Embedded newform 241.2.s.a.3.16

• Label: 241.2.s.a.3.16
• Level: $241$
• Weight: $2$
• Character: 241.3
• Analytic conductor: $1.924$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	241.s (of order \(120\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.92439468871\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(20\) over \(\Q(\zeta_{120})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{120}]$

Embedding invariants

Embedding label			3.16
Character	\(\chi\)	\(=\)	241.3
Dual form			241.2.s.a.161.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.64096 + 0.439693i) q^{2} +(2.06237 + 0.108084i) q^{3} +(0.767353 + 0.443032i) q^{4} +(0.619663 - 1.21616i) q^{5} +(3.33674 + 1.08417i) q^{6} +(-3.03296 + 2.87817i) q^{7} +(-1.33813 - 1.33813i) q^{8} +(1.25814 + 0.132236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 32 q^{2} - 32 q^{3} - 24 q^{4} - 32 q^{5} - 40 q^{6} - 48 q^{7} - 64 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/241\mathbb{Z}\right)^\times\).

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{91}{120}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.64096	+	0.439693i	1.16033	+	0.310910i	0.787098	−	0.616828i	\(-0.211583\pi\)
							0.373233	+	0.927738i	\(0.378249\pi\)
\(3\)	2.06237	+	0.108084i	1.19071	+	0.0624026i	0.637374	−	0.770554i	\(-0.280021\pi\)
							0.553338	+	0.832957i	\(0.313354\pi\)
\(5\)	0.619663	−	1.21616i	0.277122	−	0.543882i	−0.709932	−	0.704270i	\(-0.751275\pi\)
							0.987054	+	0.160388i	\(0.0512746\pi\)
\(7\)	−3.03296	+	2.87817i	−1.14635	+	1.08785i	−0.151298	+	0.988488i	\(0.548345\pi\)
							−0.995052	+	0.0993576i	\(0.968321\pi\)
\(11\)	3.74272	−	0.492739i	1.12847	−	0.148566i	0.456910	−	0.889513i	\(-0.348956\pi\)
							0.671564	+	0.740947i	\(0.265623\pi\)
\(13\)	−0.935938	+	1.96223i	−0.259583	+	0.544226i	−0.990381	−	0.138367i	\(-0.955815\pi\)
							0.730798	+	0.682593i	\(0.239148\pi\)
\(17\)	−1.97555	−	1.21062i	−0.479141	−	0.293618i	0.261879	−	0.965101i	\(-0.415658\pi\)
							−0.741020	+	0.671483i	\(0.765658\pi\)
\(19\)	−0.144596	−	1.09832i	−0.0331726	−	0.251971i	0.966821	−	0.255454i	\(-0.0822248\pi\)
							−0.999994	+	0.00348246i	\(0.998891\pi\)
\(23\)	−5.00361	−	4.27349i	−1.04333	−	0.891084i	−0.0491430	−	0.998792i	\(-0.515649\pi\)
							−0.994183	+	0.107708i	\(0.965649\pi\)
\(29\)	3.97013	−	3.21495i	0.737234	−	0.597001i	−0.185406	−	0.982662i	\(-0.559360\pi\)
							0.922640	+	0.385661i	\(0.126027\pi\)
\(31\)	−0.183560	+	7.00986i	−0.0329683	+	1.25901i	0.760072	+	0.649839i	\(0.225164\pi\)
							−0.793040	+	0.609169i	\(0.791503\pi\)
\(37\)	4.59326	+	9.62996i	0.755127	+	1.58316i	0.811228	+	0.584731i	\(0.198800\pi\)
							−0.0561010	+	0.998425i	\(0.517867\pi\)
\(41\)	1.63207	−	10.3045i	0.254887	−	1.60929i	−0.445332	−	0.895365i	\(-0.646914\pi\)
							0.700219	−	0.713928i	\(-0.253086\pi\)
\(43\)	−3.80183	−	0.912737i	−0.579773	−	0.139191i	−0.0670565	−	0.997749i	\(-0.521361\pi\)
							−0.512716	+	0.858558i	\(0.671361\pi\)
\(47\)	1.70657	+	10.7748i	0.248929	+	1.57167i	0.722786	+	0.691072i	\(0.242861\pi\)
							−0.473858	+	0.880601i	\(0.657139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	241.2.s.a.3.16	✓	640
241.161	even	120	inner	241.2.s.a.161.16	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.s.a.3.16	✓	640	1.1	even	1	trivial
241.2.s.a.161.16	yes	640	241.161	even	120	inner