Embedded newform 245.2.x.a.103.20

• Label: 245.2.x.a.103.20
• Level: $245$
• Weight: $2$
• Character: 245.103
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(26\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			103.20
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.841981 - 1.14084i) q^{2} +(-2.70668 + 0.512131i) q^{3} +(-0.00308156 - 0.00999019i) q^{4} +(1.66263 - 1.49522i) q^{5} +(-1.69471 + 3.51910i) q^{6} +(0.572619 + 2.58304i) q^{7} +(2.66268 + 0.931713i) q^{8} +(4.27120 - 1.67632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.841981	−	1.14084i	0.595370	−	0.806698i	−0.398510	−	0.917164i	\(-0.630472\pi\)
							0.993880	+	0.110466i	\(0.0352344\pi\)
\(3\)	−2.70668	+	0.512131i	−1.56270	+	0.295679i	−0.894074	−	0.447919i	\(-0.852165\pi\)
							−0.668627	+	0.743598i	\(0.733118\pi\)
\(5\)	1.66263	−	1.49522i	0.743549	−	0.668682i
							\(7\)	0.572619	+	2.58304i	0.216429	+	0.976298i
\(11\)	2.06989	−	5.27400i	0.624096	−	1.59017i								−0.172066	−	0.985085i	\(-0.555044\pi\)
							0.796162	−	0.605084i	\(-0.206861\pi\)
\(13\)	0.212818	+	0.0239788i	0.0590251	+	0.00665053i	0.141428	−	0.989949i	\(-0.454831\pi\)
							−0.0824025	+	0.996599i	\(0.526259\pi\)
\(17\)	1.92617	+	0.0720723i	0.467166	+	0.0174801i	0.269938	−	0.962878i	\(-0.412997\pi\)
							0.197227	+	0.980358i	\(0.436806\pi\)
\(19\)	−0.246021	+	0.426120i	−0.0564410	+	0.0977587i	−0.892865	−	0.450324i	\(-0.851309\pi\)
							0.836424	+	0.548082i	\(0.184642\pi\)
\(23\)	0.228339	+	6.10250i	0.0476121	+	1.27246i	0.794399	+	0.607396i	\(0.207786\pi\)
							−0.746787	+	0.665063i	\(0.768405\pi\)
\(29\)	5.53914	−	1.26427i	1.02859	−	0.234769i	0.325255	−	0.945626i	\(-0.394550\pi\)
							0.703337	+	0.710857i	\(0.251693\pi\)
\(31\)	−6.77033	+	3.90885i	−1.21599	+	0.702050i	−0.964057	−	0.265696i	\(-0.914398\pi\)
							−0.251929	+	0.967746i	\(0.581065\pi\)
\(37\)	−2.59157	+	1.36968i	−0.426051	+	0.225175i	−0.666528	−	0.745480i	\(-0.732220\pi\)
							0.240476	+	0.970655i	\(0.422696\pi\)
\(41\)	−1.36194	−	2.82809i	−0.212699	−	0.441673i	0.767136	−	0.641485i	\(-0.221681\pi\)
							−0.979834	+	0.199812i	\(0.935967\pi\)
\(43\)	2.25150	+	6.43443i	0.343351	+	0.981241i	0.978030	+	0.208465i	\(0.0668467\pi\)
							−0.634679	+	0.772776i	\(0.718868\pi\)
\(47\)	−7.08440	−	5.22853i	−1.03337	−	0.762659i	−0.0614799	−	0.998108i	\(-0.519582\pi\)
							−0.971887	+	0.235449i	\(0.924344\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.103.20	✓	624
5.2	odd	4	inner	245.2.x.a.152.20	yes	624
49.10	odd	42	inner	245.2.x.a.108.20	yes	624
245.157	even	84	inner	245.2.x.a.157.20	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.20	✓	624	1.1	even	1	trivial
245.2.x.a.108.20	yes	624	49.10	odd	42	inner
245.2.x.a.152.20	yes	624	5.2	odd	4	inner
245.2.x.a.157.20	yes	624	245.157	even	84	inner