Embedded newform 241.2.l.a.6.16

• Label: 241.2.l.a.6.16
• Level: $241$
• Weight: $2$
• Character: 241.6
• Analytic conductor: $1.924$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	241.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.92439468871\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(18\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			6.16
Character	\(\chi\)	\(=\)	241.6
Dual form			241.2.l.a.201.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.93203i q^{2} +(-2.99023 - 0.971584i) q^{3} -1.73276 q^{4} +(-2.36656 + 0.768943i) q^{5} +(1.87713 - 5.77722i) q^{6} +(0.401702 - 2.53625i) q^{7} +0.516326i q^{8} +(5.57044 + 4.04716i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 10 q^{3} - 120 q^{4} - 10 q^{5} + 6 q^{6} - 10 q^{7} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			1.93203i			1.36615i	0.730346	+	0.683077i	\(0.239359\pi\)
							−0.730346	+	0.683077i	\(0.760641\pi\)
\(3\)	−2.99023	−	0.971584i	−1.72641	−	0.560945i	−0.733487	−	0.679704i	\(-0.762108\pi\)
							−0.992923	+	0.118759i	\(0.962108\pi\)
\(5\)	−2.36656	+	0.768943i	−1.05836	+	0.343882i	−0.785944	−	0.618298i	\(-0.787823\pi\)
							−0.272415	+	0.962180i	\(0.587823\pi\)
\(7\)	0.401702	−	2.53625i	0.151829	−	0.958612i	−0.787679	−	0.616086i	\(-0.788717\pi\)
							0.939508	−	0.342526i	\(-0.111283\pi\)
\(11\)	1.86326	−	1.86326i	0.561795	−	0.561795i	−0.368022	−	0.929817i	\(-0.619965\pi\)
							0.929817	+	0.368022i	\(0.119965\pi\)
\(13\)	5.32162	−	2.71150i	1.47595	−	0.752035i	0.483577	−	0.875302i	\(-0.339337\pi\)
							0.992374	+	0.123267i	\(0.0393372\pi\)
\(17\)	−7.32310	−	1.15987i	−1.77611	−	0.281309i	−0.819587	−	0.572954i	\(-0.805797\pi\)
							−0.956526	+	0.291646i	\(0.905797\pi\)
\(19\)	5.30553	−	5.30553i	1.21717	−	1.21717i	0.248555	−	0.968618i	\(-0.420044\pi\)
							0.968618	−	0.248555i	\(-0.0799558\pi\)
\(23\)	−1.25649	−	2.46600i	−0.261997	−	0.514198i	0.722109	−	0.691779i	\(-0.243173\pi\)
							−0.984106	+	0.177581i	\(0.943173\pi\)
\(29\)	0.710155	−	0.977444i	0.131872	−	0.181507i	−0.737974	−	0.674829i	\(-0.764218\pi\)
							0.869847	+	0.493322i	\(0.164218\pi\)
\(31\)	0.168827	+	0.0267396i	0.0303222	+	0.00480257i	0.171577	−	0.985171i	\(-0.445114\pi\)
							−0.141255	+	0.989973i	\(0.545114\pi\)
\(37\)	−0.632493	−	0.322271i	−0.103981	−	0.0529810i	0.401226	−	0.915979i	\(-0.368584\pi\)
							−0.505208	+	0.862998i	\(0.668584\pi\)
\(41\)	−4.44598	−	6.11936i	−0.694345	−	0.955684i	−0.999994	−	0.00353957i	\(-0.998873\pi\)
							0.305649	−	0.952144i	\(-0.401127\pi\)
\(43\)	0.812168	+	5.12783i	0.123854	+	0.781986i	0.968929	+	0.247338i	\(0.0795558\pi\)
							−0.845075	+	0.534648i	\(0.820444\pi\)
\(47\)	3.33251	−	4.58681i	0.486097	−	0.669055i	−0.493565	−	0.869709i	\(-0.664307\pi\)
							0.979662	+	0.200654i	\(0.0643067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	241.2.l.a.6.16	✓	144
241.201	even	20	inner	241.2.l.a.201.3	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.l.a.6.16	✓	144	1.1	even	1	trivial
241.2.l.a.201.3	yes	144	241.201	even	20	inner