Embedded newform 357.2.bh.a.25.10

• Label: 357.2.bh.a.25.10
• Level: $357$
• Weight: $2$
• Character: 357.25
• 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			25.10
Character	\(\chi\)	\(=\)	357.25
Dual form			357.2.bh.a.100.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.749407 - 0.200803i) q^{2} +(-0.608761 - 0.793353i) q^{3} +(-1.21076 - 0.699034i) q^{4} +(-0.559525 - 4.25001i) q^{5} +(0.296902 + 0.716785i) q^{6} +(-1.97366 - 1.76200i) q^{7} +(1.86419 + 1.86419i) q^{8} +(-0.258819 + 0.965926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 8 q^{5}+O(q^{10}) \)

Character values

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

\(n\)	\(52\)	\(190\)	\(239\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.749407	−	0.200803i	−0.529911	−	0.141989i	−0.0160636	−	0.999871i	\(-0.505113\pi\)
							−0.513847	+	0.857882i	\(0.671780\pi\)
\(3\)	−0.608761	−	0.793353i	−0.351469	−	0.458043i
							\(5\)	−0.559525	−	4.25001i	−0.250227	−	1.90066i	−0.407661	−	0.913133i	\(-0.633655\pi\)
0.157434	−	0.987530i	\(-0.449678\pi\)
\(7\)	−1.97366	−	1.76200i	−0.745974	−	0.665975i
							\(11\)	−1.17616	−	0.154845i	−0.354626	−	0.0466874i	−0.0488911	−	0.998804i	\(-0.515569\pi\)
−0.305735	+	0.952117i	\(0.598902\pi\)
\(13\)			1.32321i			0.366994i	0.983020	+	0.183497i	\(0.0587417\pi\)
							−0.983020	+	0.183497i	\(0.941258\pi\)
\(17\)	4.09626	+	0.469777i	0.993488	+	0.113938i
							\(19\)	1.23530	+	0.330997i	0.283396	+	0.0759358i	0.397717	−	0.917508i	\(-0.369803\pi\)
−0.114321	+	0.993444i	\(0.536469\pi\)
\(23\)	0.475433	−	0.619596i	0.0991346	−	0.129195i	−0.741152	−	0.671337i	\(-0.765720\pi\)
							0.840286	+	0.542143i	\(0.182387\pi\)
\(29\)	4.66396	+	1.93188i	0.866076	+	0.358741i	0.771081	−	0.636737i	\(-0.219716\pi\)
							0.0949954	+	0.995478i	\(0.469716\pi\)
\(31\)	−4.56332	−	5.94703i	−0.819596	−	1.06812i	−0.996475	−	0.0838853i	\(-0.973267\pi\)
							0.176880	−	0.984232i	\(-0.443400\pi\)
\(37\)	−8.18938	+	1.07815i	−1.34633	+	0.177247i	−0.768991	−	0.639260i	\(-0.779241\pi\)
							−0.577336	+	0.816507i	\(0.695908\pi\)
\(41\)	3.55444	−	1.47230i	0.555110	−	0.229934i	−0.0874505	−	0.996169i	\(-0.527872\pi\)
							0.642561	+	0.766235i	\(0.277872\pi\)
\(43\)	−7.00164	−	7.00164i	−1.06774	−	1.06774i	−0.997532	−	0.0702083i	\(-0.977634\pi\)
							−0.0702083	−	0.997532i	\(-0.522366\pi\)
\(47\)	4.97122	−	2.87014i	0.725128	−	0.418653i	−0.0915094	−	0.995804i	\(-0.529169\pi\)
							0.816637	+	0.577152i	\(0.195836\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bh.a.25.10	✓	192
7.2	even	3	inner	357.2.bh.a.331.15	yes	192
17.15	even	8	inner	357.2.bh.a.151.15	yes	192
119.100	even	24	inner	357.2.bh.a.100.10	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bh.a.25.10	✓	192	1.1	even	1	trivial
357.2.bh.a.100.10	yes	192	119.100	even	24	inner
357.2.bh.a.151.15	yes	192	17.15	even	8	inner
357.2.bh.a.331.15	yes	192	7.2	even	3	inner