Embedded newform 357.2.bh.a.100.6

• Label: 357.2.bh.a.100.6
• 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			100.6
Character	\(\chi\)	\(=\)	357.100
Dual form			357.2.bh.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68695 + 0.452018i) q^{2} +(0.608761 - 0.793353i) q^{3} +(0.909440 - 0.525065i) q^{4} +(0.277214 - 2.10565i) q^{5} +(-0.668342 + 1.61352i) q^{6} +(2.14198 - 1.55304i) q^{7} +(1.17303 - 1.17303i) q^{8} +(-0.258819 - 0.965926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 8 q^{5}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68695	+	0.452018i	−1.19286	+	0.319625i	−0.800014	−	0.599981i	\(-0.795175\pi\)
							−0.392842	+	0.919606i	\(0.628508\pi\)
\(3\)	0.608761	−	0.793353i	0.351469	−	0.458043i
							\(5\)	0.277214	−	2.10565i	0.123974	−	0.941675i	−0.810524	−	0.585706i	\(-0.800817\pi\)
0.934498	−	0.355969i	\(-0.115849\pi\)
\(7\)	2.14198	−	1.55304i	0.809591	−	0.586994i
							\(11\)	−5.06864	+	0.667299i	−1.52825	+	0.201198i	−0.847294	−	0.531124i	\(-0.821770\pi\)
−0.680958	+	0.732322i	\(0.738436\pi\)
\(13\)			0.853029i			0.236588i	0.992979	+	0.118294i	\(0.0377425\pi\)
							−0.992979	+	0.118294i	\(0.962258\pi\)
\(17\)	2.60121	−	3.19902i	0.630886	−	0.775876i
							\(19\)	−2.98493	+	0.799810i	−0.684790	+	0.183489i	−0.584408	−	0.811460i	\(-0.698673\pi\)
−0.100382	+	0.994949i	\(0.532007\pi\)
\(23\)	−3.68648	−	4.80431i	−0.768683	−	1.00177i	−0.999585	−	0.0287905i	\(-0.990834\pi\)
							0.230902	−	0.972977i	\(-0.425832\pi\)
\(29\)	−3.11130	+	1.28874i	−0.577754	+	0.239314i	−0.652373	−	0.757898i	\(-0.726226\pi\)
							0.0746181	+	0.997212i	\(0.476226\pi\)
\(31\)	4.55776	−	5.93979i	0.818598	−	1.06682i	−0.177969	−	0.984036i	\(-0.556953\pi\)
							0.996568	−	0.0827822i	\(-0.0263806\pi\)
\(37\)	−3.99120	−	0.525452i	−0.656150	−	0.0863837i	−0.204902	−	0.978782i	\(-0.565688\pi\)
							−0.451248	+	0.892399i	\(0.649021\pi\)
\(41\)	2.06379	+	0.854851i	0.322310	+	0.133505i	0.537972	−	0.842963i	\(-0.319191\pi\)
							−0.215661	+	0.976468i	\(0.569191\pi\)
\(43\)	4.32276	−	4.32276i	0.659215	−	0.659215i	−0.295979	−	0.955194i	\(-0.595646\pi\)
							0.955194	+	0.295979i	\(0.0956459\pi\)
\(47\)	−1.22754	−	0.708719i	−0.179055	−	0.103377i	0.407794	−	0.913074i	\(-0.366298\pi\)
							−0.586848	+	0.809697i	\(0.699632\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bh.a.100.6	yes	192
7.4	even	3	inner	357.2.bh.a.151.19	yes	192
17.8	even	8	inner	357.2.bh.a.331.19	yes	192
119.25	even	24	inner	357.2.bh.a.25.6	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bh.a.25.6	✓	192	119.25	even	24	inner
357.2.bh.a.100.6	yes	192	1.1	even	1	trivial
357.2.bh.a.151.19	yes	192	7.4	even	3	inner
357.2.bh.a.331.19	yes	192	17.8	even	8	inner