−0.166373	−	0.179307i	0.644362	−	0.764720i	\(-0.277123\pi\)
−0.810735	+	0.585413i	\(0.800932\pi\)
\(5\)	−0.486553	+	1.57737i	−0.217593	+	0.705420i	0.779300	+	0.626652i	\(0.215575\pi\)
							−0.996893	+	0.0787687i	\(0.974901\pi\)
\(7\)	−2.63963	−	0.179924i	−0.997685	−	0.0680048i
							\(11\)	5.85165	−	0.438521i	1.76434	−	0.132219i	0.847325	−	0.531075i	\(-0.178212\pi\)
0.917014	+	0.398856i	\(0.130593\pi\)
\(13\)	−0.703466	+	1.46076i	−0.195106	+	0.405142i	−0.975454	−	0.220204i	\(-0.929328\pi\)
							0.780348	+	0.625346i	\(0.215042\pi\)
\(17\)	−3.07273	−	0.463140i	−0.745247	−	0.112328i	−0.234568	−	0.972100i	\(-0.575368\pi\)
							−0.510679	+	0.859772i	\(0.670606\pi\)
\(19\)	7.07870	+	4.08689i	1.62396	+	0.937596i	0.985845	+	0.167659i	\(0.0536208\pi\)
							0.638120	+	0.769937i	\(0.279713\pi\)
\(23\)	5.83792	−	0.879925i	1.21729	−	0.183477i	0.491182	−	0.871057i	\(-0.336565\pi\)
							0.726109	+	0.687580i	\(0.241327\pi\)
\(29\)	−0.371874	−	0.296559i	−0.0690552	−	0.0550697i	0.588354	−	0.808604i	\(-0.299776\pi\)
							−0.657409	+	0.753534i	\(0.728348\pi\)
\(31\)	2.07791	+	3.59905i	0.373205	+	0.646409i	0.990057	−	0.140670i	\(-0.0449256\pi\)
							−0.616852	+	0.787079i	\(0.711592\pi\)
\(37\)	2.79564	−	1.09721i	0.459600	−	0.180380i	−0.124234	−	0.992253i	\(-0.539647\pi\)
							0.583834	+	0.811873i	\(0.301552\pi\)
\(41\)	−0.184731	−	0.809361i	−0.0288502	−	0.126401i	0.958452	−	0.285253i	\(-0.0920777\pi\)
							−0.987302	+	0.158852i	\(0.949221\pi\)
\(43\)	3.26269	+	0.744688i	0.497556	+	0.113564i	0.463932	−	0.885871i	\(-0.346438\pi\)
							0.0336236	+	0.999435i	\(0.489295\pi\)
\(47\)	7.13856	−	4.86699i	1.04127	−	0.709923i	0.0832181	−	0.996531i	\(-0.473480\pi\)
							0.958048	+	0.286608i	\(0.0925278\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.37.5	✓	648
8.5	even	2	inner	392.2.z.a.37.26	yes	648
49.4	even	21	inner	392.2.z.a.53.26	yes	648
392.53	even	42	inner	392.2.z.a.53.5	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.37.5	✓	648	1.1	even	1	trivial
392.2.z.a.37.26	yes	648	8.5	even	2	inner
392.2.z.a.53.5	yes	648	392.53	even	42	inner
392.2.z.a.53.26	yes	648	49.4	even	21	inner