Embedded newform 245.2.s.a.202.13

• Label: 245.2.s.a.202.13
• Level: $245$
• Weight: $2$
• Character: 245.202
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.s (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			202.13
Character	\(\chi\)	\(=\)	245.202
Dual form			245.2.s.a.188.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.310080 + 0.0349376i) q^{2} +(-0.544675 - 1.55659i) q^{3} +(-1.85493 + 0.423375i) q^{4} +(1.42796 + 1.72073i) q^{5} +(0.223276 + 0.463638i) q^{6} +(1.47006 + 2.19976i) q^{7} +(1.14945 - 0.402208i) q^{8} +(0.219188 - 0.174796i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 10 q^{2} - 14 q^{3} - 14 q^{5} - 56 q^{6} - 18 q^{7} - 6 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.310080	+	0.0349376i	−0.219259	+	0.0247046i	−0.220911	−	0.975294i	\(-0.570903\pi\)
							0.00165128	+	0.999999i	\(0.499474\pi\)
\(3\)	−0.544675	−	1.55659i	−0.314468	−	0.898699i	−0.987295	−	0.158898i	\(-0.949206\pi\)
							0.672827	−	0.739800i	\(-0.265080\pi\)
\(5\)	1.42796	+	1.72073i	0.638604	+	0.769536i
							\(7\)	1.47006	+	2.19976i	0.555630	+	0.831429i
\(11\)	0.526279	−	0.659933i	0.158679	−	0.198977i								−0.696136	−	0.717910i	\(-0.745099\pi\)
							0.854815	+	0.518932i	\(0.173670\pi\)
\(13\)	0.768392	−	0.0865769i	0.213113	−	0.0240121i	−0.00476216	−	0.999989i	\(-0.501516\pi\)
							0.217876	+	0.975977i	\(0.430087\pi\)
\(17\)	0.475201	+	0.756278i	0.115253	+	0.183424i	0.899369	−	0.437190i	\(-0.144026\pi\)
							−0.784116	+	0.620614i	\(0.786883\pi\)
\(19\)	5.47678			1.25646			0.628230	−	0.778028i	\(-0.283780\pi\)
							0.628230	+	0.778028i	\(0.283780\pi\)
\(23\)	3.76651	+	2.36666i	0.785372	+	0.493482i	0.864043	−	0.503419i	\(-0.167925\pi\)
							−0.0786705	+	0.996901i	\(0.525068\pi\)
\(29\)	2.77625	+	0.633661i	0.515537	+	0.117668i	0.472371	−	0.881400i	\(-0.343398\pi\)
							0.0431662	+	0.999068i	\(0.486256\pi\)
\(31\)		−	5.35717i		−	0.962176i	−0.876672	−	0.481088i	\(-0.840242\pi\)
							0.876672	−	0.481088i	\(-0.159758\pi\)
\(37\)	−8.25742	+	5.18848i	−1.35751	+	0.852981i	−0.996819	−	0.0797041i	\(-0.974602\pi\)
							−0.360692	+	0.932685i	\(0.617460\pi\)
\(41\)	−1.66543	+	3.45830i	−0.260096	+	0.540095i	−0.989594	−	0.143888i	\(-0.954039\pi\)
							0.729498	+	0.683983i	\(0.239754\pi\)
\(43\)	−1.29759	+	3.70830i	−0.197881	+	0.565511i	−0.999476	−	0.0323558i	\(-0.989699\pi\)
							0.801596	+	0.597867i	\(0.203985\pi\)
\(47\)	−0.236685	−	2.10063i	−0.0345240	−	0.306409i	−0.999093	−	0.0425786i	\(-0.986443\pi\)
							0.964569	−	0.263830i	\(-0.0849859\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.s.a.202.13	yes	312
5.3	odd	4	inner	245.2.s.a.153.14	✓	312
49.41	odd	14	inner	245.2.s.a.237.14	yes	312
245.188	even	28	inner	245.2.s.a.188.13	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.s.a.153.14	✓	312	5.3	odd	4	inner
245.2.s.a.188.13	yes	312	245.188	even	28	inner
245.2.s.a.202.13	yes	312	1.1	even	1	trivial
245.2.s.a.237.14	yes	312	49.41	odd	14	inner