Embedded newform 357.2.bh.a.100.23

• Label: 357.2.bh.a.100.23
• Level: $357$
• Weight: $2$
• Character: 357.100
• Analytic conductor: $2.851$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 357 = 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	357.bh (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.85065935216\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(24\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			100.23
Character	\(\chi\)	\(=\)	357.100
Dual form			357.2.bh.a.25.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.47406 - 0.662922i) q^{2} +(-0.608761 + 0.793353i) q^{3} +(3.94945 - 2.28022i) q^{4} +(-0.293850 + 2.23202i) q^{5} +(-0.980180 + 2.36636i) q^{6} +(-1.00447 + 2.44766i) q^{7} +(4.63730 - 4.63730i) q^{8} +(-0.258819 - 0.965926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 8 q^{5}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/357\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(190\)	\(239\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.47406	−	0.662922i	1.74942	−	0.468757i	0.764921	−	0.644124i	\(-0.222778\pi\)
							0.984503	+	0.175368i	\(0.0561114\pi\)
\(3\)	−0.608761	+	0.793353i	−0.351469	+	0.458043i
							\(5\)	−0.293850	+	2.23202i	−0.131414	+	0.998188i	0.790954	+	0.611876i	\(0.209585\pi\)
−0.922368	+	0.386312i	\(0.873749\pi\)
\(7\)	−1.00447	+	2.44766i	−0.379654	+	0.925129i
							\(11\)	4.26117	−	0.560994i	1.28479	−	0.169146i	0.542975	−	0.839749i	\(-0.317298\pi\)
0.741817	+	0.670602i	\(0.233964\pi\)
\(13\)		−	3.57889i		−	0.992605i	−0.868150	−	0.496303i	\(-0.834691\pi\)
							0.868150	−	0.496303i	\(-0.165309\pi\)
\(17\)	−2.55091	−	3.23927i	−0.618687	−	0.785638i
							\(19\)	−2.17700	+	0.583325i	−0.499437	+	0.133824i	−0.499739	−	0.866176i	\(-0.666571\pi\)
0.000301470		1.00000i	\(0.499904\pi\)
\(23\)	−3.68136	−	4.79764i	−0.767617	−	1.00038i	−0.999617	−	0.0276623i	\(-0.991194\pi\)
							0.232000	−	0.972716i	\(-0.425473\pi\)
\(29\)	−7.92686	+	3.28341i	−1.47198	+	0.609714i	−0.967310	−	0.253598i	\(-0.918386\pi\)
							−0.504670	+	0.863312i	\(0.668386\pi\)
\(31\)	1.67311	−	2.18045i	0.300500	−	0.391620i	−0.618559	−	0.785738i	\(-0.712283\pi\)
							0.919060	+	0.394118i	\(0.128950\pi\)
\(37\)	−1.05721	−	0.139184i	−0.173804	−	0.0228817i	0.0431219	−	0.999070i	\(-0.486270\pi\)
							−0.216926	+	0.976188i	\(0.569603\pi\)
\(41\)	−0.180713	−	0.0748539i	−0.0282227	−	0.0116902i	0.368527	−	0.929617i	\(-0.379862\pi\)
							−0.396750	+	0.917927i	\(0.629862\pi\)
\(43\)	2.78924	−	2.78924i	0.425354	−	0.425354i	−0.461688	−	0.887042i	\(-0.652756\pi\)
							0.887042	+	0.461688i	\(0.152756\pi\)
\(47\)	3.50473	+	2.02346i	0.511217	+	0.295152i	0.733334	−	0.679869i	\(-0.237963\pi\)
							−0.222116	+	0.975020i	\(0.571297\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bh.a.100.23	yes	192
7.4	even	3	inner	357.2.bh.a.151.2	yes	192
17.8	even	8	inner	357.2.bh.a.331.2	yes	192
119.25	even	24	inner	357.2.bh.a.25.23	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bh.a.25.23	✓	192	119.25	even	24	inner
357.2.bh.a.100.23	yes	192	1.1	even	1	trivial
357.2.bh.a.151.2	yes	192	7.4	even	3	inner
357.2.bh.a.331.2	yes	192	17.8	even	8	inner