Embedded newform 357.2.bh.a.100.10

• Label: 357.2.bh.a.100.10
• 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			100.10
Character	\(\chi\)	\(=\)	357.100
Dual form			357.2.bh.a.25.10

$q$-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 \(\left(\mathbb{Z}/357\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bh.a.25.10	✓	192	119.25	even	24	inner
357.2.bh.a.100.10	yes	192	1.1	even	1	trivial
357.2.bh.a.151.15	yes	192	7.4	even	3	inner
357.2.bh.a.331.15	yes	192	17.8	even	8	inner