Embedded newform 242.2.g.a.5.4

• Label: 242.2.g.a.5.4
• Level: $242$
• Weight: $2$
• Character: 242.5
• Analytic conductor: $1.932$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	242.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.93237972891\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(5\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	242.5
Dual form			242.2.g.a.97.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.516397 + 0.856349i) q^{2} +(1.49802 + 1.08837i) q^{3} +(-0.466667 - 0.884433i) q^{4} +(1.49869 - 1.22548i) q^{5} +(-1.70560 + 0.720792i) q^{6} +(1.72885 - 2.24201i) q^{7} +(0.998369 + 0.0570888i) q^{8} +(0.132446 + 0.407627i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 5 q^{2} + 10 q^{3} + 5 q^{4} - q^{5} + q^{6} - 5 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{37}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.516397	+	0.856349i	−0.365148	+	0.605530i
							\(3\)	1.49802	+	1.08837i	0.864880	+	0.628372i	0.929208	−	0.369557i	\(-0.120491\pi\)
−0.0643284	+	0.997929i	\(0.520491\pi\)
\(5\)	1.49869	−	1.22548i	0.670236	−	0.548049i	−0.235509	−	0.971872i	\(-0.575676\pi\)
							0.905745	+	0.423823i	\(0.139312\pi\)
\(7\)	1.72885	−	2.24201i	0.653444	−	0.847401i	−0.342298	−	0.939592i	\(-0.611205\pi\)
							0.995741	+	0.0921907i	\(0.0293869\pi\)
\(11\)	3.26786	+	0.566621i	0.985298	+	0.170843i
							\(13\)	−1.30432	+	0.225722i	−0.361754	+	0.0626040i	−0.348502	−	0.937308i	\(-0.613309\pi\)
−0.0132524	+	0.999912i	\(0.504218\pi\)
\(17\)	−4.57062	+	0.524430i	−1.10854	+	0.127193i	−0.648871	−	0.760898i	\(-0.724759\pi\)
							−0.459667	+	0.888091i	\(0.652031\pi\)
\(19\)	−5.71750	+	3.22882i	−1.31168	+	0.740742i	−0.979302	−	0.202402i	\(-0.935125\pi\)
							−0.332382	+	0.943145i	\(0.607852\pi\)
\(23\)	−0.937309	−	0.275219i	−0.195442	−	0.0573871i	0.182546	−	0.983197i	\(-0.441566\pi\)
							−0.377989	+	0.925810i	\(0.623384\pi\)
\(29\)	1.73084	+	8.54201i	0.321408	+	1.58621i	0.736503	+	0.676434i	\(0.236476\pi\)
							−0.415095	+	0.909778i	\(0.636252\pi\)
\(31\)	−3.45739	+	1.69989i	−0.620966	+	0.305308i	−0.724455	−	0.689322i	\(-0.757908\pi\)
							0.103489	+	0.994631i	\(0.466999\pi\)
\(37\)	1.32090	+	1.35917i	0.217154	+	0.223447i	0.816953	−	0.576705i	\(-0.195662\pi\)
							−0.599798	+	0.800151i	\(0.704753\pi\)
\(41\)	−0.754273	−	8.78184i	−0.117798	−	1.37149i	−0.781691	−	0.623666i	\(-0.785643\pi\)
							0.663894	−	0.747827i	\(-0.268903\pi\)
\(43\)	−4.59406	+	2.95243i	−0.700588	+	0.450241i	−0.841836	−	0.539734i	\(-0.818525\pi\)
							0.141248	+	0.989974i	\(0.454889\pi\)
\(47\)	6.85062	−	1.59304i	0.999265	−	0.232369i	0.305206	−	0.952287i	\(-0.401275\pi\)
							0.694060	+	0.719917i	\(0.255820\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.a.5.4	✓	200
121.97	even	55	inner	242.2.g.a.97.4	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.a.5.4	✓	200	1.1	even	1	trivial
242.2.g.a.97.4	yes	200	121.97	even	55	inner